On the Possibility of Fractional Statistics
in the Two-Dimensional t-J Model
at Low Doping
Von der Fakulta¨t Mathematik und Physik
der Universita¨t Stuttgart
zur Erlangung der Wu¨rde eines Doktors der
Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung
vorgelegt von
Thorsten Beck
aus Remshalden-Geradstetten
Hauptberichter: Prof. Dr. Alejandro Muramatsu
Mitberichter: Prof. Dr. Gu¨nter Wunner
Tag der mu¨ndlichen Pru¨fung: 26. Juli 2012
Institut fu¨r Theoretische Physik III
Universita¨t Stuttgart
2012

Abstract
The purpose of this work is the derivation of an effective field theory for the low-energy
magnetic modes of the t-J model on a two-dimensional square lattice and for small val-
ues of doping, which is of relevance to the physics of high-temperature superconductors.
In particular, we address the possibility of low-energy excitations that obey fractional
spin and statistics.
For temperatures where kBT is considerably smaller than the magnetic exchange cou-
pling J and for low values of doping, the system is close to an antiferromagnetically
ordered Ne´el state and the spin correlation length takes values significantly larger than
the lattice constant a, such that a field theoretic analysis is justified. The effective model
is obtained by means of a gradient expansion around the antiferromagnetically ordered
reference state. Dimensional analysis shows that in (2 + 1) dimensions only terms up to
order O(a2) in the effective action are relevant to the behaviour at large scales, whereas
O(a3)-terms are marginal and higher order terms are irrelevant. Even though marginal,
the O(a3)-contributions may drastically influence the properties of excitations, as they
might feature a term of topological nature which endows field histories with a statistical
phase factor. In fact, all field histories can be characterized as mappings from com-
pactified spacetime, the three-sphere S3, to the order parameter space S2 and thus fall
into different homotopy classes. The topological invariant characterizing these homo-
topy classes is given by the Hopf invariant. Consequently, if the Hopf invariant emerges
in the effective field theory, the spin and statistics of low-energy excitations fractionalize.
The analysis is based on a path integral representation of the t-J model which was
obtained recently by means of Dirac quantization. After introducing a staggered quan-
tization axis, the single occupancy constraint which is inherent to the t-J model can
be taken into account exactly. We perform a long-wavelength, low-frequency gradient
expansion of the effective action and integrate over the fermionic degrees of freedom as
well as the fast-fluctuating bosonic modes. Since our derivation is based on a micro-
scopic model, we obtain an effective action where the doping dependence of the coupling
constants is made explicit.

Zusammenfassung
Die vorliegende Arbeit behandelt die Ableitung einer effektiven Feldtheorie der nieder-
energetischen magnetischen Moden des t-J Modells auf einem zweidimensionalen quadra-
tischen Gitter im Grenzfall niedriger Dotierung, welches von Bedeutung fu¨r die Physik
der unkonventionellen Kuprat-Supraleiter ist. Im Speziellen untersuchen wir das mo¨gliche
Auftreten von Anregungen mit fraktionellem Spin und fraktioneller Statistik.
Fu¨r niedrige Temperaturen, kBT hinreichend kleiner als die magnetische Austauschkop-
plung J , und bei geringer Dotierung, befindet sich das System nahe der antiferromag-
netisch geordneten Ne´el-Phase. Die Korrelationsla¨nge kann in diesem Fall als wesentlich
gro¨ßer als die Gitterkonstante a angenommen werden, was einen feldtheoretischen Zu-
gang ermo¨glicht. Unter Verwendung einer Gradientenentwicklung um den antiferromag-
netischen Referenzzustand, wird die effektive Wirkung des t-J Modells abgeleitet. Mit-
tels Dimensionsanalyse la¨ßt sich zeigen, dass lediglich Terme bis zur Ordnung O(a2)
relevant fu¨r das Verhalten auf großen Skalen sind, wohingegen Terme dritter Ordnung
als marginal, Terme ho¨herer Ordnung als irrelevant einzustufen sind. Beitra¨ge in O(a3)
ko¨nnen dennoch einen signifikanten Einfluss auf die Eigenschaften magnetischer An-
regungen haben. Sie beinhalten unter Umsta¨nden einen topologischen Term, welcher
spezifische Feldkonfigurationen mit einem statistischen Phasenfaktor versieht. In der
Tat lassen sich sa¨mtliche Feldkonfigurationen als Abbildungen der kompaktifizierten
Raumzeit, der Dreispha¨re S3, in den Raum des Ordnungsparameters S2 auffassen und
fallen somit in unterschiedliche Homotopieklassen. Die topologische Invariante, welche
diese Homotopieklassen charakterisiert, ist durch die Hopf-Invariante gegeben. Tritt sie
in der effektiven Wirkung auf, fu¨hrt dies zwangsla¨ufig zu einer Fraktionalisierung von
Spin und Statistik der niederenergetischen Anregungen.
Die Analyse basiert auf einer Pfadintegraldarstellung des t-J Modells, welche mittels
Dirac-Quantisierung ermittelt wurde. Die dem t-J Modell anhaftende Zwangsbedingung
der Einfachbesetzung la¨ßt sich durch Einfu¨hrung einer alternierenden Quantisierungs-
achse exakt behandeln. Basierend auf dieser Darstellung fu¨hren wir die Gradientenent-
wicklung durch und integrieren sowohl u¨ber die fermionischen, als auch u¨ber die schnell-
fluktuierenden bosonischen Freiheitsgrade. Wir erhalten die Kopplungskonstanten der
effektiven Wirkung explizit in Abha¨ngigkeit der Dotierung.

Acknowledgements
Having reached the point of writing these lines and reflecting upon the recent years, I feel
great relief and a fair amount of gratitude towards the people who with their knowledge
and inspiration, with their patience and kindness guided me along the way.
The time I was working on the completion of this thesis has not always been easy. In
fact, it accompanied me through a period of major changes to my personal life. How-
ever that may be, it appears quite true that the biggest challenges provide the greatest
learning experience.
All the more, I owe my deepest gratitude to my family and friends, who were able to
support me throughout these years. First and foremost, thanks are due to my mother,
who with her optimism and her endless support provided me with the much needed
endurance. I am also greatly indebted to Simone and Astrid, who helped me through
the most challenging parts of the road, dedicating a lot of time and effort in times when
it was certainly not easy.
I wish to thank all my colleagues here at the Institut fu¨r Theoretische Physik III, the
people with whom I now shared a considerable part of my life. Many of them over the
years became dear friends: Alexander Janisch, Lars Bonnes, Adam Bu¨hler and Eslam
Khalaf, to name only a few.
I consider myself exceptionally lucky for having the opportunity of working alongside a
number of extraordinarily talented people, who graduated on closely related topics. The
discussions with Felix Andraschko and Eslam Khalaf not only helped me sharpen my
physical intuition but also proved to be of major value to the completion of this work.
I always enjoyed the professional and friendly atmosphere in our discussions.
A very special thank goes to our institute’s secretary, Frau Poljak, for her kind words,
her encouragement and good advice in the more difficult phases of my work.
Finally, I want to thank my advisor, Professor Muramatsu, for giving me the opportunity
to work on this interesting project and for continuously emphasizing thoroughness and
high standards.

To Noelle

Contents
1. Introduction 1
2. Field Theory and Topology 5
2.1. The concept of homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2. The non-linear σ-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1. Action functional and classical field equation . . . . . . . . . . . . 7
2.2.2. Static field configurations and solitons . . . . . . . . . . . . . . . 8
2.3. Time evolution - the Hopf-invariant . . . . . . . . . . . . . . . . . . . . . 14
2.3.1. The Hopf invariant . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2. The spin and statistics of solitons . . . . . . . . . . . . . . . . . . 20
2.3.3. Linking of world lines . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4. CP1-representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.1. The CP1-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.2. The Hopf invariant as a total divergence . . . . . . . . . . . . . . 29
2.4.3. The massive CP1-model . . . . . . . . . . . . . . . . . . . . . . . 32
3. Field Theoretical Methods for Strongly Correlated Systems 35
3.1. A path integral representation for the partition function . . . . . . . . . 35
3.1.1. Dirac quantization of constrained systems . . . . . . . . . . . . . 38
3.2. Effective field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1. Long-wavelength effective action for the Heisenberg model . . . . 45
4. Effective Field Theory for the t-J Model at Low Doping 53
4.1. Modeling the electronic structure of the cuprates . . . . . . . . . . . . . . 54
4.2. Path integral formulation of the t-J model . . . . . . . . . . . . . . . . . 56
4.2.1. The action functional . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.2. The t-J model in the low-doping limit . . . . . . . . . . . . . . . . 59
xi
Contents
4.3. Rotating frame and staggered CP1-representation . . . . . . . . . . . . . 60
4.3.1. Transformation of the integral measure . . . . . . . . . . . . . . . 62
4.3.2. Spin part of the action . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.3. Fermionic part of the action . . . . . . . . . . . . . . . . . . . . . 69
4.3.4. Fermionic contribution in k-space . . . . . . . . . . . . . . . . . . 81
4.4. Integrating out the fermions . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4.1. Evaluating the trace . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4.2. Contribution in O(a) . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4.3. Contribution in O(a2) . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4.4. Contribution in O(a3) . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4.5. Effective action after integrating out the fermions . . . . . . . . . 126
4.5. Integrating out small bosonic fluctuations . . . . . . . . . . . . . . . . . . 128
4.5.1. Path integral over ζ . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.5.2. Contribution in O(a2) . . . . . . . . . . . . . . . . . . . . . . . . 141
4.5.3. Contribution in O(a3) . . . . . . . . . . . . . . . . . . . . . . . . 143
4.5.4. Logarithmic contribution . . . . . . . . . . . . . . . . . . . . . . . 159
4.6. Summary and discussion of results . . . . . . . . . . . . . . . . . . . . . 170
4.6.1. Effective action up to O(a3) for the t-J model at low doping . . . 171
Conclusion 175
A. Supplementaries 177
A.1. Generalized non-linear σ-model . . . . . . . . . . . . . . . . . . . . . . . 177
A.2. Gaussian functional integration . . . . . . . . . . . . . . . . . . . . . . . 179
A.3. Matsubara frequency summation . . . . . . . . . . . . . . . . . . . . . . 183
A.4. Parametrization of z and ζ and gauge fixing . . . . . . . . . . . . . . . . 189
B. Definitions 191
B.1. Densities and compressibilities . . . . . . . . . . . . . . . . . . . . . . . . 191
B.2. F - and G-fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Bibliography 194
xii
Chapter 1
Introduction
Strongly correlated electron systems and the study of phenomena arising from the col-
lective interplay of their constituents pose a great challenge to theoretical physics. The
principle difficulty lies in the fact that most of the well established perturbative methods
are unreliable or cannot even be applied to systems with strong interactions. Some of
the most intriguing unsolved puzzles of solid state physics are thus found in the field of
strong correlations.
The past decades witnessed an enormous effort in the investigation of low-dimensional
magnetic systems, as they occur for example in layered copper-oxide structures like
La2CuO4. The development was triggered mainly by the occurrence of high-temperature
superconductivity in these materials, as discovered by Bednorz and Mu¨ller in 1986 [1].
Even after 25 years of intense research, a rigorous understanding of the phenomena is
still out of reach [2, 3].
Copper-oxides in the undoped state are antiferromagnetically ordered Mott-insulators
[4], their conductivity is suppressed through localization of the conduction electrons
caused by their strong Coulomb repulsion. Nevertheless, when the parent compound is
doped with a sufficient amount of charge carriers, superconductivity arises close to the
metal-insulator transition, see Figure 1.1.
One of the most promising candidates for a minimal microscopic model that captures
the physics of copper-oxides including the superconducting instability is given by the
t-J model [5, 6]. Doping can be controlled through the electron filling factor and the
zero doping limit amounts to the case of half-filling, where the t-J model reduces to the
Heisenberg model, describing a purely magnetic system. The Heisenberg model has been
studied intensively in the past and there is excellent quantitative agreement between the
predictions of its long-wavelength effective field theory [7] and the magnetic properties
1
1. Introduction
Figure 1.1.: Schematic phase diagram of the cuprates, taken from [3]
observed in the paramagnetic phase of undoped copper-oxides [8]. However, the situ-
ation for finite doping is far less transparent. Analytical treatments of the t-J model
struggle with the incorporation of the single-occupancy constraint [6] and a numerical
analysis is hindered by the fermionic sign problem [9]. The validity of mean-field ap-
proaches based on the use of slave-particle approximations, though able to qualitatively
account for several aspects of the rich phase diagram of copper-oxide compounds [6],
remains controversial.
Recently, an effective field theory for the magnetic degrees of freedom in the low doping
regime of the t-J model was obtained by means of Dirac quantization [10]. The contin-
uum theory describes the evolution of the collinear order parameter at zero doping to
a coplanar order parameter, predicting incommensurate magnetic order at low values
of doping [11, 12] in agreement with neutron scattering experiments [13]. The single-
occupancy constraint is exactly taken into account which leads to the emergence of a
U(1)-gauge field. This gauge field attains a mass as a function of doping, opening the
possibility of deconfined spinons in the finite doping regime [14,15].
The aim of this thesis is to refine the effective theory obtained in [10] in order to address
the question whether anyonic excitations play a role in the low-energy physics of the t-J
model.
2
Anyons and fractional statistics
We are taught in elementary quantum mechanics that there are two types of particles
in nature known as bosons and fermions. The concept is implemented in form of a sym-
metrization postulate, stating that in a system of identical particles, the many-particle
states must either be symmetric or antisymmetric with respect to particle interchange,
leading respectively to Bose-Einstein or Fermi-Dirac statistics for their ensembles [16].
The connection to the particle’s spin is attributed to the celebrated spin-statistics the-
orem, valid for local relativistic quantum field theories in (3 + 1)-dimensions [17].
This is, however, only part of the story. Low-dimensional systems provide a framework
in which particles with more exotic statistics can be realized. In fact, there is an intimate
relationship between particle statistics, the number of spatial dimensions and the topol-
ogy of the configuration space C. The latter can be non-trivial since configurations where
particles occupy the same position are excluded from C and the ones related through
an interchange of identical particles are identified [18]. In a path integral picture, the
interchange of particles is represented as a closed path in C and the phase factors as-
sociated with these paths define a one-dimensional representation of its first homotopy
group [19]. It turns out that in three dimensions the phase angles φ = 0 and φ = pi are
the only possible choices [20], since the configuration space is doubly connected and its
first homotopy group is just the group of permutations. However in two dimensions, the
first homotopy group of C is given by the multiply connected braid group and thus, any
fractional value of φ is allowed [21,22]. Particles where the phase angle is different from
0 or pi were termed anyons by Frank Wilczek [23,24].
There are basically two ways of introducing anyons in model calculations. One way is to
endow charged point-like particles with a magnetic flux [23] such that anyonic phases are
obtained as a result of the Aharanov-Bohm effect. This so called flux-tube model, apart
from its use in the explanation of the fractional quantum Hall effect [25–28], turned out
to be of great value in determining the statistical properties of a non-interacting gas of
anyons [29].
Another very elegant realization of anyons in quantum field theory is given by the non-
linear σ-model, where fractional spin and statistics can arise for its particle-like solu-
tions [30]. The existence of these solutions is guaranteed by topological considerations
3
1. Introduction
and they behave as anyons, provided that another term of topological significance, the
so called Hopf term, is added to the action functional. This point will be discussed
extensively in Chapter 2.
Since the O(3)-symmetric non-linear σ-model emerges as the low-energy effective field
theory of the Heisenberg model, investigations whether the Hopf term would emerge
naturally from the Heisenberg action were underway early [31]. The analysis gave a
negative result; however, there is still the possibility that the Hopf term might arise
through doping. The subsequent question whether fractional statistics play a role in the
low-doping regime of the t-J model has not been addressed yet, mainly due to the afore-
mentioned complications in deriving an effective field theory for its magnetic degrees of
freedom. This is where we want to contribute with this work.
We will structure our analysis as follows:
• In Chapter 2, we introduce the concept of topological invariants and explain the
mechanism that leads to excitations with fractional spin and statistics.
• Chapter 3 is dedicated to a brief review of path integral methods in strongly corre-
lated electron systems and effective field theories.
• The main part of the analysis will be presented in Chapter 4, where in the first
section we establish the path integral representation of the t-J model as obtained
in [10]. The derivation of the effective action is presented in Section 4.3-4.6.
4
Chapter 2
Field Theory and Topology
In this introductory chapter, we focus on the use of topological concepts in low-dimen-
sional field theories, in particular the non-linear σ-model. After introducing some very
basic notions from algebraic topology, we discuss their application to non-trivial, particle-
like solutions of this model and investigate the mechanism that endows these solutions
with fractional spin and statistics. Our discussion will begin on a purely classical level
which is the natural starting point for more refined treatments of the quantized theory.
2.1. The concept of homotopy
Topology in general studies objects and relations as they are subjected to continuous de-
formations, that is homeomorphisms. These objects are regarded as topological spaces,
e.g. sets with a collection of subsets that define a topology, in order to obtain a notion
of continuity. Many of their properties can be mapped onto algebraic concepts, as for
instance group structures [32, 33]. Here, we will need only a small number of notions
that are common in algebraic topology and focus on the concept of
Let X, Y be topological spaces and f , g continuous functions from X to Y . A ho-
motopy between f and g is a continuous function H: X × [0, 1] → Y , such that for
all x ∈ X H(x, 0) = f(x) and H(x, 1) = g(x).
Homotopy
If there exists a homotopy between two continuous functions f and g, these functions
are said to be homotopic. Being homotopic is an equivalence relation among mappings.
Now consider the set of continuous functions that map Sd, the d-dimensional unit-
sphere, into some topological space Y . Note that these functions define generalized
loops in Y . We may choose an arbitrary base point x0 in Sd and parametrize f such
5
2. Field Theory and Topology
that x0 constitutes the start and end point of the loop. We look only at homotopies that
hold x0 constant and collect all homotopic functions f from Sd to Y into equivalence
classes [f ]. Together with a composition law that connects the start and endpoint of
loops in two given classes [f ] and [g],
[f ] · [g] = [f · g] , (2.1)
these classes form a group, the d-th homotopy group of Y : pid(Y ).
The first homotopy group is in general called the fundamental group of the topologi-
cal space Y , pi1(Y ), on account of its importance in algebraic topology.
There are homotopic invariants which characterize the homotopy class of a given map-
ping between two topological spaces and by definition, their value is invariant under
homotopies. In the following, we will be concerned with two particular homotopic invari-
ants. First of all the winding number, also called Pontryagin index which characterizes
pin(Sn) = Z, the n-th homotopy group of the n-sphere. Simply spoken, the Pontryagin
index counts the number of times one n-sphere gets wrapped around the other. Secondly,
the Hopf invariant, characterizing pi2n−1(Sn) = Z, the (2n−1)-th homotopy group of the
n-sphere. The geometrical meaning of the Hopf invariant is harder to visualize but we
will see that it is related to linking properties of curves. Note that the homotopy groups
of spheres are notoriously difficult to compute. For details we refer to the mathematical
literature, as for instance reference [34].
The mappings of relevance to our analysis represent real-space field configurations and
are thus mappings from coordinate space into the target space of a given order parameter.
As it turns out, these configurations may be non-trivial in the sense that they fall into
different homotopy sectors. We shall use homotopic invariants to characterize these
non-trivial configurations.
2.2. The non-linear σ-model
Non-linear σ-models in general may be defined for various kinds of order parameters
and underlying spaces, its simplest descendant being the O(3)-symmetric version in
which the order parameter is represented by a three-dimensional unit vector. We will
6
2.2. The non-linear σ-model
devote this section to a discussion of the classical non-linear σ-model, in particular its
topologically non-trivial solutions.
2.2.1. Action functional and classical field equation
Let us consider the non-linear σ-model in d dimensions. Its dynamics is governed by the
action functional
S = 12
∫
ddx ∂µ~n · ∂µ~n , where ~n · ~n = 1 , (2.2)
where the non-linearity enters the model via the constraint ~n · ~n = 1. The name O(3)-
NLσM refers to the fact that both the Lagrangian in (2.2) as well as the constraint are
invariant under global O(3) rotations in internal space.
Classical equations of motion
The classical field equation can be obtained by introducing a Lagrangian multiplier and
extremising the action [35]
S =
∫
ddx
(
1
2 (∂µ~n) · (∂
µ~n) + λ (~n · ~n− 1)
)
. (2.3)
Extremising S leads to the Euler-Lagrange equations
∂µ∂
µ~n+ λ~n = 0 , ~n · ~n = 1 . (2.4)
The Lagrange multiplier may be eliminated via the constraint
λ = λ~n · ~n = −~n · (∂µ∂µ~n) , (2.5)
so that we obtain a field equation which has the constraint incorporated and is explicitly
non-linear
∂µ∂
µ~n−
[
~n · (∂µ∂µ~n)
]
~n = 0 . (2.6)
We will see in the next subsection that this model yields interesting particle-like solutions
called solitons.
7
2. Field Theory and Topology
2.2.2. Static field configurations and solitons
Solitons, in general, are solutions to non-linear wave equations which are localized and
non-dispersive. In that sense, they can be viewed as particle-like configurations. The
existence of such solutions for a given field theory is by no means self-evident. Usually,
dispersive terms and non-linearities tend to distort the shape of wave packages during
their propagation. However for certain non-linear wave equations, the effects of disper-
sion and non-linearity might compensate one another so that there can be a subset of
solutions with the aforementioned property. These solutions are in general called soli-
tons [35]. In the following, we quickly review the argument that allows us to derive
soliton solutions for the O(3) non-linear σ-model. Their existence in this model can be
traced back to the non-trivial homotopy group pi2(S2), as was first discussed by Belavin
and Polyakov in 1975 in the context of two-dimensional ferromagnets [36].
We consider the non-linear σ-model in two spatial and one time dimension
S = 12g
∫
dt d2x ∂µ~n · ∂µ~n , (2.7)
where we also introduced a coupling constant which is defined such that the action has
dimension [E t]. Let us concentrate on static solutions where ∂t ~n = 0. In that case, the
field equation (2.6) reduces to
4~n−
[
~n · (4~n)
]
~n = 0 , where 4 = ∂2x + ∂2y . (2.8)
Their energy can be directly read off the action functional
Est =
1
2g
∫
d2x ∂µ~n · ∂µ~n , where now µ = x, y . (2.9)
Obviously, the trivial solution, the classical ground state, satisfies ∂µ ~n = 0 and is thus
given by some constant unit vector ~n(x, y) = ~n0. The direction of ~n is arbitrary, so there
is a set of infinitely many degenerate ground states, each one spontaneously breaking
the O(3)-symmetry.
Now let us consider the possibility of non-trivial static solutions with finite energy E >
0. It might be noteworthy that the condition for finite energy solutions is a physical
8
2.2. The non-linear σ-model
requirement and is essential to the existence of topologically non-trivial solutions. In
order for (2.9) to be finite, solutions have to be non-singular and obey the following
boundary condition
lim
r→∞~n(~r) = φ0 = const. , (2.10)
where r is the radial coordinate of the (x, y)-plane. The field approaches a constant
vector at infinity, so that a compactification of the physical space R2 into the two-sphere
S2 is not only possible but also the most adequate description. Static field configurations
are thus given by a mapping
~n : S2 → S2 , (2.11)
and these mappings fall into different homotopy sectors [34], each sector characterized
by a topological invariant which we will discuss in the next paragraph.
Pontryagin index - Winding number
The unique topological invariant [37] characterizing the homotopy sector of automor-
phisms on the two-sphere is given by the Pontryagin index, which has the following
coordinate representation
Q = 18pi
∫
d2x µν ~n · (∂µ~n× ∂ν~n) , µ, ν = x, y , (2.12)
where µν is the totally antisymmetric second-rank tensor. The Pontryagin index is also
called the winding number of the map ~n, since it counts the number of times ~n wraps the
two-sphere in physical space around the two-sphere in the target space. As such, Q takes
only integer values and characterizes the second homotopy group of the two-sphere,
pi2(S2) = Z . (2.13)
As demonstrated in [35], it is easy to prove that (2.12) provides indeed the oriented
winding number of the mapping na : S2 → S2 by considering the vector-valued surface
element of the two-sphere embedded into R3
d(S(t))a = abc dnb dnc , (2.14)
9
2. Field Theory and Topology
with abc the totally antisymmetric tensor in three dimensions and na, nb two unit vectors
in the target space, as is indicated by the subscript t in (2.14). Let na(ϕ1, ϕ2) be a
parametrization of the mapping in terms of two angular variables, designating a point
on the two-sphere in coordinate space. Then we get
d(S(t))a = abc
∂nb
∂ϕ1
∂nc
∂ϕ2
d2ϕ = 12αβabc
∂nb
∂ϕα
∂nc
∂ϕβ
d2ϕ . (2.15)
And for the Pontryagin index, we have
Q = 18pi
∫
d2x µνabc na
∂nb
∂xµ
∂nc
∂xν
= 18pi
∫
d2x µνabc na
∂nb
∂ϕr
∂ϕr
∂xµ
∂nc
∂ϕs
∂ϕs
∂xν
= 18pi
∫
d2ϕ αβabc na
∂nb
∂ϕα
∂nc
∂ϕβ
, (2.16)
since the Jacobian for the change of variables (x1, x2) to (ϕ1, ϕ2) obeys
µν
∂ϕr
∂xµ
∂ϕs
∂xν
d2x = rs d2ϕ . (2.17)
When we now plug in (2.15) into (2.16), we get
Q = 14pi
∫
na d(S(t))a =
1
4pi
∫
dS(t) , (2.18)
since na is a unit vector normal to the surface. Note that the integral is over the two-
sphere in coordinate space S2(c). Taking into account that the surface of the unit-sphere
is 4pi, we clearly see that Q gives the number of times the target sphere S2(t) is traversed
as we span the compactified coordinate space S2(c). Obviously, for this quantity to be a
well defined integer, the boundary condition (2.10) and the subsequent compactification
is crucial.
Soliton solutions for the O(3) non-linear σ-model
To conclude this section, we will derive an explicit expression for topologically non-trivial
static solutions, the solitons and elaborate on their connection to the Pontryagin index.
For the most part of this derivation, we follow the analysis given in [35].
10
2.2. The non-linear σ-model
As a first step, we derive a lower bound for the energy of static solutions in each homotopy
sector. Consider the obviously fulfilled identity
(∂µ~n± µν~n× ∂ν~n)2 ≥ 0 , (2.19)
where the indices run over two spatial coordinates, x and y, so the index position is
irrelevant. The product is performed in internal as well as in coordinate space, so we
obtain
(∂µ~n) · (∂µ~n) + µν(~n× ∂ν~n)µσ(~n× ∂σ~n) ≥ ±2 µν~n · (∂µ~n× ∂ν~n) . (2.20)
Using µνµσ = δνσ and the Lagrange identity
(~a×~b) · (~c× ~d) = (~a · ~c)(~b · ~d)− (~b · ~c)(~a · ~d) , (2.21)
we see that the first and second summand on the left-hand side of (2.20) are equal. We
integrate (2.20) over coordinate space and obtain
∫
d2x (∂µ~n) · (∂µ~n) ≥ ±
∫
d2x µν ~n · (∂µ~n× ∂ν~n) , (2.22)
which means that, given (2.9) and (2.12),
E ≥ 4pi
g
|Q| . (2.23)
Lower bounds for the energy in terms of topological charges are also called Bogomol’nyi
inequalities [38].
Solutions with a certain Q have minimal energy when the equality in (2.23) is satisfied.
Consequently, these solution have to satisfy the equality in (2.19) as well, and we can
write
∂µ~n = ±µν~n× (∂ν~n) . (2.24)
Configurations fulfilling this condition minimize the energy in a given Q-sector but also
solve the stationary field equation (2.6), which can be easily verified by calculating the
derivative of (2.24) and inserting it into (2.6).
We may further simplify the condition (2.24) by use of stereographic projection, that is,
11
2. Field Theory and Topology
we map the values of ~n which live on the two-sphere onto corresponding points in the
complex plane. The transformation is given by
ω1 =
2n1
1− n3 and ω2 =
2n2
1− n3 , (2.25)
and when we rewrite (2.24) for ω = ω1 + i ω2, we obtain
∂ω1
∂x
= ±∂ω2
∂y
and ∂ω1
∂y
= ∓∂ω2
∂x
, (2.26)
which is nothing but the Cauchy-Riemann condition for analytic functions ω(x + iy)
and ω(x− iy). Thus, any such analytic function automatically satisfies (2.24), after we
inverted the stereographic projection. Let us also rewrite (2.9) and (2.12) in terms of ω
E = 1
g
∫
d2x
|dω/dz|2
(1 + |ω|2/4)2 , |Q| =
g
4pi E . (2.27)
Any complex analytic function can be expanded in a Laurant series, a complex linear
combination of monomes of the form
ω(n)(z) =
(
z − z0
λ
)n
, (2.28)
with a complex constant z0 and λ some real parameter. We take ω(n)(z) as our prototype
solution. As it turns out, (2.28) is the static soliton solution in the Q = n homotopy
sector. This can already be anticipated from the fact that, given a value of ω, (2.28)
allows n roots for z, implicating n windings for this configuration. Alternatively, one
may insert (2.28) into (2.27) and find exactly the same result.
Finally, we consider a few explicit solutions for certain winding numbers. To do so, we
invert (2.25) and obtain
~n = 1|ω2 |2 + 1
(
ω1, ω2,
∣∣∣∣ω2
∣∣∣∣2 − 1) . (2.29)
Note that the parameter λ in (2.28) determines the spatial extent of the soliton and z0
its center. For simplicity, we will set z0 = 0 and use polar coordinates (r, ϕ) for the
coordinate plane. An explicit parametrization of solutions with the two lowest positive
winding number is thus given by
12
2.2. The non-linear σ-model
• The Q=1 soliton:
Figure 2.1.: Soliton with winding number Q = 1.
ω(z) = z
λ
,
~n(r, ϕ) =

r
λ
cosϕ
( r2λ )2+1
r
λ
sinϕ
( r2λ )2+1
( r2λ )
2−1
( r2λ )2+1
 .
• The Q=2 soliton:
Figure 2.2.: Soliton with winding number Q = 2.
ω(z) =
(
z
λ
)2
,
~n(r, ϕ) =

r2
λ2 cos 2ϕ
( r22λ2 )
2+1
r2
λ2 sin 2ϕ
( r22λ2 )
2+1
( r
2
2λ2 )
2−1
( r22λ2 )
2+1

.
Soliton solutions to the O(3) non-linear σ-model are often termed skyrmions due to
a work by T.H.R. Skyrme, where he proposed topological solitons as a model for ele-
mentary particles, describing baryons as excitations of an underlying meson field [39].
Although this idea was replaced by the concept of quarks and gluons, skyrmions prevailed
in the context of condensed matter physics. Recently, direct observation of skyrmionic
spin textures has been reported in a number of materials [40–42].
13
2. Field Theory and Topology
2.3. Time evolution - the Hopf-invariant
In the previous section, we concentrated on non-trivial static solutions of the O(3) non-
linear σ-model. Since (2.7) is Lorentz invariant, these solutions readily generalize to
solitons propagating at constant speed. In the Euclidean time formalism, which will
be important to us in the upcoming analysis, the invariance group in coordinate space
is given by O(3) instead of O(2, 1) so that time evolution resembles the generalization
to one more spatial dimension. Soliton solutions in Euclidean space-time usually are
referred to as instantons.
The topological current
Time evolution in field theory is a continuous process so that a homotopic invariant will
not change under the evolution of field configurations. One might say that the solitons
are topologically protected, so our considerations in the last section remain valid.
There is a simple geometric argument that ensures the conservation of topological charge
in the non-linear σ-model. Independently of the equation of motions, the constraint
~n · ~n = 1 implies the conservation of a current
jµ = 18pi
µνλ ~n · (∂ν~n× ∂λ~n) , µ = 0, 1, 2 . (2.30)
It is straightforward to show that
∂µj
µ = 18pi
µνλ ∂µ~n · (∂ν~n× ∂λ~n) = 0 , (2.31)
since the constraint implies that the change of ~n is always coplanar. Note that the
spatial integral over the time component j0 of this current is exactly the Pontryagin
index (2.12). Hence the name topological current for jµ and topological charge for the
Pontryagin index.
Compactification R3 → S3
From our discussion in Section 2.2.2, we know that the finite energy assumption implies
a boundary condition for all field configurations, ~n(~x) = const for |~x| → ∞, so they
can be viewed as mappings from the compactified coordinate plane S2 to the internal
14
2.3. Time evolution - the Hopf-invariant
space S2. If we now include time evolution in our description, in particular with periodic
boundary conditions, we can think of coordinate space as the direct product S1 × S2,
the one-sphere in the time domain and the spatial two-sphere. These mappings fall
into different homotopy classes, the set of which we denote [S1 × S2, S2]. In fact, it is
shown in reference [43] that all mappings β : S2 × S1 → S2 may be decomposed into
γ : S2 × S1 → S3 and η : S3 → S2 such that β = η ◦ γ. The topological invariant
characterizing η : S3 → S2 and therefore also the composition β = η ◦ γ is given by the
Hopf invariant [44].
2.3.1. The Hopf invariant
There are various related definitions of the Hopf invariant given in the literature. Prob-
ably the most natural one is the geometrical, based on the work by Heinz Hopf himself.
The idea is as follows [45]:
Given a mapping f : S3 → S2 and two distinct points u and v on S2, one may
define their inverse images f−1(u) and f−1(v) which can be shown to be closed
curves in S3, provided that f is sufficiently regular. The linking number of these
two curves is universal in the sense that for a particular mapping f , the number of
links is the same for every pair of points u and v. The Hopf invariant is then de-
fined as the linking number of the curves f−1(u) and f−1(v).
Hopf invariant
Since S3 together with the projection f : S3 → S2 can be regarded as a fiber bundle [46],
the inverse image f−1(u) of a given point u is usually termed the fiber at u.
The Hopf projection
Let us remark that in his original work [47], Hopf discussed one particular mapping
pi : S3 → S2, now known as the Hopf map or Hopf projection. He was able to show
that every point on the two-sphere has as an inverse image a distinct circle on the three-
sphere, and that these circles are linked exactly once. The Hopf projection is given by
15
2. Field Theory and Topology
the chain of homomorphisms
S3 =
{
~x ∈ R4 : |~x|2 = 1
}
z−−−→ {(z1, z2) ∈ C2 : |z1|2 + |z2|2 = 1}ypi
S2 = {~n ∈ R3 : |~n|2 = 1} ζ−1←−−− CP1 = C ∪ {∞}
, (2.32)
such that
(x1, x2, x3, x4) z−−−→ (z1, z2) = (x1 + ix2, x3 + ix4)ypi(
2 Reζ
ζ¯ζ+1 ,
2 Imζ
ζ¯ζ+1 ,
ζ¯ζ−1
ζ¯ζ+1
)
ζ−1←−−− ζ = z1
z2
. (2.33)
Note that z and ζ are merely complex representations of the three- and two-sphere,
respectively. The actual projection occurs through pi. In the literature, the Hopf map is
usually defined through the very compact relation
pi : S
3 → S2
z 7→ ~n = z¯ ~σz , (2.34)
where z is an element of the complex representation of S3 and ~σ are the Pauli matrices.
Written out explicitly, we have
n1 = z∗1z2 + z∗2z1 ,
n2 = −i (z∗1z2 − z∗2z1) ,
n3 = z∗1z1 − z∗2z2 . (2.35)
In fact, every physicist will recognize this as the spinor representation of a unit vector ~n.
As such, for a given ~n, the vectors z are determined only up to gauge transformations
z → z eiΛ , (2.36)
with an arbitrary Λ ∈ R.
As we already mentioned, the Hopf projection has Hopf invariant H = 1. In fact, we
may verify this statement by considering two arbitrary points on the two-sphere, say for
16
2.3. Time evolution - the Hopf-invariant
instance ~n1 = (1, 0, 0) and ~n2 = (−1, 0, 0). Their inverse images in S3 are given by two
circles z1 = 1/
√
2
(
eiΛ, eiΛ
)
and z2 = 1/
√
2
(
eiΛ,−eiΛ
)
. These may be stereographically
projected into R3, where we obtain the two curves
γ1 : Λ 7→
(
cos Λ√
2 + sin Λ
,
sin Λ√
2 + sin Λ
,
cos Λ√
2 + sin Λ
)
,
γ2 : Λ 7→
(
cos Λ√
2− sin Λ ,
sin Λ√
2− sin Λ ,−
cos Λ√
2− sin Λ
)
, (2.37)
which obviously link exactly once, as sketched in Figure 2.3.
Figure 2.3.: Two Hopf fibers.
Hopf invariant
We already defined the Hopf invariant H in terms of its geometrical interpretation and
argued that the Hopf projection has H = 1. However, it would be desirable to have an
analytic expression for H, so that we can easily derive the invariant for a given mapping
~n : S3 → S2. It turns out that such an expression can in fact be constructed from the
topological current we defined in (2.30) [30].
Note that the existence of a conserved current enables us to introduce a vector potential
Aµ, such that
jµ = µνλ∂νAλ . (2.38)
The potential is defined up to gauge transformations which leave the current invariant
Aµ → Aµ + ∂µΛ , (2.39)
17
2. Field Theory and Topology
with an arbitrary real-valued field Λ. The Hopf invariant may then be defined in terms
of the current and its potential
Hopf invariant
H = − 12pi
∫
d3x jµAµ , (2.40)
where the integration runs over S3, the compactified coordinate space. Note that, since
jµ and Aµ are determined from the field configurations ~n(xµ), see (2.30) and (2.38),
the Hopf invariant can in principle be computed for any given mapping ~n : S3 → S2.
A rigorous proof that (2.40) is indeed equivalent to the geometric definition given in
Section 2.3.1 can be found in the literature [48].
Let us make a few physical remarks on this expression for the Hopf invariant. Actually,
(2.40) resembles the well known coupling term in the covariant formulation of electro-
dynamics. The situation is however different to electrodynamics, where Aµ has its own
dynamics, even in the absence of matter currents, as conditioned by the kinetic term
F µνFµν in the action functional. The kinetic term, together with the matter coupling
gives rise to Maxwell’s equation. Most importantly, the electromagnetic potential is
not the source of the matter currents as is the case in (2.38). It solely determines the
electromagnetic field strength through Fµν = ∂µAν − ∂νAµ.
Indeed, we are also able to define a field strength from the topological gauge potential
Fµν = ∂µAν − ∂νAµ ,
(∗F )µ = 12
µνλFνλ = µνλ∂νAλ , (2.41)
which identifies jµ as the dual field strength tensor, which in three dimensions is a vector.
With (2.41), we may recast the Hopf invariant into
H = − 12pi
∫
d3x µνλAµ ∂νAλ = − 14pi
∫
d3x µνλAµFνλ , (2.42)
where it assumes the form of a Chern-Simons term [49].
18
2.3. Time evolution - the Hopf-invariant
Homotopic invariant
For H to be a homotopic invariant, it must be insensitive to continuous deformations of
the mapping ~n. A continuous deformation can be thought of as subsequent infinitesimal
deformations. Thus, a local variation in the field configurations ~n → ~n + δ~n should
not lead to a change in H. We show that the Hopf term (2.40) is indeed a homotopic
invariant by explicitly demonstrating that its variation vanishes. Consider
δH = − 12pi
∫
d3x
(
δAµ j
µ + Aµ δjµ
)
. (2.43)
From (2.38), we obtain
δjµ = µνλ∂ν δAλ , (2.44)
whereas, using (2.30), we get
δjµ = µνλabc
(
δna ∂νn
b ∂λn
c + na ∂ν(δnb) ∂λnc + na ∂νnb ∂λ(δnc)
)
= 2 µνλabc na ∂ν(δnb) ∂λnc
= µνλ ∂ν
(
2 abc na δnb ∂λnc
)
. (2.45)
This, together with (2.44), gives us the variation of the vector potential
δAµ = 2 abc na δnb ∂λnc . (2.46)
So we get for the variation of H
δH = − 12pi
∫
d3x
(
δAµ j
µ + Aµ µνλ∂ν(δAλ)
)
= − 12pi
∫
d3x
(
µνλ∂νAλ δAµ − µνλ∂νAµ δAλ
)
= − 1
pi
∫
d3x jµ δAµ . (2.47)
When we now plug in (2.30) and (2.46), we find that
δH = − 14pi2
∫
d3x µνλabcijk n
a ∂νn
b ∂λn
c ni δnj ∂µn
k = 0 , (2.48)
since µνλ∂νnb ∂λnc ∂µnk = 0. Thus, H is a homotopic invariant.
19
2. Field Theory and Topology
2.3.2. The spin and statistics of solitons
Let us return to a more physical discussion of the Hopf invariant and how it can be uti-
lized to implement fractional statistics in quantum field theory. The following argument
follows a celebrated analysis by Wilzcek and Zee [30], connecting the spin and statistics
of solitons to the Hopf invariant. We consider therefore a system in (2 + 1) dimensions,
governed by the O(3)-symmetric non-linear σ-model action
Sσ =
1
2g
∫
dt d2x ∂µ~n · ∂µ~n . (2.49)
As we saw in Section 2.2.2, the classical field theory exhibits soliton solutions which are
characterized by the homotopy sector of the mapping ~n : S2 → S2 and its corresponding
homotopic invariant, the Pontryagin index Q, see (2.12).
Consider now a soliton state with Q = 1 and a quasi-static time evolution in which
the soliton slowly rotates around its center. According to Feynman, at the end of the
rotation the wave function will have acquired a phase eiS [50] , where S is given by
the action corresponding to this particular process. It is also well known, that for
quantum mechanical states, a spatial rotation around the z-axis is realized by the action
of an unitary operator, constructed from the generator of the rotation which is given
by the angular momentum operator along this axis Jˆz [16]. Eigenstates of the angular
momentum operator will therefore acquire a phase eiJz , where Jz corresponds to the
projected angular momentum of the state. Thus, we have the correspondence
exp
(
iS
)
= exp
(
iJz
)
. (2.50)
From a classical point of view, the rotation of the soliton may be realized by the action
of a SO(3) rotation matrix R(t) on the vector field ~n. If we perform the rotation in-
finitesimally slow, the period of a 2pi rotation T →∞ and the contribution to the action
(2.49) vanishes as 1/T → 0. Thus, the left-hand side of (2.50) is 1. The soliton has no
intrinsic angular momentum.
Now let us add an additional term SH to the action, which is proportional to the Hopf
invariant
S = Sσ + θH = Sσ − θ2pi
∫
dt d2x jµAµ . (2.51)
20
2.3. Time evolution - the Hopf-invariant
The angular variable θ ∈ [0, 2pi) is usually called the statistical angle. If we perform
again the 2pi-rotation of the soliton, the Hopf term, which is of the order T 0 = 1, will
eventually contribute to the action, even in the limit of an adiabatic rotation. The na-
ture of this contribution is determined by the field history, which can be understood by
applying the geometric interpretation of the Hopf invariant.
For the argument to work, it is crucial that we consider field histories that start from and
evolve into the classical ground state, such that the compactification of the coordinate
space into S3 can be performed and field histories map S3 → S2, see the discussion on
page 14. Under this condition, solitons may still be realized through the creation of a
soliton-antisoliton pair at a given time t′ > ti which we assume to annihilate at a later
time t′′ < tf . Note that the total topological charge of the system is zero at all times.
Consider then a process where after the pair creation the soliton is adiabatically rotated
by 2pi. After the rotation, we let the soliton pair annihilate again such that at tf the
system is in the ground state.
Figure 2.4.: Rotation of a single skyrmion by pi/2.
We may visualize this process by following the trace of the inverse images of two points
on the two-sphere of ~n(t, x). Take for instance ~n = (−1, 0, 0) and ~n′ = (0, 1, 0) as in
Figure 2.4. Indeed for the pair creation and annihilation process, their inverse images
21
2. Field Theory and Topology
r = (t, x, y) = f−1(~n) and r′ = (t′, x′, y′) = f−1(~n′) are closed curves and their projected
world lines may be sketched as in Figure 2.5 (a)-(c). There we can easily see that a
2pi-rotation leads to exactly one link in the world lines of r and r′. Thus, recalling the
geometrical interpretation of the Hopf invariant in Section 2.3.1, we get a contribution
of θ to the action functional. The soliton in the non-linear σ-model with Hopf term has
angular momentum θ.
Figure 2.5.: Linking of fibers: (a) no rotation, (b) 2pi-rotation, (c) 4pi-rotation.
The same reasoning can be applied to a process which involves the exchange of two soli-
tons, determining their statistics. For this case, one has to create two soliton-antisoliton
pairs and subsequently exchange the two solitons before both pairs annihilate again.
Such a process is sketched in Figure 2.6 (b), where we see that the exchange leads again
to a linking number of one. Accordingly, the solitons acquire a phase of ei θ under an
exchange of particles.
Figure 2.6.: Linking of fibers: (a) no exchange, (b) soliton exchange.
The argument can obviously be generalized to more complicated field histories and
22
2.3. Time evolution - the Hopf-invariant
exchange processes. We therefore established that the Hopf term, if it is added to the
action functional of the non-linear σ-model as in (2.49), will endow the solitons of the
theory with fractional spin and statistics.
2.3.3. Linking of world lines
There is an alternative way of showing that the Hopf term will lead to fractional statis-
tics which does not depend on the creation and annihilation of soliton-antisoliton pairs.
In this picture, we explicitly connect the Gaussian linking integrals of two skyrmion
world lines to the Hopf invariant [51,52].
The analysis is most conveniently performed in the analytic continuation of the theory to
imaginary time. In this Wick rotated representation, the Minkowski metric is replaced
by the Euclidean metric, and we will use the vector notation ~r = (τ, x, y) for space and
imaginary time coordinates, such that the current conservation in (2.31) now reads
~∇ ·~j = 0 . (2.52)
Thus, ~j(~x) is a source-free vector field and we may introduce a vector potential ~A, such
that, in analogy to (2.38),
~j = ~∇× ~A , (2.53)
where ~A is again defined up to gauge transformations
~A→ ~A+ ~∇Λ . (2.54)
We will now give an explicit expression for the vector potential ~A. To do so, we first
choose the Coulomb-gauge in which ~∇ · ~A = 0. We obtain
~∇×~j = ~∇× (~∇× ~A) (2.55)
= ~∇ (~∇ · ~A)− ~∇2 ~A = −4 ~A , (2.56)
a Poisson-like equation that can be solved for ~A:
~A(~r) = 14pi
∫
d3r′
~∇~r′ ×~j(~r′)
|~r − ~r′|
23
2. Field Theory and Topology
= − 14pi
∫
d3r′ ~∇~r′
(
1
|~r − ~r′|
)
×~j(~r′) = 14pi
∫
d3r′ ~j(~r′)× ~r − ~r
′
|~r − ~r′|3 , (2.57)
where we used integration by parts. This makes the non-locality of the potential and its
dependence on the matter fields explicit.
Let us now consider two solitons with negligible spatial extent λ → 0. The topological
current for such a configuration reads
~j(~r) =
∫
dϕ
(
I1
d~γ1(ϕ)
dϕ
δ
(
~r − ~γ1(ϕ)
)
+ I2
d~γ2(ϕ)
dϕ
δ
(
~r − ~γ2(ϕ)
))
, (2.58)
γa(ϕ) being the world line of soliton a, parametrized in ϕ. I1 and I2 are the respective
current strengths and for solitons with winding number Q = 1, they normalize to 2pi.
For such line currents we have
~j d3r = 2pid~γ1(ϕ)
dϕ
dϕ+ 2pid~γ2(ϕ)
dϕ
dϕ = 2pi(d~γ1 + d~γ2) , (2.59)
independent of the parametrization. Thus, we get for (2.57)
~A(~r) = 12
∫
γ1
~dγ1 × ~r − ~r1|~r − ~r1|3 +
1
2
∫
γ2
~dγ2 × ~r − ~r2|~r − ~r2|3 . (2.60)
Note that due to our boundary conditions, the world lines are closed curves in S1 × S2.
For better visualization, we may project them into a torus T = S1×D2 or into R3 with
the plains at t = ±∞ identified, see Figure 2.7. Hence, the integrals in (2.60) run over
closed curves γa and we may use Stoke’s theorem to rewrite them as integrals over the
enclosed surface Σa which have normal vectors ~na
~A(~r) = 12
(
~∇ ·
∫
Σ1
~dn1
~r − ~r1
|~r − ~r1|3 +
~∇ ·
∫
Σ2
~dn2
~r − ~r2
|~r − ~r2|3
)
(2.61)
= 12
(
~∇Ω1(~r) + ~∇Ω2(~r)
)
, (2.62)
where Ωa is nothing but the solid angle of the surface Σa, viewed from the point ~r. Thus,
the vector potential ~A describes the local change in solid angles of the two surfaces, seen
from the point ~r, in the sense that it is given by the sum of gradients, pointing into the
direction of steepest ascent.
24
2.3. Time evolution - the Hopf-invariant
From these considerations, let us derive the Hopf invariant (2.40) for our line currents
(2.58):
H = − 12pi
∫
d3r ~A ·~j = − 12pi
∫
γ1
~dγ1 ~A(~r)−
1
2pi
∫
γ2
~dγ2 ~A(~r)
= − 14pi
∫
γ1
~dγ1
(
~∇Ω1(~r) + ~∇Ω2(~r)
)
− 14pi
∫
γ2
~dγ2
(
~∇Ω1(~r) + ~∇Ω2(~r)
)
. (2.63)
The further evaluation depends on the type of world lines we are considering. Let us
assume they look as in Figure 2.7 (a) or 2.7 (c). Then the change in solid angle of Ω1,
as seen from the curve γ1, is perpendicular to the tangent vector dγ1. The same holds
for Ω2, seen from γ2. Thus, we are left with the second and third summand in (2.63),
which after writing out Ω is given by
H = − 14pi
∫
γ1
∫
γ2
(
(~r1 − ~r2)× ~dγ2
)
· ~dγ1
|~r1 − ~r2|3 −
1
4pi
∫
γ1
∫
γ2
(
(~r2 − ~r1)× ~dγ1
)
· ~dγ2
|~r1 − ~r2|3
= 2 · 14pi
∫
γ1
∫
γ2
(
(~r1 − ~r2)× ~dγ1
)
· ~dγ2
|~r1 − ~r2|3 = 2N , (2.64)
where we recognized the well known Gaussian linking integral [53] which measures the
number of times the curves γ1 and γ2 link. Thus, the value of N for the world lines in
Figure 2.7 (a) is zero, whereas for Figure 2.7 (c), depicting a double exchange of the
solitons, we obtain N = 1.
Figure 2.7.: Soliton world lines: (a) trivial, (b) exchange, (c) double exchange.
The simple exchange process shown in Figure 2.7 (b) is special since, due to the exchange,
25
2. Field Theory and Topology
we only have a single world line. The Hopf invariant in that case reads
H = − 14pi
∫
γ
~dγ ~∇Ω(~r) (2.65)
where the integral counts the number of times the loop pierces its own surface so that
for the process shown in Figure 2.7 (b), the integral evaluates to 1.
In conclusion, the analysis leads to exactly the same result for the statistics of solitons
as the one in Section 2.3.2, where we examined extended solitons.
2.4. CP1-representation
We saw in Section 2.3.1 that the Hopf invariant H, defined through (2.40) is a non-local
quantity, which makes a Lagrangian containing H rather unusual. This was due to the
form of the gauge potential Aµ, see (2.38) and (2.57). However, the non-locality is not an
inherent property of the Hopf invariant, but rather results from the representation that
is being used. It was shown in a work by Wu and Zee [54], that in CP1-representation
the Hopf term actually acquires a local form. We will see in the following, that a repre-
sentation in terms of complex projective variables is in many respects the most natural
one for the non-linear σ-model.
The complex projective line CP1 can be described by homogeneous coordinates, a
pair of complex numbers where two points that differ by an overall complex scaling
factor are identified [46]. An equivalent definition is in terms of a complex unit vector
z = (z1, z2) , |z1|2 + |z2|2 = 1 and z1, z2 ∈ C , (2.66)
where we identify z and eiΛz for Λ ∈ R. The two complex numbers z1 and z2 together
with the unimodular constraint may be seen as a complex representation of the S3, which
was already discussed in Section 2.3.1. Together with the aforementioned identification,
CP1 is isomorphic to the coset space S3 modulo U(1)
CP1 ∼ S3/U(1) :=
{
[s] : s ∈ S3
}
, (2.67)
e.g. the set of equivalence classes [s], where the equivalence relation is given through
26
2.4. CP1-representation
the action of U(1) on the complex representation of S3. Moreover, we may use the Hopf
projection pi defined in (2.34), to map any representative of [s] ∈ S3/U(1) to the S2.
Due to the identification, pi : S3/U(1) → S2 is an isomorphism. Consequently, CP1 is
isomorphic to S2 and thus a faithful representation of the two-sphere.
2.4.1. The CP1-model
We will see in this section, that the O(3) non-linear σ-model when written in terms of
CP1-variables assumes the form of an U(1) gauge theory which in general is called the
CP1-model. The transition to the new representation is realized by means of the Hopf
projection, see also (2.34)
~n = z¯ ~σ z , z = (z1, z2) with z¯z = 1 , (2.68)
so that the real vector field n ∈ S2 is now expressed in terms of a two component spinor
z ∈ C2 where elements z and eiΛz are identified. By means of this transformation, the
action of the non-linear σ-model becomes
Sσ =
1
2g
∫
d3x ∂µ~n · ∂µ~n = 2
g
∫
d3x
(
∂µz¯ ∂
µz + (z¯ ∂µz)(z¯ ∂µz)
)
(2.69)
and is indeed invariant with respect to the local gauge transformations
z → eiΛ(xµ)z , (2.70)
as can be easily verified. We recall that there was a conserved topological current,
defined in (2.30), which by means of the identity
σiσjσm = δijσm + i
∑
k
ijkσkσm , (2.71)
in CP1-representation reads
jµ = − i4pi
µνλ∂ν z¯ ∂λz . (2.72)
In (2.38), we defined a gauge potential as the source of this current which turned out to
be non-local in ~n. However now, due to the simple form of the topological current, the
27
2. Field Theory and Topology
gauge potential may be directly read off (2.72) and we obtain
Aµ = −i (z¯ ∂µz) , (2.73)
so that in CP1, the gauge potential is a local quantity. Gauge invariance of the current
is of course still provided and we note that under the gauge transformation (2.70), the
potential transforms like an abelian gauge potential
Aµ → A′µ = −i
[(
z¯e−iΛ
)
∂µ
(
zeiΛ
)]
= Aµ + ∂µΛ , (2.74)
such that Aµ will serve as the connection in the gauge covariant representation of the
CP1-model
Sσ =
2
g
∫
d2x Dµz D
µz , Dµ = ∂µ − i Aµ , (2.75)
which by means of (2.73) is equivalent to (2.69).
One might pause here for a moment to appreciate how well things fit together in this
representation. Originally, we introduced the potential only as a source of the conserved
topological current and pointed to its possible gauge transformations. We were able to
define a homotopic invariant in terms of this potential but the construction was not very
elegant in the sense that it resulted in a non-local expression. Now, after changing to CP1
representation, actually introducing yet another redundancy to the description in form
of a local gauge invariance of the order parameter, the topological current’s potential
turns out to be the gauge connection for the theory and on top of that assumes a local
form in terms of the order parameter.
The Hopf invariant in CP1-representation
Collecting the results from (2.72) and (2.73), we may rewrite (2.40) as
Hopf invariant in CP1-representation
H = 18pi2
∫
d3x µνλ(z¯ ∂µz)(∂ν z¯ ∂λz) , (2.76)
so that we obtained a representation of the Hopf invariant which is local in the fields.
28
2.4. CP1-representation
Note that from the constraint z¯z = 1, we obtain via differentiation the identity
z¯ ∂αz + z ∂αz¯ = 0 , (2.77)
such that
Aµ = −i(z¯ ∂µz) = − i2(z¯ ∂µz − z ∂µz¯) (2.78)
is a real quantity. The current may also be rewritten as
jµ = − i8pi
µνλ
(
∂ν z¯ ∂λz − ∂νz ∂λz¯
)
, (2.79)
such that it is obviously real, which implies that the Hopf invariant is a real quantity as
well. In fact,
H = 18pi2
∫
d3x µνλ(z¯ ∂µz)(∂ν z¯ ∂λz)
= 132pi2
∫
d3x µνλ
(
z¯ ∂µz − z ∂µz¯
)(
∂ν z¯ ∂λz − ∂νz ∂λz¯
)
. (2.80)
Note also that partial derivatives are symmetric with respect to permutations, so that
(2.76) reduces to
H = 18pi2
∫
d3x µνλ
(
z∗1 ∂µz1 ∂νz
∗
2 ∂λz2 + z∗2 ∂µz2 ∂νz∗1 ∂λz1
)
. (2.81)
2.4.2. The Hopf invariant as a total divergence
Topological invariants can be expressed as integrals over total divergences. Actually, this
property is crucial for them to be a homotopic invariant and it follows directly from their
differential-form definitions, where they are expressed as integrals over a given manifold
of forms of maximal rank [45]. Consequently, their contribution to an action functional
may be expressed as a surface term that has no effect on the classical equations of
motion. However in quantum field theory, the contribution from the surface terms are
converted into the phases of the initial and final wave functions, as we already discussed
in Section 2.3.2.
The fact that the Hopf term is a total divergence is hard to see in the representation
(2.40). However with a little effort, it can be made explicit in the CP1 representation,
29
2. Field Theory and Topology
as was pointed out in [54]. Let us decompose z1 and z2 into real- and imaginary part,
z1 = (y1 + i y2) , z2 = (y3 + i y4) , (2.82)
so that for (2.81), we get
H = i4pi2
∫
d3x µνλ
[
(y1 ∂µy1 + y2 ∂µy2) ∂νy3 ∂λy4 + i (y1 ∂µy2 − y2 ∂µy1) ∂νy3 ∂λy4
+ (y3 ∂µy3 + y4 ∂µy4) ∂νy1 ∂λy2 + i (y3 ∂µy4 − y4 ∂µy3) ∂νy1 ∂λy2
]
, (2.83)
where we used again the antisymmetry of the epsilon tensor. At this point we consider
the constraint z¯ z = 1 written in terms of y1, . . . , y4
y21 + y22 + y23 + y24 = 1 , (2.84)
which upon differentiation leads to
y1∂µy1 + y2∂µy2 + y3∂µy3 + y4∂µy4 = 0 . (2.85)
When we plug this into H, we get
H = i4pi2
∫
d3x µνλ
[
i (y1 ∂µy2 − y2 ∂µy1) ∂νy3 ∂λy4
+ i (y3 ∂µy4 − y4 ∂µy3) ∂νy1 ∂λy2
]
. (2.86)
Now we eliminate all occurrences of ∂µy4 by means of the constraint (2.85),
∂µy4 = − 1
y4
(
y1∂µy1 + y2∂µy2 + y3∂µy3
)
, (2.87)
so that we obtain
H = 14pi2
∫
d3x
1
y4
µνλ
[
y21 ∂µy1 ∂νy2 ∂λy3 + y22 ∂µy1 ∂νy2 ∂λy3
+ y23 ∂µy1 ∂νy2 ∂λy3 + y24 ∂νy1 ∂λy2 ∂µy3
]
, (2.88)
30
2.4. CP1-representation
and using (2.84), this reads
H = 14pi2
∫
d3x
1
y4
µνλ ∂µy1 ∂νy2 ∂λy3 . (2.89)
Finally, using again the constraint, we are able to write this as a total derivative
H = 14pi2
∫
d3x µνλ ∂λ
[
f(y) ∂µy1 ∂νy2
]
, (2.90)
where
f(y) = arcsin
(
y3√
1− y21 − y22
)
. (2.91)
We share the opinion in [45], that this form is not particularly inspiring. Though it
explicitly shows that the Hopf density can be expressed as a total divergence.
Total derivatives, integration by parts and Fourier analysis
A total divergence is of course nothing but a sum of total derivatives and we just saw
that the Hopf density may be expressed in terms of a sum of derivatives. This in turn
might lead to an uneasy feeling regarding the tools commonly used in the derivation of
field theories, especially when we are searching for topological invariants.
Among these tools are the use of Fourier analysis or integration by parts with the care-
less neglection of boundary terms. So one might suspect that this neglection can lead to
the disappearance of topological invariants, which otherwise would have emerged. An
obvious solution to this problem is to keep all boundary terms in the derivation.
However, the issue is more delicate, since the concept of Fourier series relies on the
assumption that functions are periodic, or rapidly decreasing in the case of Fourier
transforms. The integrals over rapidly decreasing functions do not yield any boundary
terms at all. One might argue that for periodic functions, there might still be jumps
or kinks at the edge of their periodicity interval, such that an integral over this interval
may in fact yield boundary terms. However, any finite order Fourier series of such a
function is continuous, and at the discontinuity it converges to the mean value.
Thus, information about boundary values is lost by the use of Fourier analysis. A direct
consequence of this is that the integral over the Fourier transform of a total derivative
31
2. Field Theory and Topology
will always vanish
∫
dx ∂xf(x) ∼
∫
dx ∂x
∑
k
e−ikxf(k)
=
∫
dx
∑
k
(−i k) e−ikxf(k)
= −i∑
k
δk,0 f(k)
= 0 . (2.92)
However, we know from our discussion in Section 2.3.3 that the Hopf invariant is com-
pletely determined by fields inside the space-time manifold. This of course is a conse-
quence of the divergence theorem. Note that the total divergence is only apparent in
the particular representation we derived in (2.90). In fact, Fourier analysis of the Hopf
term in its conventional representation (2.76) is not harmful at all.
2.4.3. The massive CP1-model
We saw in (2.69), that the O(3)-symmetric non-linear σ-model can be equivalently for-
mulated in terms of CP1-fields. There is a generalization of this model which will turn
out to be relevant to our analysis of the doped antiferromagnet [10]. It is usually termed
the massive CP1-model and, different to the conventional CP1-model, its action is defined
in terms of two coupling constants
S =
∫
d3x
∑
µ
2
gµ
(
∂µz¯ ∂µz + γµ (z¯ ∂µz)2
)
, z¯z = 1 . (2.93)
The massive-CP1-model is not gauge invariant, which can be easily checked by means
of (2.70). Only for the specific value γµ = 1, the global U(1)-invariance becomes a local
gauge invariance and the action simplifies to the conventional CP1-model. Consequently,
for γµ 6= 1 the model cannot be written in terms of a real vector field ~n.
This model was previously analyzed in the context of frustrated antiferromagnets [14,
55,56] where it describes fluctuations around a incommensurate coplanar ordered state,
see also reference [12]. This can be understood in terms of a SO(3) matrix-valued
representation of the massive CP1-model. We construct SO(3)-matrices out of g ∈
32
2.4. CP1-representation
SU(2) by means of the identity [57]
Rab = 12 Tr
(
σag σbg†
)
. (2.94)
The rotation matrix Rab can then expressed in terms of three orthogonal unit vectors
R = (~n1, ~n2, ~n3), so that (2.93) becomes
S =
∫
d3x
∑
µ
1
2gµ
[
∂µ~n1 · ∂µ~n1 + ∂µ~n2 · ∂µ~n2 −
(
1 + γµ
)
(~n1 · ∂µ~n2)2
]
, (2.95)
showing that a coplanar configuration is generally favored. For γµ = 1, the usual O(3)
non-linear σ-model is obtained by means of the identity
∂µ~n3 · ∂µ~n3 = ∂µ~n1 · ∂µ~n1 + ∂µ~n2 · ∂µ~n2 − 2 (~n1 · ∂µ~n2)2 . (2.96)
Third homotopy group of SO(3)
As was pointed out in reference [10], the long-wavelength effective field theory obtained
in a second order gradient expansion for the t-J model at low doping is given by the
massive CP1-model. In the case of zero doping, the model reduces to the O(3) non-linear
σ-model, correctly describing the Heisenberg limit, but for finite doping, the order pa-
rameter generally takes values in SO(3).
On account of this, we need to discuss the different homotopy groups of SO(3). The
homotopies of classical groups where analyzed by Bott [58], and the simple case of
relevance to us is given through
pi1(SO(3)) ' Z2 , pi2(SO(3)) ' 0 , pi3(SO(3)) ' Z . (2.97)
Since pi2(SO(3)) ' 0, there are no skyrmions in the model, unlike in the O(3) non-linear
σ-model with its vector-valued order parameter. Consequently, the non-triviality of the
third homotopy group does not imply fractional spin and statistics for the excitations.
Nevertheless, if the topological invariant characterizing these different homotopy sectors
emerged in the effective action, it would have consequences on the quantum statistical
properties at low energies, see the discussion on the 1D Heisenberg model in Chapter 3.
The homotopic invariant characterizing the third homotopy group of SO(3) was dis-
33
2. Field Theory and Topology
cussed in [59] and is given by
H = 124pi2
∫
d3x µνλ Tr
[(
R−1∂µR
)(
R−1∂νR
)(
R−1∂λR
)]
. (2.98)
To see how this invariant manifests itself in terms of CP1-variables, we may use identity
(2.94) and an explicit parametrization for the SU(2)-matrices
g =
 z1 −z∗2
z2 z
∗
1
 , z∗1z1 + z∗2z2 = 1 . (2.99)
After some straightforward algebra we obtain for (2.98)
H = − 1
pi2
∫
d3x µνλ (z¯ ∂µz)(∂ν z¯ ∂λz) , (2.100)
which apart from the numerical factor is exactly the Hopf invariant in CP1-representation.
Note that we obtained the SO(3) valued order parameter by virtue of a mapping from
SU(2) to SO(3). It is well known that this mapping is not an isomorphism but rather
two-to-one, in the sense that R(g) = R(−g) and SO(3) ' SU(2)/Z2. Although SU(2)
and SO(3) have the same Lie-algebra and are thus locally equivalent, they differ in their
global structure. This can be readily seen from the fact that the fundamental group of
SU(2) is different from SO(3). Indeed, we have
pi1(SU(2)) ' 0 , pi2(SU(2)) ' 0 , pi3(SU(2)) ' Z , (2.101)
as opposed to (2.97). Thus, for a discussion of global properties like homotopy sectors,
it seems more appropriate to stay in the representation in terms of SU(2)-matrices.
On the other hand, the discussion appears somewhat academic, given the fact that the
topological invariant characterizing pi3(SU(2)) as obtained through (2.99) evaluates to
the Hopf invariant in CP1-representation as well. Further details on the SU(2) matrix
model can be found in Appendix A.1.
34
Chapter 3
Field Theoretical Methods for
Strongly Correlated Systems
Before we turn to the main part of this work, we want to give a quick overview on some
of the analytical methods used in the study of strongly correlated electron systems. The
methods we describe here form the basis of our upcoming analysis of the doped antifer-
romagnet.
First and foremost, strongly correlated electron systems are many-particle systems.
Their dynamics is described through a second quantized Hamiltonian and is governed
by the non-relativistic Schro¨dinger equation which we are not able to solve exactly. Our
goal in fact is to reduce the complexity of the microscopic system to what is absolutely
necessary, find suitable methods of approximation that still capture the essential physics
and derive from that predictions on macroscopic quantities, i.e. ground state properties,
phase diagrams, information about low-energy excitations or the behaviour of response
functions, which ultimately allows us to draw conclusions on actual experiments and
to understand the underlying physical mechanism. A quantity that is instrumental in
all this is the quantum partition function which incorporates the statistical information
about a given many-body system. It serves as a generating functional for correlation
functions and allows us to analyze its behaviour in terms of an effective field theory.
3.1. A path integral representation for the partition
function
In order to obtain a path-integral description for the quantum partition function, we
need to make the transition from an operator based Hamiltonian description to one in
terms of field variables. This can be achieved by means of an appropriate set of eigen-
35
3. Field Theoretical Methods for Strongly Correlated Systems
states.
Let us consider a quantum statistical system in equilibrium. The partition function in
the grand canonical ensemble is defined as
Z = Tr e−β(Hˆ−µNˆ) =
∫
dα 〈α|e−β(Hˆ−µNˆ)|α〉 , (3.1)
where |α〉 is an arbitrary complete set of states. Given the partition function, we obtain
the grand canonical potential through
Ω(µ, V, T ) = − 1
β
lnZ , (3.2)
from which in principle all thermodynamic quantities can be derived via differentiation.
Note that the expression for Z resembles the time evolution operator in quantum me-
chanics
U(xf , tf ;xi, ti) = 〈xf |e− i~ Hˆ(tf−ti)|xi〉 , (3.3)
provided that it is evaluated in imaginary time
− i
~
t = τ
~
, τ ∈ (0, ~β) (3.4)
and summed over a complete set of initial and final states. This enables us to construct
a functional integral representation for the partition function, much in the spirit of
Feynman’s path integral for the evolution operator [60]. In particular, we subdivide the
finite interval into M infinitesimal steps of size  = ~β
M
, and evaluate the Gibbs-operator
in (3.1) for each step
Z =
∫
dα 〈α|e− i~ (Hˆ−µNˆ)e− i~ (Hˆ−µNˆ) . . . e− i~ (Hˆ−µNˆ)|α〉 . (3.5)
For a many-particle Hamiltonian expressed in terms of creation and annihilation oper-
ators, one may evaluate the trace in terms of coherent states [61]. They are defined as
eigenstates of the annihilation operator, aα|φ〉 = φα|φ〉 and may be constructed from
the vacuum through
|φα〉 = exp(φαaˆ† − φ∗αaˆ) |0〉 = exp
(
− φ
∗
αφα
2
)
exp(φαaˆ†) |0〉 . (3.6)
36
3.1. A path integral representation for the partition function
Note that in the bosonic case, the eigenvalues φα are complex numbers, whereas in the
fermionic case the anticommutation relations require them to be Graßmann numbers [62].
Any arbitrary state can be expanded with respect to |φα〉 since they form an overcomplete
set
1 = 1
N
∫ ∏
α
dφ∗αdφα e−
∑
α
φ∗αφα |φ〉〈φ| . (3.7)
The operator trace in terms of bosonic (fermionic) coherent states can be derived as
TrA = 1
Nb/f
∫ ∏
α
dφ∗αdφα e−
∑
α
φ∗αφα〈±φ|A|φ〉 , (3.8)
so that when we express the partition function (3.1) for bosons (fermions) as an evolution
in imaginary time, (3.3), we have to impose periodic (antiperiodic) boundary conditions.
Consequently, by inserting the resolution of unity (3.7) for each time step, the partition
function is expressed in terms of a coherent state functional integral
Z =
∫
φα(~β)=±φα(0)
D(φ∗α, φα) e−
1
~SE(φ
∗
α,φα) , (3.9)
where the integration is understood as a limiting process
∫
D(φ∗α, φα) =
1
Nb/f
lim
M→∞
∫ M∏
k=1
∏
α
dφ∗α,kdφα,k , (3.10)
and SE is given in its discrete formulation by
SE(φ∗α, φα) = 
M∑
k=2
[∑
α
(
~φ∗α,k
φα,k − φα,k−1

− µφ∗α,kφα,k−1
)
+H(φ∗α,k, φα,k−1)
]
+ 
[∑
α
(
~φ∗α,1
φα,1 − φα,0

− µφ∗α,1φα,0
)
+H(φ∗α,1, φα,0)
]
. (3.11)
We assume the evolution of states to be continuous so that in the limit of  → 0 we
may substitute the quotients by derivatives, the Riemann sum by an integral and use
the trajectory notation. In this limit, the boundary term vanishes and (3.11) becomes
the Euclidean action
SE(φ∗α, φα) =
~β∫
0
dτ
[∑
α
φ∗α(τ)
(
~
∂
∂τ
− µ
)
φα(τ) +H
(
φ∗α(τ), φα(τ)
)]
. (3.12)
37
3. Field Theoretical Methods for Strongly Correlated Systems
We conclude that a coherent state functional integral representation for the partition
function of a bosonic (fermionic) many-particle system is given by
Partition function for bosonic (fermionic) many-particle systems
Z =
∫
φα(~β)=±φα(0)
D(φ∗α, φα) exp
− 1
~
~β∫
0
dτ
[∑
α
φ∗α(τ)
(
~
∂
∂τ
− µ
)
φα(τ) +H
(
φ∗α(τ), φα(τ)
)] .
(3.13)
Let us compare this expression with the coherent state integral for the evolution operator
[61]. The probability of finding a state φf at time tf for a system initially prepared in
state φi at time ti is given by
U(φ∗α,f , tf ;φα,i, ti) =
φα(tf )∫
φα(ti)
D(φ∗α, φα) exp
(
i
~
S(φ∗α, φα)
)
, (3.14)
where S in the continuum notation is given through
S = −i~∑
α
φ∗α(tf )φα(tf ) +
tf∫
ti
dt
[
i~
∑
α
φ∗α(t)
∂
∂t
φα(t)−H
(
φ∗α(t), φα(t)
)]
. (3.15)
We note that the transition from (3.15) to (3.12) can be realized through a Wick rotation,
e.g. by substituting t→ τ and → i or equivalently by ∂
∂t
→ i ∂
∂τ
and dt→ −i dτ .
3.1.1. Dirac quantization of constrained systems
The coherent-state method for obtaining a path integral formulation may be applied to
any Hamiltonian where the relevant degrees of freedom are given in terms of canonical
creation and annihilation operators. However, complications arise if the operators in the
Hamiltonian obey more complicated algebras, a situation typically encountered in the
description of magnetic systems by an effective Hamiltonian.
For these situations, the method of coherent-states can be generalized to an arbitrary
algebra [63] and a path integral representation for simple Hamiltonians as the Heisenberg
Hamiltonian may be easily performed. Notice however, that for the more complicated
38
3.1. A path integral representation for the partition function
t-J model, the relevant degrees of freedom respect the graded Lie algebra spl(2, 1) and
the method of coherent states becomes notoriously difficult [64].
Thus, one might approach the problem from another side. After all, the only thing
needed to derive a path integral representation is a classical action functional. Unfortu-
nately, neither for the Heisenberg nor the t-J model, the classical action and in particular
the correct form of constraints is a priori known. Of course, we may always take an ed-
ucated guess.
However then, it is mandatory to check whether this classical action functional together
with the constraints reproduces the correct quantum theory from which we started. This
may be achieved conveniently by use of Dirac’s quantization method for constrained sys-
tems [65,66], a generalization of the canonical quantization process to systems which are
subjected to constraints.
The program is performed in a set of simple steps [67]:
• The classical problem together with a set of primary constraints {φ(i)} is formulated
in terms of Lagrangian multipliers and the Hamilton function is derived by means of
a Legendre transformation.
• A consistency condition is applied, such that the constraints are respected at all
times
φ˙(i) = {φ(i), H|M}+ λj{φ(i), φ(j)}
∣∣∣
M
, (3.16)
where the curly brackets are the usual Poisson brackets and M denotes the con-
strained subspace. The condition may yield additional requirements, called secondary
constraints.
• The constraints have to be separated into two classes. φ(i) is called a first-class
constraint if
{φ(i), φ(j)}
∣∣∣
M
= 0 ∀ j (3.17)
and second-class otherwise. We denote first-class constraints by ϕ(k) and second-
class constraints by φ˜(l). The {ϕ(k)} amount to an actual degeneracy of the system,
a gauge degree of freedom. The Lagrangian multipliers have to be fixed manually,
such that from the {ϕ(k)}, we obtain another set of second-class constraints {θ(k)},
the gauge fixing conditions.
39
3. Field Theoretical Methods for Strongly Correlated Systems
• One replaces the conventional Poisson brackets by the so-called Dirac brackets, de-
fined through
{
a, b
}
D
=
{
a, b
}
−
{
a, φ˜(i)
}{
φ˜(i), φ˜(j)
}−1{
φ˜(j), b
}
. (3.18)
The Dirac brackets take the role of Poisson brackets in the reduced phase space. Con-
sequently, the quantization is performed by replacing
{
a, b
}
D
→ −i [aˆ, bˆ] . (3.19)
Finally, after we analyzed all constraints and made sure the classical action leads to
the correct quantum theory, we may construct a path integral formulation based on
the classical action functional. Starting from a representation in terms of variables in
the reduced phase space, one obtains the following representation in the unconstrained
coordinates
Path integral for a constrained Hamiltonian system
Z =
∫
DqDp
m∏
k=1
δ(θ(k))δ(ϕ(k))
m′∏
l=1
δ(φ˜(l))
· |det{θ(a), ϕ(b)}| | det{φ˜(a), φ˜(b)}|1/2 exp
(
i
∫
dt
[
pq˙ −H(q, p)
])
. (3.20)
This generalization of Feynman’s path integral construction was obtained by Faddeev
[68] and Senjanovic [69]. It can readily be extended to the case of infinitely many degrees
of freedom.
Dirac quantization of the Heisenberg model
Let us illustrate the idea for a relatively simple model with constraints. The isotropic
spin-12 Heisenberg model with antiferromagnetic next-neighbour coupling is defined through
the Hamiltonian
H = J
∑
<i,j>
~Si · ~Sj , J > 0 , (3.21)
40
3.1. A path integral representation for the partition function
where the spin operators fulfill the su(2) algebra of quantum mechanical angular mo-
menta
[Sai , Sbj ] = i δij abc Sci . (3.22)
In anticipation of our analysis in Chapter 4, we use a representation in terms of Hubbard
X-operators [70–72] which represent transitions between states of a reduced Hilbert
space
Xˆαβi = |αi〉〈βi| , (3.23)
with |αi〉 ∈
{
|+〉, |−〉
}
on each site i. The X-operators obey the following Lie-algebra
[
Xˆαβi , Xˆ
γδ
j
]
= δij
(
δβγXˆαδi − δαδXˆγβi
)
(3.24)
and a completeness relation
Xˆ++ + Xˆ−− = 1 . (3.25)
They are related to the conventional spin-operators by
Sx = 12
(
Xˆ+− + Xˆ−+
)
,
Sy = − i2
(
Xˆ+− − Xˆ−+
)
,
Sz = 12
(
Xˆ++ − Xˆ−−
)
, (3.26)
such that we obtain for (3.21)
H = J
∑
<i,j>
~Si · ~Sj = J2
∑
<i,j>
(
Xˆαβi Xˆ
βα
j −
1
2
)
. (3.27)
In order to obtain a path integral representation for this model, we use a classical
Lagrangian that was proposed in [73]
L(X, X˙) = i X
−+X˙+− −X+−X˙−+
1 +X++ −X−− −H(X) , (3.28)
plus the following constraints
φ(1) = X+−X−+ + 14
(
X++ −X−−
)2 − 14 = 0
φ(2) = X++ +X−− − 1 = 0 , (3.29)
41
3. Field Theoretical Methods for Strongly Correlated Systems
Obviously, φ(1) fixes the magnitude of spins to 12 and φ
(2) represents the completeness
condition. From the generalized momenta
Παβ = ∂L
∂X˙αβ
, (3.30)
we obtain
φ(3) = Π++ = 0
φ(4) = Π+− − iX
−+
1 + (X++ −X−−) = 0
φ(5) = Π−+ + iX
+−
1 + (X++ −X−−) = 0
φ(6) = Π−− = 0 . (3.31)
Note that these are all primary constraints and there are no secondary constraints im-
plied by the consistency condition. Now we consider the Poisson brackets
{
φ(i), φ(j)
}
=
∑
k
α,β
(
∂φ(i)
∂Xαβk
∂φ(j)
∂Xαβk
− ∂φ
(j)
∂Xαβk
∂φ(i)
∂Xαβk
)
(3.32)
and their determinant, which evaluates to
det
{
φ(i), φ(j)
}
= −4
[
2X+−X−+ + (X++ −X−−)(1 +X++ −X−−)
]2
(
1 +X++ −X−−
)4
∣∣∣∣∣
φ(1)=0
= −1 . (3.33)
Thus, the constraints are all second-class and we may directly derive the inverse of the
respective Poisson brackets
{
φ(i), φ(j)
}−1
to form the Dirac brackets
{
A,B
}
D
=
{
A,B
}
−
{
A, φ(i)
}{
φ(i), φ(j)
}−1{
φ(j), B
}
. (3.34)
Finally, we may calculate the Dirac brackets of our field variables Xab. We obtain
{
X++, X+−
}
D
= −iX+− ,
{
X++, X−+
}
D
= iX−+{
X++, X−−
}
D
= 0 ,
{
X+−, X−+
}
D
= −i
(
X++ −X−−
)
42
3.1. A path integral representation for the partition function
{
X+−, X−−
}
D
= −iX+− ,
{
X−+, X−−
}
D
= iX−+ , (3.35)
and taking into account the correspondence relation
[
Xˆαβi , Xˆ
γδ
j
]
= i
{
Xαβi , X
αβ
j
}
D
, (3.36)
we recovered precisely the commutation relation for the X-operators (3.24). Now that
the correct algebra for Xαβ was confirmed we may construct the path integral following
reference [73]. The functional integral representation of the quantum partition function
for a system at inverse temperature β is given by
Z =
∫
DX δ
(
φ(1)
)
δ
(
φ(2)
) ∣∣∣det{φ(i), φ(j)}∣∣∣ exp(− ∫ β
0
dτL(X, X˙)
)
, (3.37)
where the imaginary time Lagrangian is given by
L(X, X˙) =
∑
i
X−+i X˙
+−
i −X+−i X˙−+i
1 +X++i −X−−i
+H(X) . (3.38)
We may now integrate over X−−, taking care of φ(2), and perform a change of variables
nx = X+− +X−+ ,
ny = −X+− +X−+ ,
nz = X++ −X−− . (3.39)
The Jacobian for this transformation, as well as det
{
φ(i), φ(j)
}
are constant and thus
can be absorbed in the normalization of the functional integral. We obtain for
L(n, n˙) = i2
∑
i
(nx)i(n˙y)i − (ny)i(n˙x)i
1 + (nz)i
+ J4
∑
<i,j>
~ni · ~nj , (3.40)
and by introducing a vector potential
~A(~n) =
(
− ny1 + nz ,
nx
1 + nz
, 0
)
, (3.41)
such that ~∇n × ~A(~n) = ~n, we get the same expression for the partition function that is
obtained via the method of coherent states [51]
43
3. Field Theoretical Methods for Strongly Correlated Systems
Partition function for the quantum Heisenberg model
Z =
∫
D~n exp
(
−
∫
dτ
[
i
2
∑
i
~A(~ni) · ∂τ~ni + J4
∑
i,j
~ni · ~nj
])
. (3.42)
3.2. Effective field theory
A basic approach in studying the behaviour of magnetic systems is to consider small
deviations from a fixed global magnetic order. As long as fluctuations are small, such a
mean-field approximation remains valid. However, since it explicitely breaks the model’s
rotational symmetry, the approach is highly biased.
In situations where fluctuations are large, the system might develop more exotic forms
of order or even become completely disordered. The size of fluctuations is typically
enhanced by low dimensionalilty or frustration due to competing interactions. In order
to study these strongly fluctuating systems, we will use an effective field theory. The
only assumptions in this approach is the existence of a large spin correlation length ξ,
considerably larger than the lattice constant a, such that
lim
ξ→∞
ξ
a
→∞ ⇐⇒ lim
a→0
ξ
a
→∞ . (3.43)
In these cases, there is an intermediate scale a << Γ−1 << ξ where, as long as the
system is close to an antiferromagnetically ordered state, we may introduce a local order
parameter with short range antiferromagnetic order.
Strongly correlated antiferromagnets feature large correlation lengths due to the huge
exchange coupling J , which typically is of the size of 103 K [74]. As long as the temper-
ature T << J the correlation length is large, such that the continuum approximation
a→ 0 is clearly justified.
The first step in our analysis is to distinguish contributions that get amplified by the
rescaling from those which get suppressed. This can be done very transparently in the
continuum formulation by means of simple power counting. The second step consists of
44
3.2. Effective field theory
separating the fields into slow and fast degrees of freedom. This will be accomplished by
use of Haldane’s mapping. Thereupon, we integrate out the fast degrees of freedom that
represent fluctuations, such that we obtain a long-wavelength effective action. Physical
quantities can then be obtained in terms of a renormalization group treatment using
perturbative expansions, for instance in 1/N [75].
In order to make the analysis more transparent, we once more demonstrate the concept
for the simple Heisenberg model, before in Chapter 4 we come to the more involved
analysis of the t-J-model. For the Heisenberg model in one- and two spatial dimensions,
the effective action turns out to assume the form of a O(3) non-linear σ model with an
additional topological term in the 1D-case. The exposition follows in parts the reviews
in [76,77].
3.2.1. Long-wavelength effective action for the Heisenberg model
Let us consider the antiferromagnetic isotropic Heisenberg model for arbitrary (integer
or half-integer) spin s with next neighbour interactions
H = J
∑
<i,j>
~Si · ~Sj , J > 0 , (3.44)
and its partition function in the canonical ensemble [51]
Z =
∫
D~n exp
[
−
∫
dτ
(
i s
∑
i
~A(~ni) ∂τ~ni + Js2
∑
<i,j>
~ni · ~nj
)]
. (3.45)
Note that for the case of s = 12 , this coincides with (3.42).
Haldane’s mapping and gradient expansion
The separation into slow and fast degrees of freedom is achieved via Haldane’s mapping
[78]. For an order parameter exhibiting local antiferromagnetic order, we have
~ni → (−1)i
√
1− a2~l2i ~ni + a~li , (3.46)
where the alternating factor (−1)i realizes the staggering of the configurations. Con-
sequently, we may assume ~ni and ~li to be smooth and slowly varying. However, the
45
3. Field Theoretical Methods for Strongly Correlated Systems
decomposition is constructed such that in the limit of small lattice constant a, the
li-field will represent small fluctuations perpendicular to the direction of the order pa-
rameter, given by the unit vector ~ni.
The constraint |~ni|2 = 1 translates into
|~ni|2 = 1 , ~ni ·~li = 0 , (3.47)
and the measure transforms as
D~ni → D~niD~li δ(~n2i − 1) δ(~ni ·~li)J , (3.48)
where J is the Jacobian of the transformation (3.46).
In the resulting action functional, we expand the square roots in powers of a and replace
the fields on next neighbour sites j by their gradient expansion in terms of fields at site
i. Depending on the number of dimensions d = D+1, where D is the spatial dimension,
terms up to a certain order in a can be neglected, since their contribution gets suppressed
in the limit of a→ 0. Schematically, we have
∫
dτ
∑
<i,j>
~ni · ~nj =
∫
dτ
∑
i
~ni ·
∞∑
l=0
al ∂lµ~ni
−→
∞∑
l=0
al−D
∫
dτ
∫
dDx ~n · ∂lµ~n −→
∞∑
l=0
al−d
∫
ddx ~n · ∂lµ~n , (3.49)
where in the last step we performed an appropriate rescaling of the time coordinate. We
see that the expansion yields for
l > d ⇒ irrelevant terms,
l = d ⇒ marginal terms,
l < d ⇒ relevant terms. (3.50)
In the following, we give the results for the 1D-case, as were first obtained in [78] and
the 2D-case which was discussed in [31].
46
3.2. Effective field theory
1D spin chain:
For the 1D chain, the expansion is performed up to 2nd-order in a. The interaction
term in (3.45) translates into
Js2
∑
i
~ni · ~ni+1 → Js2
∑
i
1
2
(
a2(∂x~ni)2 + 4a2(~li)2
)
, (3.51)
and for the Berry phase we obtain
is
∑
i
~A(~ni) · ∂τ~ni → is
∑
i
(
(−1)i ~A(~ni) · ∂τ~ni − a~li · (~ni × ∂τ~ni)
)
. (3.52)
Note that the alternating sum has to be evaluated as
N∑
i
(−1)i ~A(~ni) · ∂τ~ni =
N/2∑
i
~A(~n2i) · ∂τ~n2i −
N/2∑
i
~A(~n2i−1) · ∂τ~n2i−1
= 12
N/2∑
i
(2a)
~A(~n2i) · ∂τ~n2i − ~A(~n2i−1) · ∂τ~n2i−1
a
= 12
∫
dx ∂x
(
~A(~n) · ∂τ~n
)
(3.53)
and by means of (3.41) may be written as
= −14
∫
dx dτ µν ~n · (∂µ~n× ∂ν~n) = −2pi Q , (3.54)
where Q is exactly the Pontryagin index, see (2.12). So when we sum up the results
from (3.51), (3.53) and (3.54), we obtain an effective action
S =
∫
dτ dx
(
Js2a
2
[
4~l 2 + (∂x~n)2
]
− i s~l · (~n× ∂τ~n)
)
− i s 2pi Q . (3.55)
Now we integrate out the fluctuations. The Gaussian functional integral over ~l can be
performed [79], such that we arrive at the long-wavelength effective action
S =
∫
dτ dx
(
1
8Ja (∂τ~n)
2 + Js
2a
2 (∂x~n)
2
)
− i s 2pi Q , (3.56)
and through a rescaling of the time coordinate τ → 2Jsa x0 = c x0, such that dτ = c dx0
and ∂τ = 1/c ∂x0 , we obtain the simple form of the O(3) non-linear σ-model
47
3. Field Theoretical Methods for Strongly Correlated Systems
Long-wavelength effective field theory in D = 1
S = 12g
∫
d2x ∂µ~n · ∂µ~n− i s 2pi Q , g = 2
s
. (3.57)
Ignoring for a moment the effect of the topological term Q, (3.57) has the form of a
free theory. However, we have to take into account the constraint which generates in-
teractions between the fields. The renormalization group analysis shows that even the
ground state of the one-dimensional non-linear σ-model is disordered due to quantum
fluctuations. For low temperatures, the spin correlation length for the integer spin-chain
is finite and temperature independent. The spectrum is gapped.
However, in the spin-1/2 case the result is different: T = 0 is a critical point of the model,
the excitations are gapless and the correlation function decays as a power law [80]. At
T 6= 0 a finite correlation length appears ξ ∼ 1/T . In order to understand this, let us
consider the effect of the topological term. We know from our discussion in Section 2.2.2
that Q takes only integer values, so the contribution is purely imaginary. Since Q occurs
in the exponent of the partition function, it has to be considered modulo 2pi. Thus,
it gives a contribution only for half-integer s and in this case, the partition function
decomposes into a sum over different topological sectors, each one weighted with an
alternating sign
Z =
∞∑
k=−∞
(−1)k Zk . (3.58)
This form makes it very plausible, that the topological term has a tremendous effect on
the thermodynamic properties of the half-integer spin chain. In fact, Haldane conjec-
tured that all half-integer chains are critical at zero temperature on account of Q [81].
The conjecture is supported by a rigorous proof that the spin-12 Heisenberg model is
critical [80], which to some extent could be generalized to arbitrary half-integer spins as
well [82]. Additionally, there are various numerical studies [83, 84] on the half-integer
Heisenberg chains, confirming the result.
2D square lattice:
The derivation of the effective action in case of the 2D square lattice is similar to the
procedure in the 1D-case, yet we have to expand the fields up to third order in a. This
48
3.2. Effective field theory
opens the possibility for another topological term to emerge. As we have seen in (2.40),
the Hopf invariant features three derivatives and may thus appear in a third order gra-
dient expansion.
For the interaction term, all third order contributions vanish, owing to the alternating
sign in (3.46). We eventually obtain
Js2
∑
<i,j>
~ni ~nj =
Js2
2
∑
<i,j>
[
(∂x~ni)2 + (∂y~ni)2 + 8 (~li)2
]
, (3.59)
and for the Berry phase, we have
A(~ni) · ∂τ~ni = (−1)p+qs ~A(~ni) · ∂τ~ni + a~li · (~ni × ∂τ~ni) . (3.60)
When we derive the continuum limit for the first summand we get
∫
dτ
∑
i
(−1)p+qs ~A(~ni) · ∂τ~ni = s
∑
p,q
(−1)p+q
∫
dτ ~A(~n) · ∂τ~n
= pis
∫
dy ∂yQ(y) = pis
∫
dx ∂xQ(x) , (3.61)
where Q(x) and Q(y) are winding numbers of the ~n field in the x- and y-plane, respec-
tively. Thus, we rediscovered the topological term that arose in the 1D-case. This time,
since for the 2D-case we have an alternating sum of spin chains, we obtained the deriva-
tive of Q(x) or Q(y) which is zero. So the term does not contribute to the effective action.
Consequently, when going to the continuum we obtain
S =
∫
dτ d2x
(
4Js2~l 2 − is
a
(~n× ∂τ~n)~l + Js
2
2
[
(∂x~n)2 + (∂y~n)2
])
, (3.62)
and after integrating out the fluctuations we obtain the long-wavelength effective action
S =
∫
dτ d2x
(
1
16Ja2 (∂τ~n)
2 + Js
2
2
[
(∂x~n)2 + (∂y~n)2
])
, (3.63)
which after rescaling the time derivative corresponds again to the O(3) non-linear σ-
model, this time in three dimensions
49
3. Field Theoretical Methods for Strongly Correlated Systems
Long-wavelength effective field theory in D = 2
S = 12g
∫
d3x ∂µ~n · ∂µ~n , g = 2
√
2 a
s
. (3.64)
Thus, we find that in the long-wavelength limit of the two-dimensional antiferromagnetic
Heisenberg model on the square lattice no topological term, in particular no Hopf term,
emerges [31].
Still, the (2 + 1)-dimensional non-linear σ-model has an interesting phase diagram even
without topological term [7,85], see Figure 3.1 . Depending on the value of the coupling
Figure 3.1.: Crossover diagram of the 2D quantum non-linear σ-model, adopted from
[85]. The lines marked (1) and (2) are two possible experimental paths for
which the staggered spin-spin correlation length will behave very differently.
constant, a continuous symmetry may be broken for T = 0 but is restored at finite
temperature through thermal fluctuations, in agreement with Mermin-Wagner’s theorem
50
3.2. Effective field theory
[86]. One can distinguish three qualitatively different regimes for the coupling flow.
• Renormalized classical: This region is characterized by a diverging correlation length
as T → 0 and a true long-range ordered state at T = 0. Quantum fluctuations
merely lead to a renormalization of the classical parameters, like spin stiffness and
susceptibility.
• Quantum critical: This is the region where thermal and quantum fluctuations are
equally important, |g − gc| ∼ T . The correlation length is of order 1/T and decays
as an inverse power of the distance.
• Quantum disordered: This region is disordered even at T = 0 due to the effect of
quantum fluctuations. The low-energy excitations are spin-1 bosons with a spectral
gap.
On the experimental side, there is excellent agreement of the spin correlation lengths
measured for instance in La2CuO4 [8] or Sr2CuO2Cl2 [87] via neutron scattering, when
compared to theoretical values calculated with an appropriate coupling constant in the
renormalized classical regime [7] and Monte-Carlo simulations of the spin-12 Heisenberg
antiferromagnet [88,89].
The occurrence of a Ne´el ordered phase in real materials which survives up to finite val-
ues of temperature is attributed to the effect of interlayer couplings in the real material.
Admittedly, those couplings are only of the order 10−5 times the intra-layer couplings
but they can still help the broken symmetry state to survive into a finite temperature
region, provided that the in-layer correlation length is sufficiently large.
51

Chapter 4
Effective Field Theory for
the t-J Model at Low Doping
From a theoretical point of view, it has become increasingly obvious that unconventional
superconductivity is a very tough problem.1
The aim of this work is the derivation of an effective field theory for the low lying mag-
netic degrees of freedom of the t-J model at low values of doping, with a focus on the
possibility of fractional statistics of its excitations. The t-J model might be seen as a
minimal model for the behaviour of strongly correlated, doped antiferromagnets, as they
occur in the high-temperature superconducting copper-oxide materials, in particular the
comparatively simple La2−δSrδCuO4-compounds.
This chapter will be structured as follows: First we give a short review on the arguments
why the t-J model is supposedly a good model for the description of La2−δSrδCuO4. Af-
ter that, we set up the path-integral formulation for the t-J model in order to derive
an effective field theory [10]. This is by no means trivial, mainly for two reasons. First
of all, the Hamiltonian of the t-J model is not bilinear in the electronic creation and
annihilation operators and secondly, since the model arises from the strong coupling
limit of the Hubbard model, doubly occupied sites are strictly forbidden, thus we are
dealing with a constrained system in a reduced Hilbert space with either no occupa-
tion or singly occupied sites. The problem of quantization was addressed by several
authors [10, 73, 90], using different kinds of quantization schemes. We follow up on a
work of our own group [10].
Finally in Section 4.3, we present the actual analysis performed on this path-integral
formulation, which mainly consists of
1quoting Michael R. Norman in ‘The challenge of unconventional superconductivity’, Science, 332
(6026): 196-200 (2011)
53
4. Effective Field Theory for the t-J Model at Low Doping
a) the introduction of a staggered quantization axis and the separation of fast and slow
varying spin degrees of freedom in a third order gradient expansion,
b) the integration over fermionic degrees of freedom,
c) the integration over fluctuating spin degrees of freedom.
4.1. Modeling the electronic structure of the cuprates
All copper oxide superconductors are fairly complicated chemical compounds and the
question arises how a reasonable Hamiltonian would look like that captures the essential
physics but is still simple enough to derive prediction from it. Since we want our effective
field theory to be explicitly derived from a microscopic model, we add a brief discussion
on their electronic structure.
Figure 4.1.: La2CuO4 and its relevant orbitals, taken from [76].
Cuprate superconductors are composed of stacked copper-oxide planes that show very
small interlayer coupling so that tunneling processes from one layer to another are highly
suppressed. Thus, at sufficiently low energies, the electronic degrees of freedom are con-
fined to only two dimensions. The electronic properties of CuO-materials are mainly
attributed to the half-filled 3dx2−y2 orbital of copper and the doubly occupied oxygen
orbitals, 2px and 2py, see Figure 4.1. The orbitals are strongly hybridized and from the
54
4.1. Modeling the electronic structure of the cuprates
view of band structure calculations would allow hole conduction. This is prevented by
the strong Coulomb repulsion, the ’Mottness’ of the system, which leads to the local-
ization of holes. The undoped parent compounds are well described by the Heisenberg
model, describing the interaction of localized magnetic moments on the copper sites.
We may dope the system by adding impurities to the parent compound. For instance in
La2−δSrδCuO4, Lanthanum existing in the oxidation state La3+ is replaced by Strontium
which appears in oxidation state Sr2+. Consequently, there is one electron less con-
tributed to the conduction band. This additional hole will reside in a hybridized state of
copper and oxygen orbitals, forming a spin singlet which, due to the antiferromagnetic
superexchange [91] between O and Cu sites, is lower in energy than the symmetric spin
states. Two holes occupying the same copper site are energetically highly unfavourable
as well, such that the singlet will move through the crystal just like a hole would in a
single-band Hubbard model with large on-site repulsion. This picture was obtained by
Zhang and Rice [92] and is well supported by experiment [93].
Conclusively, an appropriate effective model for the electronic degrees of freedom of the
CuO-based high-Tc superconductors is given by the two-dimensional t-J model on the
square lattice, which arises from the dynamics of the Zhang-Rice singlet in the limit of
strong coupling. The model is defined through the Hamiltonian
The t-J model Hamiltonian
Ht-J = −
∑
<i,j>
σ
tij c˜
†
iσ c˜jσ +
J
2
∑
<i,j>
(
~Si · ~Sj − 14 n˜in˜j
)
− µ∑
i
n˜i , (4.1)
where
c˜†iσ = (1− ni−σ)c†iσ , n˜i =
∑
σ
c˜†iσ c˜iσ , ~Si =
∑
σσ′
c†iσ~σσ,σ′ciσ′ . (4.2)
The operators c†iσ, ciσ are canonical creation and annihilation operators for electrons with
spin σ = ± on site i. c˜†iσ and c˜iσ project out the doubly occupied states. t denotes the
hopping amplitude, J the antiferromagnetic exchange coupling and we added a chemical
potential term ∼ µ.
55
4. Effective Field Theory for the t-J Model at Low Doping
4.2. Path integral formulation of the t-J model
As was already pointed out, obtaining a path integral representation and deriving the
effective field theory for this model is highly non-trivial. Fermionic and spin degrees
of freedom cannot be treated separately, the constraint for single occupancy must be
respected at all times and except for the kinetic and the chemical potential term, all
contributions are quartic combinations of field operators.
A common procedure is the use of Hubbard X-operators [70–72] in order to obtain
a bilinear form. These operators represent transitions between states of the reduced
Hilbert space, so they automatically respect the single occupancy constraint
Xˆαβi = |αi〉〈βi| , (4.3)
with |αi〉 ∈
{
|0〉, |+〉, |−〉
}
on each site i. The X-operators for the t-J model fulfill the
graded Lie-algebra spl(2, 1), given through
[
Xˆαβi , Xˆ
γδ
j
]
± = δij
(
δβγXˆαδi ± δαδXˆγβi
)
, (4.4)
where the −(+) sign refers to the commutator (anticommutator). (4.4) is to be under-
stood as an anticommutator (+), only when all operators are fermionic, e.g. for Xˆα0,
Xˆ0α, otherwise as a commutator. Written in terms of Hubbard X-operators the t-J
model Hamiltonian becomes a bilinear
Ht-J = −
∑
<i,j>
σ
tij Xˆ
σ0
i Xˆ
0σ
j +
J
4
∑
<i,j>
σ,σ¯
(
Xˆσσ¯i Xˆ
σ¯σ
j − Xˆσσi Xˆ σ¯σ¯j
)
− µ∑
i,σ
Xˆσ0i Xˆ
0σ
i . (4.5)
4.2.1. The action functional
We will consider in the following a path integral representation of the t-J model derived
from a classical action found by Wiegmann via the method of coherent states [90]. The
same action functional was obtained in reference [73] in the supersymmetric extension of
the Faddeev-Jackiw symplectic quantization formalism [94]. More recently, the path in-
tegral representation was derived [10] in a manifestly rotational invariant form by means
of Dirac quantization [66], see also the discussion on Dirac quantization in Chapter 3.
We follow up on the real-time action functional obtained in [10], expressed in terms
56
4.2. Path integral formulation of the t-J model
of X-fields. Bosonic fields are represented by complex variables, fermionic fields by
anticommuting Graßmann numbers [62]
S = −i
∫
dt
∑
i
(1 + ρi)ui − 1
(2− vi)2 − 4ρi − u2i
(
X−+i X˙
+−
i −X+−i X˙−+i
)
+ i2
∫
dt
∑
i,σ
(
Xσ0i X˙
0σ
i −X0σi X˙σ0i
)
−H(X) . (4.6)
The first two summands represent the Berry phase term and H was defined in (4.5). We
also introduced the definitions
ρi = X0+i X+0i +X0−i X−0i ,
ui = X++i −X−−i ,
vi = X++i +X−−i , (4.7)
and the constraints are defined through
φ
(a)
i = X++i +X−−i + ρ− 1 ,
φ
(b)
i = X+−i X−+i +
1
4u
2
i −
(
1− 12vi
)2
+ ρi ,
φ
(c)
i = X00i −
(
X0+i X
+0
i +X0−i X−0i
)
,
φ
(d)
i = X0+i X+−i −X0−i X++i ,
φ
(e)
i = X+0i X−+i −X−0i X++i . (4.8)
A detailed analysis of the constraints can be found in reference [10]. All of them turn
out to be second-class, such that the generating functional can be constructed as
Z =
∫
DX ∏
i
δ
[
φ
(a)
i
]
δ
[
φ
(b)
i
]
δ
[
φ
(c)
i
]
δ
[
φ
(d)
i
]
δ
[
φ
(e)
i
]
(1 + 2ρi)
1
2 exp
(
− S
)
, (4.9)
where the factor (1 + 2ρi)
1
2 results from the superdeterminant of constraints. It can be
absorbed by a shift of the chemical potential.
Since X00 does not contribute to the action, we can eliminate φ(c) by performing the
57
4. Effective Field Theory for the t-J Model at Low Doping
integral over X(00). Furthermore, we apply the change of variables
X++ = (1− ρ)(1 + Ωz)/2 , X−− = (1− ρ)(1− Ωz)/2 ,
X+− = (1− ρ)(Ωx − iΩy)/2 , X−+ = (1− ρ)(Ω+ + iΩy)/2 ,
X+0 = ψ+ , X−0 = ψ− ,
X0+ = ψ∗+ , X0− = ψ∗− , (4.10)
where we introduced a real vector field ~Ω and new Grassmann fields ψ± and ψ∗±, so that
ρ = ψ∗+ψ+ + ψ∗−ψ−. In this new set of variables the constraint φ(a) is automatically
satisfied, and the integral over X−− can be performed. The action now takes the simple
form
S =
∫
dt
(
1
2
∑
i
ΩxΩ˙y − ΩyΩ˙x
1 + Ωz
+ i
∑
i,σ
ψ∗iσψ˙iσ −H(~Ω, ψ, ψ∗)
)
. (4.11)
Finally, we present the imaginary time path integral representation of the t-J model [10],
based on the action functional (4.6) and the constraints in (4.8)
Z =
∫
D~ΩDψDψ∗ ∏
i
δ
[
φ
(1)
i
]
δ
[
φ
(2)
i
]
δ
[
φ
(3)
i
]
exp
(
− S
)
, (4.12)
where the action in imaginary time is obtained through a Wick rotation of (4.11)
S =
β∫
0
dτ
(
− i2
∑
i
ΩxΩ˙y − ΩyΩ˙x
1 + Ωz
+
∑
i,σ
ψ∗iσψ˙iσ −
∑
<i,j>
σ
tijψ
∗
iσψjσ
+ J8
∑
<i,j>
(1− ρi)(1− ρj)
(
~Ωi · ~Ωj − 1
)
− µ∑
i
ρi
)
, (4.13)
with β = 1/kBT , the inverse temperature. The remaining constraints can be reformu-
lated elegantly in matrix notation
φ(1) = (~Ω2 − 1) ,
φ(2) = (1− ~Ω · ~σ)ψ ,
φ(3) = ψ∗(1− ~Ω · ~σ) . (4.14)
Note that the bosonic constraint φ(1) fixes the amplitude of the spin field and the two
fermionic constraints φ(2), φ(3) correspond to the single occupancy condition.
58
4.2. Path integral formulation of the t-J model
4.2.2. The t-J model in the low-doping limit
For small values of doping, we may neglect the term proportional to ρiρj in (4.13), so that
we obtain an action that is bilinear in the fermionic degrees of freedom ψi. Moreover,
we allow for 2nd-nearest neighbour hopping along the diagonal with amplitude t′ and
along the principal axis, denoted by t′′. The resulting path integral, which will serve as
the basis for all further considerations has the following form [95]:
Summary 1 (Path-integral formulation of the t-J model at low doping)
Z =
∫
D~Ω Dψ∗Dψ δ
[
φ(1)
]
δ
[
φ(2)
]
δ
[
φ(3)
]
exp
[
− (SS + SF )
]
, (4.15)
where
SS =
∫
dτ
(
− i2
∑
i
~A(~Ωi) · ∂τ ~Ωi + J8
∑
<i,k>
~Ωi · ~Ωk
)
(4.16)
is the action of a pure Heisenberg model. ~A is the vector potential of a magnetic
monopole, (~∇× ~A) · ~Ω = 1, defined through
~A =
(
− Ωy1 + Ωz ,
Ωx
1 + Ωz
, 0
)
, (4.17)
and the fermionic part is given by
SF =
∫
dτ
(∑
i,σ
ψ∗iσ∂τψiσ + t
∑
<i,k>
σ
ψ∗iσψkσ + t′
∑
<<i,k>>
σ
ψ∗iσψkσ
+ t′′
∑
<<<i,k>>>
σ
ψ∗iσψkσ −
J
4
∑
<i,k>
σ
ψ∗iσψiσ ~Ωi · ~Ωk + µ
∑
i,σ
ψ∗iσψiσ
)
. (4.18)
The constraints read
φ(1) = (~Ω2 − 1) ,
φ(2) = (1− ~Ω · ~σ)ψ ,
φ(3) = ψ∗(1− ~Ω · ~σ) . (4.19)
59
4. Effective Field Theory for the t-J Model at Low Doping
4.3. Rotating frame and staggered CP1-representation
Following the approach in [10], we assume local antiferromagnetic order, since at low
doping the system is close to a Ne´el ordered state in which neighbouring spins are
aligned in opposite directions. The state may be described in terms of two sub-lattices.
On sublattice A all spins point up, on the other they are pointing down.
Figure 4.2.: Ne´el ordered reference state.
We introduce a local quantization axis for the fermions ψi which points in the direction
of the order parameter ~Ω. The rotation is implemented via U ∈ SU(2) which fulfills
U †i ~Ωi · ~σ Ui = (−1)iσz . (4.20)
This is accomplished by
U =
 z1 −z∗2
z2 z
∗
1
 , z¯z = 1 , (4.21)
where in order to fulfill (4.20) we need to distinguish between even and odd lattice sites.
For the case where i = p+ q is even, we have the usual CP1-transformation for ~Ω
Ωa = z¯ σa z , (4.22)
60
4.3. Rotating frame and staggered CP1-representation
while for i odd the transformation has to be chosen such that
Ωa = zα σyαβσaβγσ
y
γδ z
∗
δ . (4.23)
The fermions consequently get transformed as
ψi = Uiχi , ψ∗i = χ∗iU
†
i . (4.24)
We may verify that this rotation fulfills (4.20) by use of an explicit parametrization for
z¯ and z in terms of spherical polar-coordinates, see Appendix A.4 and also [10].
Slow CP1-variables in the rotating reference frame
We introduce cells j, each containing an even (A) and an odd (B) site. The cells will be
addressed in a rotated basis such that
x′ = 1√
2
(x+ y) , y′ = 1√
2
(y − x) . (4.25)
Figure 4.3.: Rotated basis related to cells.
For each cell we define new fields
z˜j =
1
2
(
zBj + zAj
)
,
aζj =
1
2
(
zBj − zAj
)
, (4.26)
with the inverse transformation
zAj = z˜j − aζj ,
61
4. Effective Field Theory for the t-J Model at Low Doping
zBj = z˜j + aζj . (4.27)
In these new variables the constraint z¯z = 1 transforms into
¯˜zζ + ζ¯ z˜ = 0 ,
¯˜zz˜ + a2ζ¯ζ = 1 . (4.28)
Here, we introduce new fields
z˜ = z
√
1− a2ζ¯ζ , (4.29)
such that the second constraint is satisfied automatically and
z¯ζ + ζ¯z = 0 . (4.30)
In summary, the transformation for even (A) and odd sites (B) reads as follows:
Result 1 (Staggered CP1-representation)
~ΩAj =
(
z¯j
√
1− a2ζ¯jζj − aζ¯j
)
~σ
(
zj
√
1− a2ζ¯jζj − aζj
)
,
~ΩBj =
(
zj
√
1− a2ζ¯jζj + aζj
)
σy ~σ σy
(
z¯j
√
1− a2ζ¯jζj + aζ¯j
)
. (4.31)
4.3.1. Transformation of the integral measure
In order to derive the change of the integral measure due to the transformation into the
rotating reference frame and the staggered CP1-representation, we first need to assess
the effect on the fermionic constraints, as is done in reference [10].
Fermionic constraints in the rotating reference frame
From (4.19) we have
φ(2) = (1− ~Ω · ~σ)ψ ,
φ(3) = ψ∗(1− ~Ω · ~σ) . (4.32)
62
4.3. Rotating frame and staggered CP1-representation
By introducing new fermions χ = U †ψ and χ∗ = ψ∗U , the constraints translate into a
simple relation for even sites
δ
[
φ(2)
]
δ
[
φ(3)
]
= 2 δ
[
χ∗−
]
δ
[
χ−
]
(4.33)
and for odd sites
δ
[
φ(2)
]
δ
[
φ(3)
]
= 2 δ
[
χ∗+
]
δ
[
χ+
]
, (4.34)
which implies that χ− = 0 on even sites and χ+ = 0 on odd sites. We can therefore work
with spinless fermions χA on even sites and χB on odd sites, where A and B denote the
two sublattices such that the fermionic constraints are exactly taken into account.
Bosonic constraint in the staggered CP1-representation
Now we derive the constraint δ
(
|~ΩA| − 1
)
δ
(
|~ΩB| − 1
)
in terms of the new variables z
and ζ. From (4.31) we easily obtain
(~ΩA)2 =
[ (
1− a2ζ¯ζ
)
z¯z + a
(
a ζ¯ζ −
√
1− a2ζ¯ζ
(
z¯ζ + ζ¯z
))]2
, (4.35)
(~ΩB)2 =
[ (
1− a2ζ¯ζ
)
z¯z − a
(
a ζ¯ζ −
√
1− a2ζ¯ζ
(
z¯ζ + ζ¯z
))]2
. (4.36)
So we get for
δ
(
|~ΩA| − 1
)
δ
(
|~ΩB| − 1
)
= 12a δ
(
(1− a2ζ¯ζ)z¯z + a2ζ¯ζ − 1
)
δ
(√
1− a2ζ¯ζ
(
z¯ζ + ζ¯z
) )
= 1
2a (1− a2ζ¯ζ)3/2 δ
(
z¯z − 1
)
δ
(
z¯ζ + ζ¯z
)
' 12a
(
1 + 32a
2ζ¯ζ +O(a4)
)
δ
(
z¯z − 1
)
δ
(
z¯ζ + ζ¯z
)
, (4.37)
where we expanded with respect to powers of a and used the following identities for the
δ-functions:
δ(x1 − x0)δ(x2 − x0) = δ(x1 − x0)δ(x2 − x1) , δ(a x) = δ(x)|a| . (4.38)
63
4. Effective Field Theory for the t-J Model at Low Doping
The integral measure
From (4.15) we have for the measure
∫
D~Ω Dψ∗Dψ δ
[
φ(1)
]
δ
[
φ(2)
]
δ
[
φ(3)
]
. (4.39)
After introducing even (A) and odd (B) sites and rewriting the fermionic constraints in
terms of the new quantization axis as in (4.34), the measure becomes
∫
D~ΩAD~ΩB D(χ∗A, χA)D(χ∗B, χB) δ
(
|~ΩA| − 1
)
δ
(
|~ΩB| − 1
)
· δ
(
χ∗A−
)
δ
(
χA−
)
δ
(
χ∗B+
)
δ
(
χB+
)
, (4.40)
where we neglected constant prefactors from (4.34) and the Jacobian of the unitary
transformation U , since they can be absorbed in the normalization.
Going over to the staggered CP1-representation, we get from (4.37)
∫
D(z¯, z)D(ζ¯ , ζ) D(χ∗A, χA)D(χ∗B, χB)
1
2a
(
1 + 32a
2ζ¯ζ +O(a4)
)
J
· δ
(
z¯z − 1
)
δ
(
z¯ζ + ζ¯z
)
δ
(
χ∗A−
)
δ
(
χA−
)
δ
(
χ∗B+
)
δ
(
χB+
)
, (4.41)
where J denotes the Jacobian of the transformation from (~ΩA, ~ΩB) to (z, aζ) which we
will evaluate at a later stage, see Section 4.5.
4.3.2. Spin part of the action
Berry phase:
For the spin’s Berry phase, we get from (4.16)
B = − i2
∑
i
~A[~Ωi] · ∂τ ~Ωi = i2
∑
i
ΩiyΩ˙ix − ΩixΩ˙iy
1 + Ωiz
= i2
∑
i even
ΩiyΩ˙ix − ΩixΩ˙iy
1 + Ωiz
+ i2
∑
i odd
ΩiyΩ˙ix − ΩixΩ˙iy
1 + Ωiz
. (4.42)
Inserting (4.22) or (4.23), resp., we go over to CP1-representation, which then yields
64
4.3. Rotating frame and staggered CP1-representation
B =
∑
j
(
zA∗j1 ∂τz
A
j1 − zAj1∂τzA∗j1 − 2 · |zAj1|2 z¯Aj ∂τzAj
2 · |zAj1|2
+
zBj2∂τz
B∗
j2 − zB∗j2 ∂τzBj1 + 2 · |zBj2|2 z¯Bj ∂τzBj
2 · |zBj2|2
)
, (4.43)
where we used z¯AzA = 1 and z¯BzB = 1. Writing zA and zB in polar representation,
z1 = |z1| eiϕ1 , z2 = |z2| eiϕ2 , (4.44)
we see that part of the contribution is a total time derivative:
B =
∑
j
(
2i |zAj1|2 ϕ˙Aj1 − 2 |zAj1|2 z¯Aj ∂τzAj
2 |zAj1|2
+
−2i |zBj2|2 ϕ˙Bj2 + 2 |zBj2|2 z¯Bj ∂τzBj
2 |zBj2|2
)
= −∑
j
((
z¯Aj ∂τz
A
j − z¯Bj ∂τzBj
)
− i ∂τ
(
ϕAj1 − ϕBj2
))
. (4.45)
Let us consider the effect of this total derivative in more detail. Since the functional
integral we are calculating represents a partition function, we know that the state with
which we start at τ = 0 must be the same state as at τ = β. That means, z(0) = z(β)
modulo a phase of the form ei 2pin and the phase of the z(τ) might wind around several
times in the periodicity interval. So the integral over the time derivative in (4.45) equals
the difference of winding numbers of zA1 and zB1
i
β∮
0
dτ ∂τϕ
A
1 (τ) = i 2pin , i
β∮
0
dτ ∂τϕ
B
2 (τ) = i 2pim . (4.46)
Thus, the total derivative in the Berry phase contributes to the action by an integer
multiple of 2pii, such that it holds no contribution to the partition function at all.
We continue with the first two summands in (4.45) and write them in terms of slow and
fast varying field variables z and ζ
zAj (τ) = zj
√
1− a2ζ¯ζ − aζj , zBj (τ) = zj
√
1− a2ζ¯ζ + aζj . (4.47)
We expand the square root up to second order in a where we obtain
B = −∑
j
([
(1− 12a
2 ζ¯jζj) z¯j − aζ¯j
]
∂τ
[
(1− 12a
2 ζ¯jζj) zj − ζj
]
65
4. Effective Field Theory for the t-J Model at Low Doping
−
[
(1− 12a
2 ζ¯jζj) z¯j + aζ¯j
]
∂τ
[
(1− 12a
2 ζ¯jζj) zj + ζj
])
, (4.48)
so that we get
Result 2 (Berry phase)
B =
∑
j
2a
(
z¯j∂τζj + ζ¯j∂τzj
)
︸ ︷︷ ︸
O(a2)
+O(a4) . (4.49)
Note that a time derivative must be taken as ∼ a.
Interaction term:
I = J8
∑
< i, k >
~Ωi ~Ωk =
J
8
∑
< i, k >
i even
~Ωi ~Ωk +
J
8
∑
< i, k >
i odd
~Ωi ~Ωk . (4.50)
For the case of next neighbours, if i is even then k has to be odd and vice versa. The
respective multi-indices j′ are summarized in Table 4.1.
old lattice rotated lattice
neighbour’s position i even i odd
right (l,m) (l + 1,m− 1)
upper (l,m+ 1) (l + 1,m)
left (l − 1,m+ 1) (l,m)
lower (l − 1,m) (l,m− 1)
Table 4.1.: Cell address j′ of next neighbours, see also Figure 4.3.
We consider both sums separately.
i even, k odd:
In this case we insert (4.22) for Ωi and (4.23) for Ωk
∑
< i, k >
i even
~Ωi ~Ωk =
∑
< i, k >
i even
(
z¯iσ
azi
)(
zkασ
y
αβσ
a
βγσ
y
γδz
∗
kδ
)
66
4.3. Rotating frame and staggered CP1-representation
=
∑
< i, k >
i even
2
(
zk1zi2 − zk2zi1
)(
z∗k1z
∗
i2 − z∗k2z∗i1
)
− 1 . (4.51)
We switch to cell indices instead of site indices
∑
< i, k >
i even
~Ωi ~Ωk =
∑
j,j′
2
(
zA∗j1 z
B∗
j′2 − zA∗j2 zB∗j′1
)(
zAj1z
B
j′2 − zAj2zBj′1
)
− 1 . (4.52)
By using (4.27) we go over to the new field variables and expand the square root up to
third order in a. Finally, we perform the gradient expansion such that
f
(
~x+
√
2a~eα
)
'
3∑
k=0
1
k!
(√
2a∂α
)k
f(~x) . (4.53)
We have to distinguish the different cases of next neighbours, since each one yields
another expansion vector (see Table 4.1).
• Right neighbour:
The first one is easy, since cell j′ is the same as cell j. We obtain
(
~Ωi~Ωk
)(1)
e
= −1 + 2a2G∗jGj +O(a4) , (4.54)
where we used the definitions in Appendix B.2.
• Upper neighbour:
Here we have
zj′ = zj +
√
2a ∂yzj + a2 ∂2yzj +
√
2
3 a
3 ∂3yzj , (4.55)
and similarly for ζj′ . We find
(
~Ωi ~Ωk
)(2)
e
= −1 + 2a2
(√
2F ∗jy +G∗j
) (√
2Fjy +Gj
)
+ a3
[(√
2F ∗jy +G∗j
)(
2Fjyy +
√
2∂yGj
)
+
(
2F ∗jyy +
√
2∂yG∗j
)(√
2Fjy +Gj
)]
. (4.56)
67
4. Effective Field Theory for the t-J Model at Low Doping
• Left neighbour:
(
~Ωi ~Ωk
)(3)
e
= −1 + 2a2
(
−√2F ∗jx +
√
2F ∗jy +G∗j
) (
−√2Fjx +
√
2Fjy +Gj
)
+ a3
[(
−√2F ∗jx +
√
2F ∗jy +G∗j
)
·
(
2Fjxx − 4Fjxy + 2Fjyy −
√
2∂xGj +
√
2∂yGj
)
+
(
2F ∗jxx − 4F ∗jxy + 2F ∗jyy −
√
2∂xG∗j +
√
2∂yG∗j
)
·
(
−√2Fjx +
√
2Fjy +Gj
)]
. (4.57)
• Lower neighbour:
(
~Ωi ~Ωk
)(4)
e
= −1 + 2a2
(
−√2F ∗jx +G∗j
) (
−√2Fjx +Gj
)
+ a3
[(
−√2F ∗jx +G∗j
)(
2Fjxx −
√
2∂xGj
)
+
(
2F ∗jxx −
√
2∂xG∗j
)(
−√2Fjx +Gj
)]
. (4.58)
Thus, we get for the even part
4∑
a=1
(
~Ωi ~Ωk
)(a)
e
= −4 + 4a2
[
2
(
G∗G+ F ∗xFx + F ∗yFy
)
−
(
F ∗xFy + F ∗yFx
)
−√2
(
F ∗x − F ∗y
)
G−√2G∗ (Fx − Fy)
]
+ a3
[ (
−√2F ∗x +G∗
) (
2Fxx −
√
2∂xG
)
+
(
2F ∗xx −
√
2∂xG∗
) (
−√2Fx +G
)
+
(
−√2F ∗x +
√
2F ∗y +G∗
)(
2Fxx − 4Fxy + 2Fyy −
√
2∂xG+
√
2∂yG
)
+
(
2F ∗xx − 4F ∗xy + 2F ∗yy −
√
2∂xG∗ +
√
2∂yG∗
)(
−√2Fx +
√
2Fy +G
)
+
(√
2F ∗y +G∗
)(
Fyy +
√
2∂yG
)
+
(
F ∗yy +
√
2∂yG∗
)(√
2Fy +G
)]
. (4.59)
i odd, k even:
68
4.3. Rotating frame and staggered CP1-representation
After inserting (4.23) for Ωi and (4.22) for Ωk we get:
∑
< i, k >
i odd
~Ωi ~Ωk =
∑
j,j′
2
(
zA∗j′1z
B∗
j2 − zA∗j′2zB∗j1
)(
zAj′1z
B
j2 − zAj′2zBj1
)
− 1 , (4.60)
which is like (4.52) with A ↔ B. Again, we expand the fields with cell coordinate j′
with respect to the coordinate vector (x, y), obtaining
(
~Ωi ~Ωk
)(1...4)
o
. In the end, we get
for the odd part
4∑
a=1
(
~Ωi ~Ωk
)(a)
o
= −4 + 4a2
[
2
(
G∗G+ F ∗xFx + F ∗yFy
)
−
(
F ∗xFy + F ∗yFx
)
−√2
(
F ∗x − F ∗y
)
G−√2G∗ (Fx − Fy)
]
− a3
[ (
−√2F ∗x +G∗
) (
2Fxx −
√
2∂xG
)
+
(
2F ∗xx −
√
2∂xG∗
) (
−√2Fx +G
)
+
(
−√2F ∗x +
√
2F ∗y +G∗
)(
2Fxx − 4Fxy + 2Fyy −
√
2∂xG+
√
2∂yG
)
+
(
2F ∗xx − 4F ∗xy + 2F ∗yy −
√
2∂xG∗ +
√
2∂yG∗
)(
−√2Fx +
√
2Fy +G
)
+
(√
2F ∗y +G∗
)(
2Fyy +
√
2∂yG
)
+
(
2F ∗yy +
√
2∂yG∗
)(√
2Fy +G
)]
. (4.61)
Finally, we sum up all contributions to the interaction term, such that
Result 3 (Spin interaction)
I = J
∑
j
[
− 1 + a2
(
2
(
G∗G+ F ∗xFx + F ∗yFy
)
−
(
F ∗xFy + F ∗yFx
)
−√2
[(
F ∗x − F ∗y
)
G+G∗ (Fx − Fy)
] )]
+O(a4) . (4.62)
There is no contribution in O(a3), since all a3-terms canceled.
4.3.3. Fermionic part of the action
Rotating the quantization axis, see eq. (4.21), leads to new fermionic variables
ψi = Uiχi , ψ∗i = χ∗iU
†
i . (4.63)
69
4. Effective Field Theory for the t-J Model at Low Doping
Moreover, the fermionic constraints (4.33) and (4.33) enable us to express the action in
terms of spinless fermions χA and χB, defined on even and odd lattice sites, respectively.
χ −→
 χA
0
 for i even χ −→
 0
χB
 for i odd . (4.64)
Temporal derivative:
The contribution in (4.18) featuring a temporal derivative reads
T =
∑
i,σ
ψ∗iσ∂τψiσ =
∑
i even
σ
ψ∗iσ∂τψiσ +
∑
i odd
σ
ψ∗iσ∂τψiσ
=
∑
i even
σ
(
χ∗iAU
†
i,(+σ) ∂τUi,(σ+)χiA + χ∗iA∂τχiA
)
+
∑
i odd
σ
(
χ∗iBU
†
i,(−σ) ∂τUi,(σ−)χiB + χ∗iB∂τχiB
)
. (4.65)
Note that
U †i,(σσ′) ∂τUi,(σ′σ′′) =
 z¯i∂τzi −z∗i1∂τz∗i2 + z∗i2∂τz∗i1
zi1∂τzi2 − zi2∂τzi1 −z¯i∂τzi
 , (4.66)
so we obtain
T =
∑
i even
σ
(
χ∗iA∂τχiA + iχ∗iAAAi,τχiA
)
+
∑
i odd
σ
(
χ∗iB∂τχiB − iχ∗iBABi,τχiB
)
, (4.67)
with the gauge potential Aµ = −i z¯ ∂µz. Now we switch to cell indices instead of site
indices and go over to the new fields z and ζ.
• On even sites
iAAjτ = z¯Aj ∂τzAj '
[(
1− 12a
2ζ¯ζ
)
z¯j − aζ¯j
]
∂τ
[(
1− 12a
2ζ¯ζ
)
zj − aζ¯j
]
= z¯j∂τzj − a
[
z¯j∂τζj + ζ¯j∂τzj
]
+ a2
[1
2(ζ¯j∂τζ − ζj∂τ ζ¯j)− ζ¯ζ z¯j∂τzj
]
+ a
3
2
[
ζ¯jzj ∂τ (ζ¯ζ) + ζ¯ζ ζ¯j∂τζj + ζ¯ζ ζ¯j∂τzj
]
, (4.68)
70
4.3. Rotating frame and staggered CP1-representation
which, after introducing the definitions
Cjµ ≡ z¯j∂µζj + ζ¯j∂µzj Ljµ ≡ 12
(
ζ¯j∂µζj − ζj∂µζ¯j
)
− iAjµζ¯ζ , (4.69)
reads
iAAjτ = iAjτ − aCjτ + a2Ljτ +
a3
2
[
ζ¯ζCjτ +
1
2 ζ¯jzj∂τ (ζ¯ζ)
]
︸ ︷︷ ︸
O(a4)
. (4.70)
• On odd sites
iABjτ = z¯Bj ∂τzBj
' z¯j∂τzj + a
[
z¯j∂τζj + ζ¯j∂τzj
]
+ a2
[1
2(ζ¯j∂τζ − ζj∂τ ζ¯j)− ζ¯ζ z¯j∂τzj
]
− a
3
2
[
ζ¯jzj ∂τ (ζ¯ζ) + ζ¯ζ ζ¯j∂τζj + ζ¯ζ ζ¯j∂τzj
]
= iAjτ + aCjτ + a2Ljτ − a
3
2
[
ζ¯ζCjτ +
1
2ζjzj∂τ (ζ¯ζ)
]
︸ ︷︷ ︸
O(a4)
. (4.71)
Summarizing these terms, we get for the temporal derivative
T =
∑
i,σ
ψ∗iσ∂τψiσ =
∑
j
χ∗j
[
∂τ + iσzAjτ − aCjτ + a2σzLjτ
]
χj +O(a4) , (4.72)
where we defined the spinor
χj =
 χjA
χjB
 , (4.73)
and the σz-matrices take care of the alternating signs in the even- and odd-cases. We
may introduce the gauge covariant derivative and write
Result 4 (Time derivative)
T =
∑
j
χ∗j
[
Dτ − aCτ + a2σzLτ
]
χj , Dµ = ∂µ + iσzAµ . (4.74)
71
4. Effective Field Theory for the t-J Model at Low Doping
Next neighbour hopping
K(t) = t
∑
< i, k >
σ
ψ∗iσψkσ = t
∑
< i, k >
χ∗i U
†
i Uk χk
= t
∑
< i, k >
i even
χ∗iA U
†
i,(+σ)Uk,(σ−) χkB + t
∑
< i, k >
i odd
χ∗iB U
†
i,(−σ)Uk,(σ+) χkA . (4.75)
Since the first kinetic contribution describes hopping among next neighbours, we can
apply the same consideration as in Section 4.3.2. The respective indices j′ for the next
neighbours can be read of Table 4.1.
i even, k odd:
For the product of rotations matrices, we obtain
U †i,(+σ)Uk,(σ−) = −
(
z∗j1z
∗
j′2 − z∗j2z∗j′1
)
− a
[(
z∗j1ζ
∗
j′2 − z∗j2ζ∗j′1
)
−
(
ζ∗j1z
∗
j′2 − ζ∗j2z∗j′1
)]
+ a2
[(
ζ∗j1ζ
∗
j′2 − ζ∗j2ζ∗j′1
)
− 12
(
ζ¯jζj + ζ¯j′ζj′
) (
z∗j2z
∗
j′1 − z∗j1z∗j′2
)]
+ a
3
2
[
ζ¯jζj
(
z∗j1ζ
∗
j′2 − z∗j2ζ∗j′1
)
− ζ¯j′ζj′
(
ζ∗j1z
∗
j′2 − ζ∗j2z∗j′1
) ]
. (4.76)
Again, we must be careful to choose the proper expansion vector from Table 4.1, de-
pending on whether k is the right, upper, left or lower neighbour. As a result we obtain
• Right neighbour:
[
U †(+σ)U(σ−)
](t,1)
j
= −aG∗ + a
3
2 ζ¯ζ G
∗ .
(4.77)
• Upper neighbour:
[
U †(+σ)U(σ−)
](t,2)
j
= −a
[√
2F ∗y +G∗
]
− a2F ∗yy + a3
[
−
√
2
3
(
z∗1∂
3
yz
∗
2 − z∗2∂3yz∗1
)
−
(
z∗1∂
2
yζ
∗
2 − z∗2∂2yζ∗1
)
+
(
ζ∗1∂
2
yz
∗
2 − ζ∗2∂2yz∗1
)
+
√
2 (ζ∗1∂yζ∗2 − ζ∗2∂yζ∗1 )
+
√
2ζ¯ζ (z∗1∂yz∗2 − z∗2∂yz∗1) + ζ¯ζ (z∗1ζ∗2 − z∗2ζ∗1 )
]
. (4.78)
72
4.3. Rotating frame and staggered CP1-representation
• Left neighbour:
[
U †(+σ)U(σ−)
](t,3)
j
= a
[√
2
(
F ∗x − F ∗y
)
−G∗
]
− a2
[
F ∗xx − 2F ∗xy + F ∗yy
]
+ a3
[√
2
3
(
z∗1
[
∂3xz
∗
2 − 3∂2x∂yz∗2 + 3∂x∂2yz∗2 − ∂3yz∗2
]
− z∗2
[
∂3xz
∗
1 − 3∂2x∂yz∗1 + 3∂x∂2yz∗1 − ∂3yz∗1
] )
−
(
z∗1
[
∂2xζ
∗
2 − 2∂x∂yζ∗2 + ∂2yζ∗2
]
− z∗2
[
∂2xζ
∗
1 − 2∂x∂yζ∗1 + ∂2yζ∗1
])
+
(
ζ∗1
[
∂2xz
∗
2 − 2∂x∂yz∗2 + ∂2yz∗2
]
− ζ∗2
[
∂2xz
∗
1 − 2∂x∂yz∗1 + ∂2yz∗1
] )
−√2 (ζ∗1 [∂xζ∗2 − ∂yζ∗2 ]− ζ∗2 [∂xζ∗1 − ∂yζ∗1 ])
−√2 ζ¯ζ (z∗1 [∂xz∗2 − ∂yz∗2 ]− z∗2 [∂xz∗1 − ∂yz∗1 ]) + ζ¯ζ (z∗1ζ∗2 − z∗2ζ∗1 )
]
. (4.79)
• Lower neighbour:
[
U †(+σ)U(σ−)
](t,4)
j
= a
[√
2F ∗x −G∗
]
− a2 F ∗xx + a3
[√
2
3
(
z∗1∂
3
xz
∗
2 − z∗2∂3xz∗1
)
−
(
z∗1∂
2
xζ
∗
2 − z∗2∂2xζ∗1
)
+
(
ζ∗1∂
2
xz
∗
2 − ζ∗2∂2xz∗1
)
−√2 (ζ∗1∂xζ∗2 − ζ∗2∂xζ∗1 )
−√2 ζ¯ζ (z∗1∂xz∗2 − z∗2∂xz∗1) + ζ¯ζ (z∗1ζ∗2 − z∗2ζ∗1 )
]
, (4.80)
where F and G are defined in Appendix B.2.
i odd, k even:
U †i,(−σ)Uk,(σ+) = (zj1zj′2 − zj2zj′1)− a
[
(zj1ζj′2 − zj2ζj′1)− (ζj1zj′2 − ζj2zj′1)
]
− a2
[
(ζj1ζj′2 − ζj2ζj′1)− 12
(
ζ¯jζj + ζ¯j′ζj′
)
(zj2zj′1 − zj1zj′2)
]
+ a
3
2
[
ζ¯jζj (zj1ζj′2 − zj2ζj′1)− ζ¯j′ζj′ (ζj1zj′2 − ζj2zj′1)
]
. (4.81)
The result for the right (1), upper (2), left (3) and lower (4) neighbour read as follows:
• Right neighbour:
73
4. Effective Field Theory for the t-J Model at Low Doping
[
U †(−σ)U(σ+)
](t,1)
j
= a
[√
2 (Fx − Fy)−G
]
+ a2
[
Fxx − 2Fxy + Fyy
]
+ a3
[√
2
3
(
z1
[
∂3xz2 − 3∂2x∂yz2 + 3∂x∂2yz2 − ∂3yz2
]
− z2
[
∂3xz1 − 3∂2x∂yz1 + 3∂x∂2yz1 − ∂3yz1
] )
−
(
z1
[
∂2xζ2 − 2∂x∂yζ2 + ∂2yζ2
]
− z2
[
∂2xζ1 − 2∂x∂yζ1 + ∂2yζ1
])
+
(
ζ1
[
∂2xz2 − 2∂x∂yz2 + ∂2yz2
]
− ζ2
[
∂2xz1 − 2∂x∂yz1 + ∂2yz1
])
−√2 (ζ1 [∂xζ2 − ∂yζ2]− ζ2 [∂xζ1 − ∂yζ1])
−√2 ζ¯ζ (z1 [∂xz2 − ∂yz2]− z2 [∂xz1 − ∂yz1]) + ζ¯ζ (z1ζ2 − z2ζ1)
]
. (4.82)
• Upper neighbour:
[
U †(−σ)U(σ+)
](t,2)
j
= a
[√
2Fx −G
]
+ a2Fxx + a3
[√
2
3
(
z1∂
3
xz2 − z2∂3xz1
)
−
(
z1∂
2
xζ2 − z2∂2xζ1
)
+
(
ζ1∂
2
xz2 − ζ2∂2xz1
)
−√2 (ζ1∂xζ2 − ζ2∂xζ1)
−√2 ζ¯ζ (z1∂xz2 − z2∂xz1) + ζ¯ζ (z1ζ2 − z2ζ1)
]
. (4.83)
• Left neighbour:
[
U †(−σ)U(σ+)
](t,3)
j
= −aG+ a3 ζ¯ζ G .
(4.84)
• Lower neighbour:
[
U †(−σ)U(σ+)
](t,4)
j
= a
[
−√2Fy −G
]
+ a2Fyy + a3
[
−
√
2
3
(
zj1∂
3
yzj2 − zj2∂3yzj1
)
−
(
zj1∂
2
yζj2 − zj2∂2yζj1
)
+
(
ζj1∂
2
yzj2 − ζj2∂2yzj1
)
+
√
2 (ζj1∂yζj2 − ζj2∂yζj1)
+
√
2 ζ¯jζj (zj1∂yzj2 − zj2∂yzj1) + ζ¯jζj (zj1ζj2 − zj2ζj1)
]
. (4.85)
In summary,
74
4.3. Rotating frame and staggered CP1-representation
Result 5 (Next neighbour hopping)
K(t) = t
∑
j
(
χ∗jA
4∑
a=1
[
U †(+σ)U(σ−)
](t,a)
j
χjB + χ∗jB
4∑
a=1
[
U †(−σ)U(σ+)
](t,a)
j
χjA
)
. (4.86)
Second-nearest neighbour hopping along the diagonal
The contribution from nearest neighbour hopping along the diagonal is
K(t
′) = t′
∑
<<i, k>>
σ
ψ∗iσψkσ = t′
∑
<<i, k>>
χ∗i U
†
i Uk χk (4.87)
= t′
∑
<<i, k>>
i even
χ∗iA U
†
i,(+σ)Uk,(σ+) χkA + t′
∑
<<i, k>>
i odd
χ∗iB U
†
i,(−σ)Uk,(σ−) χkB . (4.88)
Note that for these second-nearest neighbours, the hopping takes place between sites of
the same parity. So if i is even/odd, so is k. The addresses of the neighbouring cells are
summarized in Table 4.2.
old lattice rotated lattice
neighbour’s position i even i odd
upper-right (l + 1,m) (l + 1,m)
upper-left (l,m+ 1) (l,m+ 1)
lower-left (l − 1,m) (l − 1,m)
lower-right (l,m− 1) (l,m− 1)
Table 4.2.: Cell address j′ of nearest neighbours along the diagonal, see also Figure 4.3.
i even, k even:
The rotation matrices for the respective neighbours read
• Upper-right case:
75
4. Effective Field Theory for the t-J Model at Low Doping
[
U †(+σ)U(σ+)
](t′,1)
j
= 1 +
√
2 a z¯∂xz + a2
[
z¯∂2xz −
√
2
(
z¯∂xζ + ζ¯∂xz
)]
+ a3
[√
2
3 z¯∂
3
xz −
(
z¯∂2xζ + ζ¯∂2xz
)
+
√
2 ζ¯∂xζ −
√
2 ζ¯ζ z¯∂xz
]
. (4.89)
• Upper-left case:
[
U †(+σ)U(σ+)
](t′,2)
j
= 1 +
√
2 a z¯∂yz + a2
[
z¯∂2yz −
√
2
(
z¯∂yζ + ζ¯∂yz
)]
+ a3
[√
2
3 z¯∂
3
yz −
(
z¯∂2yζ + ζ¯∂2yz
)
+
√
2 ζ¯∂yζ −
√
2 ζ¯ζ z¯∂yz
]
. (4.90)
• Lower-left case:
[
U †(+σ)U(σ+)
](t′,3)
j
= 1−√2 a z¯∂xz + a2
[
z¯∂2xz +
√
2
(
z¯∂xζ + ζ¯∂xz
)]
+ a3
[
−
√
2
3 z¯∂
3
xz −
(
z¯∂2xζ + ζ¯∂2xz
)
−√2 ζ¯∂xζ +
√
2 ζ¯ζ z¯∂xz
]
. (4.91)
• Lower-right case:
[
U †(+σ)U(σ+)
](t′,4)
j
= 1−√2 a z¯∂yz + a2
[
z¯∂2yz +
√
2
(
z¯∂yζ + ζ¯∂yz
)]
+ a3
[
−
√
2
3 z¯∂
3
yz −
(
z¯∂2yζ + ζ¯∂2yz
)
−√2 ζ¯∂yζ +
√
2 ζ¯ζ z¯∂yz
]
. (4.92)
i odd, k odd:
• Upper-right case:
[
U †(−σ)U(σ−)
](t′,1)
j
= 1 +
√
2 a z∂xz¯ + a2
[
z∂2xz¯ +
√
2
(
z∂xζ¯ + ζ∂xz¯
)]
+ a3
[√
2
3 z∂
3
xz¯ + z∂2xζ¯ + ζ∂2xz¯ +
√
2 ζ∂xζ¯ −
√
2 ζ¯ζ z∂xz¯
]
. (4.93)
• Upper-left case:
[
U †(−σ)U(σ−)
](t′,2)
j
= 1 +
√
2 a z∂yz¯ + a2
[
z∂2y z¯ +
√
2
(
z∂y ζ¯ + ζ∂yz¯
)]
76
4.3. Rotating frame and staggered CP1-representation
+ a3
[√
2
3 z∂
3
y z¯ + z∂2y ζ¯ + ζ∂2y z¯ +
√
2 ζ∂y ζ¯ −
√
2 ζ¯ζ z∂yz¯
]
. (4.94)
• Lower-left case:
[
U †(−σ)U(σ−)
](t′,3)
j
= 1−√2 a z∂xz¯ + a2
[
z∂2xz¯ −
√
2
(
z∂xζ¯ + ζ∂xz¯
)]
+ a3
[
−
√
2
3 z∂
3
xz¯ + z∂2xζ¯ + ζ∂2xz¯ −
√
2 ζ∂xζ¯ +
√
2 ζ¯ζ z∂xz¯
]
. (4.95)
• Lower-right case:
[
U †(−σ)U(σ−)
](t′,4)
j
= 1−√2 a z∂yz¯ + a2
[
z∂2y z¯ −
√
2
(
z∂y ζ¯ + ζ∂yz¯
)]
+ a3
[
−
√
2
3 z∂
3
y z¯ + z∂2y ζ¯ + ζ∂2y z¯ −
√
2 ζ∂y ζ¯ +
√
2 ζ¯ζ z∂yz¯
]
. (4.96)
Such that
Result 6 (Second-nearest neighbour hopping along the diagonal)
K(t
′) = t′
∑
j
(
χ∗jA
4∑
a=1
[
U †(+σ)U(σ−)
](t′,a)
j
χjA + χ∗jB
4∑
a=1
[
U †(−σ)U(σ+)
](t′,a)
j
χjB
)
. (4.97)
Second-nearest neighbour hopping along the principal axis
The second-nearest neighbour hopping along the principal axis contributes to the kinetic
energy through
K(t
′′) = t′′
∑
<<<i, k>>>
σ
ψ∗iσψkσ = t′′
∑
<<<i, k>>>
χ∗i U
†
i Uk χk (4.98)
= t′′
∑
<<<i, k>>>
i even
χ∗iA U
†
i,(+σ)Uk,(σ+) χkA + t′′
∑
<<<i, k>>>
i odd
χ∗iB U
†
i,(−σ)Uk,(σ−) χkB . (4.99)
Again, either i and k are both even or odd indices.
77
4. Effective Field Theory for the t-J Model at Low Doping
old lattice rotated lattice
neighbour’s position i even i odd
right (l + 1,m− 1) (l + 1,m− 1)
upper (l + 1,m+ 1) (l + 1,m+ 1)
left (l − 1,m+ 1) (l − 1,m+ 1)
lower (l − 1,m− 1) (l − 1,m− 1)
Table 4.3.: Cell address j′ of 2nd-nearest neighbours along the p.a., see also Fig. 4.3.
i even, k even:
• Right case:
[
U †(+σ)U(σ+)
](t′′,1)
j
= 1 +
√
2 a z¯
[
∂xz − ∂yz
]
+ a2
[
z¯
[
∂2xz − 2 ∂x∂yz + ∂2yz
]
−√2
(
z¯
[
∂xζ − ∂yζ
]
+ ζ¯
[
∂xz − ∂yz
]) ]
+ a3
[√2
3 z¯
[
∂3xz − 3 ∂2x∂yz + 3 ∂x∂2yz − ∂3yz
]
− z¯
[
∂2xζ − 2 ∂x∂yζ + ∂2yζ
]
− ζ¯
[
∂2xz − 2 ∂x∂yz + ∂2yz
]
+
√
2 ζ¯
[
∂xζ − ∂yζ
]
−√2 ζ¯ζ z¯
[
∂xz − ∂yz
]]
. (4.100)
• Upper case:
[
U †(+σ)U(σ+)
](t′′,2)
j
= 1 +
√
2 a z¯
[
∂xz + ∂yz
]
+ a2
[
z¯
[
∂2xz + 2 ∂x∂yz + ∂2yz
]
−√2
(
z¯
[
∂xζ + ∂yζ
]
+ ζ¯
[
∂xz + ∂yz
]) ]
+ a3
[√2
3 z¯
[
∂3xz + 3 ∂2x∂yz + 3 ∂x∂2yz + ∂3yz
]
− z¯
[
∂2xζ + 2 ∂x∂yζ + ∂2yζ
]
− ζ¯
[
∂2xz + 2 ∂x∂yz + ∂2yz
]
+
√
2 ζ¯
[
∂xζ + ∂yζ
]
−√2 ζ¯ζ z¯
[
∂xz + ∂yz
]]
. (4.101)
• Left case:
[
U †(+σ)U(σ+)
](t′′,3)
j
= 1−√2 a z¯
[
∂xz − ∂yz
]
+ a2
[
z¯
[
∂2xz − 2 ∂x∂yz + ∂2yz
]
+
√
2
(
z¯
[
∂xζ − ∂yζ
]
+ ζ¯
[
∂xz − ∂yz
]) ]
78
4.3. Rotating frame and staggered CP1-representation
+ a3
[
−
√
2
3 z¯
[
∂3xz − 3 ∂2x∂yz + 3 ∂x∂2yz − ∂3yz
]
− z¯
[
∂2xζ − 2 ∂x∂yζ + ∂2yζ
]
− ζ¯
[
∂2xz − 2 ∂x∂yz + ∂2yz
]
−√2 ζ¯
[
∂xζ − ∂yζ
]
+
√
2 ζ¯ζ z¯
[
∂xz − ∂yz
]]
. (4.102)
• Lower case:
[
U †(+σ)U(σ+)
](t′′,4)
j
= 1−√2 a z¯
[
∂xz + ∂yz
]
+ a2
[
z¯
[
∂2xz + 2 ∂x∂yz + ∂2yz
]
+
√
2
(
z¯
[
∂xζ + ∂yζ
]
+ ζ¯
[
∂xz + ∂yz
]) ]
+ a3
[
−
√
2
3 z¯
[
∂3xz + 3 ∂2x∂yz + 3 ∂x∂2yz + ∂3yz
]
− z¯
[
∂2xζ + 2 ∂x∂yζ + ∂2yζ
]
− ζ¯
[
∂2xz + 2 ∂x∂yz + ∂2yz
]
−√2 ζ¯
[
∂xζ + ∂yζ
]
+
√
2 ζ¯ζ z¯
[
∂xz + ∂yz
]]
. (4.103)
i odd, k odd:
Although the cell-address in the odd case matches the one in the even case, we have to
insert the expansion for the z-fields on odd sites.
U †i,(−σ)Uk,(σ−) = zj z¯j′ + a
(
zj ζ¯j′ + ζj z¯j′
)
+ a2
(
ζj ζ¯j′ − 12
(
ζ¯j′ζj′ + ζ¯jζj
)
zj z¯j′
)
− a
3
2
(
ζ¯jζj zj ζ¯j′ + ζ¯j′ζj′ ζj z¯j′
)
. (4.104)
• Right case:
[
U †(−σ)U(σ−)
](t′′,1)
j
= 1 +
√
2 a z
[
∂xz¯ − ∂yz¯
]
+ a2
[
z
[
∂2xz¯ − 2 ∂x∂yz¯ + ∂2y z¯
]
+
√
2
(
z
[
∂xζ¯ − ∂y ζ¯
]
+ ζ
[
∂xz¯ − ∂yz¯
]) ]
+ a3
[√
2
3 z
[
∂3xz¯ − 3 ∂2x∂yz¯ + 3 ∂x∂2y z¯ − ∂3y z¯
]
+ z
[
∂2xζ¯ − 2 ∂x∂y ζ¯ + ∂2y ζ¯
]
+ ζ
[
∂2xz¯ − 2 ∂x∂yz¯ + ∂2y z¯
]
+
√
2 ζ
[
∂xζ¯ − ∂y ζ¯
]
−√2 ζ¯ζ z
[
∂xz¯ − ∂yz¯
]]
. (4.105)
• Upper case:
[
U †(−σ)U(σ−)
](t′′,2)
j
= 1 +
√
2 a z
[
∂xz¯ + ∂yz¯
]
+ a2
[
z
[
∂2xz¯ + 2 ∂x∂yz¯ + ∂2y z¯
]
79
4. Effective Field Theory for the t-J Model at Low Doping
+
√
2
(
z
[
∂xζ¯ + ∂y ζ¯
]
+ ζ
[
∂xz¯ + ∂yz¯
]) ]
+ a3
[√
2
3 z
[
∂3xz¯ + 3 ∂2x∂yz¯ + 3 ∂x∂2y z¯ + ∂3y z¯
]
+ z
[
∂2xζ¯ + 2 ∂x∂y ζ¯ + ∂2y ζ¯
]
+ ζ
[
∂2xz¯ + 2 ∂x∂yz¯ + ∂2y z¯
]
+
√
2 ζ
[
∂xζ¯ + ∂y ζ¯
]
−√2 ζ¯ζ z
[
∂xz¯ + ∂yz¯
]]
. (4.106)
• Left case:
[
U †(−σ)U(σ−)
](t′′,3)
j
= 1−√2 a z
[
∂xz¯ − ∂yz¯
]
+ a2
[
z
[
∂2xz¯ − 2 ∂x∂yz¯ + ∂2y z¯
]
−√2
(
z
[
∂xζ¯ − ∂y ζ¯
]
+ ζ
[
∂xz¯ − ∂yz¯
]) ]
+ a3
[
−
√
2
3 z
[
∂3xz¯ − 3 ∂2x∂yz¯ + 3 ∂x∂2y z¯ − ∂3y z¯
]
+ z
[
∂2xζ¯ − 2 ∂x∂y ζ¯ + ∂2y ζ¯
]
+ ζ
[
∂2xz¯ − 2 ∂x∂yz¯ + ∂2y z¯
]
−√2 ζ
[
∂xζ¯ − ∂y ζ¯
]
+
√
2 ζ¯ζ z
[
∂xz¯ − ∂yz¯
]]
. (4.107)
• Lower case:
[
U †(−σ)U(σ−)
](t′′,4)
j
= 1−√2 a z
[
∂xz¯ + ∂yz¯
]
+ a2
[
z
[
∂2xz¯ + 2 ∂x∂yz¯ + ∂2y z¯
]
−√2
(
z
[
∂xζ¯ + ∂y ζ¯
]
+ ζ
[
∂xz¯ + ∂yz¯
]) ]
+ a3
[
−
√
2
3 z
[
∂3xz¯ + 3 ∂2x∂yz¯ + 3 ∂x∂2y z¯ + ∂3y z¯
]
+ z
[
∂2xζ¯ + 2 ∂x∂y ζ¯ + ∂2y ζ¯
]
+ ζ
[
∂2xz¯ + 2 ∂x∂yz¯ + ∂2y z¯
]
−√2 ζ
[
∂xζ¯ + ∂y ζ¯
]
+
√
2 ζ¯ζ z
[
∂xz¯ + ∂yz¯
]]
, (4.108)
such that
Result 7 (Second-nearest neighbour hopping along the principal axis)
K(t
′′) = t′′
∑
j
(
χ∗jA
4∑
a=1
[
U †(+σ)Uσ−
](t′′,a)
j
χjA + χ∗jB
4∑
a=1
[
U †(−σ)Uσ+
](t′′,a)
j
χjB
)
. (4.109)
80
4.3. Rotating frame and staggered CP1-representation
Fermion-dressed spin interaction:
The interaction term in (4.18), featuring fermionic as well as spin degrees of freedom,
reads
F =
∑
<i, k>
σ
ψ∗iσψiσ~Ωi~Ωk =
∑
i even
χ∗iAχiA
∑
k nn.i
~Ωi~Ωk +
∑
i odd
χ∗iBχiB
∑
k nn.i
~Ωi~Ωk . (4.110)
the occurring terms were already obtained in Section 4.3.2, see (4.59) and (4.61).
Result 8 (Fermion-dressed spin interaction)
F =
∑
j
[
χ∗jA (4.59)χjA + χ∗jB (4.61)χjB
]
. (4.111)
4.3.4. Fermionic contribution in k-space
In order to integrate out the fermions, we go over to the field’s Fourier representation
and express all terms as bilinear forms in χ. For the fermionic fields we use
χjA/B(τ) =
√
2
N
1
β
∑
n,~k
e−iνnτ e−i~k·(~xj+~xA/B) χA/B(νn, ~k) , (4.112)
with the inverse transformation
χA/B(νn, ~k) =
√
2
N
∫
dτ eiνnτ
∑
j
ei~k·(~xj+~xA/B) χjA/B(τ) . (4.113)
Note that depending on whether ~x points to an even or an odd site, we have to insert
the corresponding expansion vectors pointing to the specific site in a given cell j
~xA =
 0
0
 ~xB =
√
2a
2
 1
−1
 . (4.114)
We are working in imaginary time, so the fermionic Green’s function is anti-periodic in
τ and the Matsubara frequencies are given by νn = (2n+ 1)pi/β. For the bosonic z- and
81
4. Effective Field Theory for the t-J Model at Low Doping
ζ-fields we take
zj(τ) =
√
2
N
1
β
∑
m,~q
e−iωmτ e−i ~q ~xj z(ωm, ~q) (4.115)
and
z(ωm, ~q) =
√
2
N
∫
dτ eiωmτ
∑
j
ei ~q ~xj zj(τ) , (4.116)
and similarly for the ζ-fields. We write ~q for the wave vector and ωm = 2mpi/β for the
Matsubara frequencies in order to distinguish them from the fermionic variables.
Temporal derivative
From (4.72) we know that
T =
∫
dτ
∑
j
χ∗j
[
∂τ + iσzAjτ − aCjτ + a2σzLjτ
]
χj . (4.117)
First order in a: Remember that every temporal derivative counts as an additional
a. So in first order we have
T(1) =
∫
dτ
∑
j
(
χ∗jA ∂τχjA + χ∗jB ∂τχjB + χ∗jAz¯ ∂τz χjA − χ∗jB z¯ ∂τz χjB
)
. (4.118)
Now we implement the Fourier transformation for χ. For the first summand we get
∫
dτ
∑
j
χ∗jA ∂τχjA
= −2i
Nβ2
∑
n,~k
n′, ~k′
χ∗A(~k, νn) νn′
(∫
dτ ei(νn−νn′ )τ
∑
j
ei(~k−~k′)·(~xj+~xA)
)
χA(~k′, νn′) , (4.119)
and with
1
β
∫
dτ ei(νn−νn′ )τ = δνn,νn′ and
2
N
∑
j
ei(~k−~k′)·~xj = δ~k,~k′ (4.120)
we get
∫
dτ
∑
j
χ∗jA ∂τχjA = −
i
β
∑
n,~k
n′, ~k′
νn δ~k,~k′ δνn,νn′χ
∗
A(~k, νn)χA(~k′, νn′) . (4.121)
82
4.3. Rotating frame and staggered CP1-representation
So, the first two summands in (4.118) read
∫
dτ
∑
j
χ∗j ∂τχj =
∑
k,k′
χ∗(k)
(
− i νn
β
δk,k′
)
χ(k′) , (4.122)
where we introduced the shorthand notation k for (νn, ~k). This term is diagonal in k
and thus contributes to the fermionic free Green’s function.
Now we examine the last two summands in (4.118). The term on sublattice A is given
by:
∫
dτ
∑
j
χ∗jAz¯ ∂τz χjA
= 2
Nβ2
∑
n,~k
n′, ~k′
χ∗A(~k, νn)
∫
dτ ei(νn−νn′ )τ
∑
j
ei(~k−~k′)·(~xj+~xA) z¯ ∂τz χA(~k′, νn′) . (4.123)
The z-term in between reads in Fourier-space:
z¯ ∂τz =
−2i
Nβ2
∑
m, ~q
m′, ~q′
ei(ωm−ωm′ )τ ei(~q−~q′)·~xj z¯(ωm, ~q)ω′m z(ωm′ , ~q′) . (4.124)
Inserting this into (4.123) and noting that
1
β
∫
dτ ei(νn−νn′+ωm−ωm′ )τ = δωm′ ,νn−νn′+ωm ,
2
N
∑
j
ei(~k−~k′+~q−~q′)·~xj = δ~q′,~k−~k′+~q , (4.125)
we obtain
(4.123) = −2i
Nβ3
ei(~k−~k′)·~xA
∑
n,~k
n′, ~k′
χ∗A(~k, νn)
[∑
m,~q
z¯(ωm, ~q) ·
(
ωm + νn − νn′
)
· z(ωm + νn − νn′ , ~q + ~k − ~k′)
]
χA(νn′ , ~k′) . (4.126)
The term on sublattice B is calculated similarly, so that for (4.118) we get
83
4. Effective Field Theory for the t-J Model at Low Doping
Result 9 (Temporal Derivative in O(a))
T(1) =
∑
k,k′
χ∗(k)
[
− i νn
β
δk,k′1 + Σ(1,τ)0 1 + Σ(1,τ)z σz
]
χ(k′) , (4.127)
where
Σ(1,τ)0 =
1
2
(
ei(~k−~k′)·~xA − ei(~k−~k′)·~xB
)
Σ˜(1,τ)(k, k′) ,
Σ(1,τ)z =
1
2
(
ei(~k−~k′)·~xA + ei(~k−~k′)·~xB
)
Σ˜(1,τ)(k, k′) (4.128)
and
Σ˜(1,τ)(k, k′) = − 2i
Nβ3
∑
m,~q
(
ωm + νn− νn′
)
· z¯(ωm, ~q)z(ωm + νn− νn′ , ~q+~k−~k′) . (4.129)
Second order in a: Here we have
T(2) = −a ·
∫
dτ
∑
j
χ∗j Cjτ χj = −
∫
dτ
∑
j
χ∗j
[
z¯j∂τζj + ζ¯j∂τzj
]
χj . (4.130)
Taking the Fourier-transform of the χ-fields we get
(4.130) = − 2a
Nβ2
∑
n,~k
n′, ~k′
(
χ∗A(νn, ~k)
∫
dτ ei(νn−νn′ )τ
∑
j
ei(~k−~k′)·(~xj+~xA)
·
[
z¯j∂τζj + ζ¯j∂τzj
]
χA(νn′ , ~k′) + χB-part
)
. (4.131)
Fourier-transformation of the z- and ζ-fields gives
z¯j∂τζj + ζ¯j∂τzj =
2
Nβ2
∑
m,~q
z¯(ωm, ~q)eiωmτei~q~xj
∂τ
 ∑
m′,~q′
ζ(ωm′ , ~q′)e−iωm′τe−i~q
′~xj

+ 2
Nβ2
∑
m,~q
ζ¯(ωm, ~q)eiωmτei~q~xj
∂τ
 ∑
m′,~q′
z(ωm′ , ~q′)e−iωm′τe−i~q
′~xj
 , (4.132)
so that, after summing over j, τ and ~q′, ω′, we end up with
84
4.3. Rotating frame and staggered CP1-representation
Result 10 (Temporal Derivative in O(a2))
T(2) =
∑
k,k′
χ∗(k)
(
Σ(2,τ)0 1+ Σ(2,τ)z σz
)
χ(k′) , (4.133)
where
Σ(2,τ)0 =
1
2
(
ei(~k−~k′)·~xA + ei(~k−~k′)·~xB
)
Σ˜(2,τ)(k, k′) ,
Σ(2,τ)z =
1
2
(
ei(~k−~k′)·~xA − ei(~k−~k′)·~xB
)
Σ˜(2,τ)(k, k′) (4.134)
and
Σ˜(2,τ)(k, k′) = 2ai
Nβ3
∑
m,~q
(
ωm + νn − νn′
)[
z¯(ωm, ~q)ζ(ωm + νn − νn′ , ~q + ~k − ~k′)
+ ζ¯(ωm, ~q)z(ωm + νn − νn′ , ~q + ~k − ~k′)
]
.
Third order in a: Here we have
T(3) = a2
∫
dτ
∑
j
χ∗j σ
zLjτ χj
= a2
∫
dτ
∑
j
χ∗j
[1
2
(
ζ¯j∂τζj − ζj∂τ ζ¯j
)
− ζ¯ζ
(
z¯j∂τzj
)]
σz χj . (4.135)
And after taking the Fourier-transform of the χ-fields we get
= 2a
2
Nβ2
∑
n,~k
n′, ~k′
χ∗A(νn, ~k)
∫
dτ ei(νn−νn′ )τ
∑
j
ei(~k−~k′)·(~xj+~xA)
[1
2
(
ζ¯j∂τζj − ζj∂τ ζ¯j
)
− ζ¯ζ
(
z¯j∂τzj
)]
χA(νn′ , ~k′)− χB-part . (4.136)
We do not transform the z- and ζ-fields, since it will not be necessary for the following
analysis. In the end we have
85
4. Effective Field Theory for the t-J Model at Low Doping
Result 11 (Temporal Derivative in O(a3))
T(3) =
∑
k,k′
χ∗(k)
(
Σ(3,τ)0 1+ Σ(3,τ)z σz
)
χ(k′) , (4.137)
where
Σ(3,τ)0 =
1
2
(
ei(~k−~k′)·~xA − ei(~k−~k′)·~xB
)
Σ˜(3,τ)(k, k′) ,
Σ(3,τ)z =
1
2
(
ei(~k−~k′)·~xA + ei(~k−~k′)·~xB
)
Σ˜(3,τ)(k, k′) (4.138)
and
Σ˜(3,τ)(k, k′) = 2a
2
Nβ2
∫
dτ ei(νn−νn′ )τ
∑
j
ei(~k−~k′)·~xj
[1
2
(
ζ¯j∂τζj − ζj∂τ ζ¯j
)
− ζ¯ζ
(
z¯j∂τzj
)]
.
(4.139)
Next neighbour term
There are eight different contributions, coming from (4.77)-(4.80) and (4.82)-(4.85).
When we write χjA(B) in terms of its Fourier representation, we need to be careful
with the expansion vector xj′ , which differs for the respective next neighbours, see Table
4.1. Let us also define two vectors pointing to the neighbouring cell in the m and l
direction, see Figure 4.3
~x1 =
√
2a (1, 0) , ~x2 =
√
2a (0, 1) . (4.140)
All next neighbour contributions are non-diagonal and we only get contributions to the
self-energy part of the fermionic Green’s function.
K(t) =
∑
k,k′
χ∗(k)
(
Σ(t)+ (k, k′)σ+ + Σ
(t)
− (k, k′)σ−
)
χ(k′) . (4.141)
First order in a: We look at the first order terms proportional to σ+ in (4.141).
1. Right neighbour
2t
Nβ2
ei (~k~xA−~k′~xB)
∫
dτ ei (νn−νn′ )τ ·∑
j
ei(~k−~k′)·~xj ·
[
− 2a
(
z∗j1ζ
∗
j2 − z∗j2ζ∗j1
)]
. (4.142)
86
4.3. Rotating frame and staggered CP1-representation
After implementing the Fourier-Transform for the z-fields we get
= −8at
N2β4
ei (~k~xA−~k′~xB)
∫
dτ ei (νn−νn′ )τ
∑
j
ei(~k−~k′)·~xj
· ∑
m, ~q
m′, ~q′
ei(~q+~q′)~xj ei(ωm+ω′m)τ
(
z∗1(ωm, ~q)ζ∗2 (ωm′ , ~q′)− z∗2(ωm, ~q)ζ∗1 (ωm′ , ~q′)
)
= − 4ta
Nβ3
ei (~k~xA−~k′~xB)
∑
m,~q
(
z∗1(ωm, ~q)ζ∗2 (νn′ − νn − ωm, ~k′ − ~k − ~q)
− z∗2(ωm, ~q)ζ∗1 (νn′ − νn − ωm, ~k′ − ~k − ~q)
)
. (4.143)
2. Upper neighbour
2t
Nβ2
ei (~k~xA−~k′~xB) e−i~k′~x2
∫
dτ ei (νn−νn′ )τ ·∑
j
ei(~k−~k′)·~xj
·
[
−√2a
(
z∗j1∂yz
∗
j2 − z∗j2∂yz∗j1
)
− 2a
(
z∗j1ζ
∗
j2 − z∗j2ζ∗j1
)]
= − 4ta
Nβ3
ei (~k~xA−~k′~xB) e−i~k′~x2
∑
m,~q
[
z∗1(ωm, ~q)ζ∗2 (νn′ − νn − ωm, ~k′ − ~k − ~q)
− z∗2(ωm, ~q)ζ∗1 (νn′ − νn − ωm, ~k′ − ~k − ~q)
+ i√
2
[
k′y − ky − qy
][
z∗1(ωm, ~q)z∗2(νn′ − νn − ωm, ~k′ − ~k − ~q)
− z∗2(ωm, ~q)z∗1(νn′ − νn − ωm, ~k′ − ~k − ~q)
]]
. (4.144)
3. Left neighbour
2t
Nβ2
ei (~k~xA−~k′~xB) ei~k′(~x1−~x2)
∫
dτ ei (νn−νn′ )τ ·∑
j
ei(~k−~k′)·~xj
·
[√
2a
(
z∗j1(∂x − ∂y)z∗j2 − z∗j2(∂x − ∂y)z∗j1
)
− 2a
(
z∗j1ζ
∗
j2 − z∗j2ζ∗j1
)]
= − 4ta
Nβ3
ei (~k~xA−~k′~xB) ei~k′(~x1−~x2)
∑
m,~q
[
z∗1(ωm, ~q)ζ∗2 (νn′ − νn − ωm, ~k′ − ~k − ~q)
− z∗2(ωm, ~q)ζ∗1 (νn′ − νn − ωm, ~k′ − ~k − ~q)
− i√
2
[(
k′x − k′y
)
−
(
kx − ky
)
−
(
qx − qy
)]
·
[
z∗1(ωm, ~q)z∗2(νn′ − νn − ωm, ~k′ − ~k − ~q)
87
4. Effective Field Theory for the t-J Model at Low Doping
− z∗2(ωm, ~q)z∗1(νn′ − νn − ωm, ~k′ − ~k − ~q)
]]
. (4.145)
4. Lower neighbour
2t
Nβ2
ei (~k~xA−~k′~xB) ei~k′~x1
∫
dτ ei (νn−νn′ )τ ·∑
j
ei(~k−~k′)·~xj
·
[√
2a
(
z∗j1∂xz
∗
j2 − z∗j2∂xz∗j1
)
− 2a
(
z∗j1ζ
∗
j2 − z∗j2ζ∗j1
)]
= − 4ta
Nβ3
ei (~k~xA−~k′~xB) ei~k′~x1
∑
m,~q
[
z∗1(ωm, ~q)ζ∗2 (νn′ − νn − ωm, ~k′ − ~k − ~q)
− z∗2(ωm, ~q)ζ∗1 (νn′ − νn − ωm, ~k′ − ~k − ~q)
− i√
2
[
k′x − kx − qx
][
z∗1(ωm, ~q)z∗2(νn′ − νn − ωm, ~k′ − ~k − ~q)
− z∗2(ωm, ~q)z∗1(νn′ − νn − ωm, ~k′ − ~k − ~q)
]]
. (4.146)
We sum up the terms leading to
Σ(1,t)+ = −
8ta
Nβ3
ei (~k~xA−~k′~xB)ei
√
2a
2 (k
′
x−k′y)
∑
m,~q
[
2 cos
(√2a k′x
2
)
cos
(√2 a k′y
2
)
·
(
z∗1(ωm, ~q) ζ∗2 (νn′ − νn − ωm, ~k′ − ~k − ~q)
− z∗2(ωm, ~q) ζ∗1 (νn′ − νn − ωm, ~k′ − ~k − ~q)
)
+ i√
2
(
e−i
√
2a
2 k
′
y cos
(√2 a k′x
2
)(
k′y − ky − qy
)
− ei
√
2a
2 k
′
x cos
(√2 a k′y
2
)(
k′x − kx − qx
))
·
(
z∗1(ωm, ~q) z∗2(νn′ − νn − ωm, ~k′ − ~k − ~q)
− z∗2(ωm, ~q) z∗1(νn′ − νn − ωm, ~k′ − ~k − ~q)
)]
. (4.147)
Next we consider the first order terms proportional to σ−. Deriving the right, upper, left
and lower neighbour terms as we did in the last paragraph, leads to a total contribution
Σ(1,t)− = −
8ta
Nβ3
ei (~k~xB−~k′~xA)e−i
√
2a
2 (k
′
x−k′y)
∑
m,~q
[
2 cos
(√2a k′x
2
)
cos
(√2 a k′y
2
)
88
4.3. Rotating frame and staggered CP1-representation
·
(
z1(ωm, ~q) ζ2(νn − νn′ − ωm, ~k − ~k′ − ~q)
− z2(ωm, ~q) ζ1(νn − νn′ − ωm, ~k − ~k′ − ~q)
)
− i√
2
(
ei
√
2a
2 k
′
y cos
(√2 a k′x
2
)(
ky − k′y − qy
)
− e−i
√
2a
2 k
′
x cos
(√2 a k′y
2
)(
kx − k′x − qx
))
·
(
z1(ωm, ~q) z2(νn − νn′ − ωm, ~k − ~k′ − ~q)
− z2(ωm, ~q) z1(νn − νn′ − ωm, ~k − ~k′ − ~q)
)]
. (4.148)
Finally we have to combine (4.147) and (4.148) and obtain
Result 12 (Next neighbour contribution in O(a))
K(t,1) =
∑
k,k′
χ∗(k)
(
Σ(1,t)+ (k, k′)σ+ + Σ
(1,t)
− σ
−(k, k′)
)
χ(k′) , (4.149)
where Σ(1,t)+/− is given by (4.147) and (4.148), respectively.
Second order in a: First we look for the second order terms proportional to σ+ in
result (4.141) and call this contribution Σ(2,t)+ . Once more, we consider the right, upper,
left and lower neighbour, sum up their contributions and obtain
Σ(2,t)+ =
4ta2
Nβ3
ei (~k~xA−~k′~xB)ei
√
2a(k′x−k′y)
∑
m,~q
[
ei
√
2a
2 k
′
y cos
(√2a k′y
2
)(
k′x − kx − qx
)2
−
(
k′x − kx − qx
)(
k′y − ky − qy
)
+ e−i
√
2a
2 k
′
x cos
(√2a k′x
2
)(
k′y − ky − qy
)2]
·
[
z∗1(ωm, ~q) z∗2(νn′ − νn − ωm, ~k′ − ~k − ~q)
− z∗2(ωm, ~q) z∗1(νn′ − νn − ωm, ~k′ − ~k − ~q)
]
. (4.150)
Then we look at the second order terms proportional to σ− in (4.141) and call this
contribution Σ(2,t)− . We obtain
89
4. Effective Field Theory for the t-J Model at Low Doping
Σ(2,t)− = −
4ta2
Nβ3
ei (~k~xB−~k′~xA)e−i
√
2a(k′x−k′y)
∑
m,~q
[
e−i
√
2a
2 k
′
y cos
(√2a k′y
2
)(
kx − k′x − qx
)2
−
(
kx − k′x − qx
)(
ky − k′y − qy
)
+ ei
√
2a
2 k
′
x cos
(√2a k′x
2
)(
ky − k′y − qy
)2]
·
[
z1(ωm, ~q) z2(νn − νn′ − ωm, ~k − ~k′ − ~q)
− z2(ωm, ~q) z1(νn − νn′ − ωm, ~k − ~k′ − ~q)
]
. (4.151)
Then we combine (4.150) and (4.151) to get the O(a2)-contribution for the next neigh-
bours.
Result 13 (Next neighbour self-energy contribution in O(a2))
K(t,2) =
∑
k,k′
χ∗(k)
(
Σ(2,t)+ (k, k′)σ+ + Σ
(2,t)
− (k, k′)σ−
)
χ(k′) , (4.152)
where Σ(2,t)+/− is given by (4.150) and (4.151), respectively.
Third order in a: We will see in Section 4.4 that the O(a3)-term does not contribute
to the effective action.
2nd-nearest neighbours along the diagonal
From (4.89)-(4.92) and (4.93)-(4.96) we get eight different contributions. Again, we
express the fields in Fourier space. Since this 2nd-nearest neighbour term contains
diagonal as well as off-diagonal parts in k, there is are contributions to the free Green’s
functions well.
K(t
′) =
∑
k,k′
χ∗(k)
(
1(~k)
β
δkk′1+ Σ(t
′)
A (k, k′)γ+ + Σ
(t′)
B (k, k′)γ−
)
χ(~k′) , (4.153)
where we defined
1(~k) = 2t′
[
cos(
√
2akx) + cos(
√
2aky)
]
, (4.154)
and γ± = (1± σz)/2, so that χA is multiplied by χA and χB by χB.
90
4.3. Rotating frame and staggered CP1-representation
First order in a: First we examine the part proportional to γ+ and call it Σ(1,t
′)
A . It
consists of
1. Upper-right neighbour
2t′
Nβ2
ei (~k−~k′)~xA e−i~k′~x1
∫
dτ ei (νn−νn′ )τ ·∑
j
ei(~k−~k′)·~xj ·
[√
2a z¯j∂xzj
]
= −2
√
2t′a i
Nβ3
ei (~k−~k′)~xA e−i~k′~x1
∑
m,~q
z¯(ωm − νn + νn′ , ~q − ~k + ~k′) · qx · z(ωm, ~q) . (4.155)
2. Upper-left neighbour
2t′
Nβ2
ei (~k−~k′)~xA e−i~k′~x2
∫
dτ ei (νn−νn′ )τ ·∑
j
ei(~k−~k′)·~xj ·
[√
2a z¯j∂yzj
]
= −2
√
2t′a i
Nβ3
ei (~k−~k′)~xA e−i~k′~x2
∑
m,~q
z¯(ωm − νn + νn′ , ~q − ~k + ~k′) · qy · z(ωm, ~q) . (4.156)
3. Lower-left neighbour
2t′
Nβ2
ei (~k−~k′)~xA ei~k′~x1
∫
dτ ei (νn−νn′ )τ ·∑
j
ei(~k−~k′)·~xj ·
[
−√2a z¯j∂xzj
]
= 2
√
2t′a i
Nβ3
ei (~k−~k′)~xA ei~k′~x1
∑
m,~q
z¯(ωm − νn + νn′ , ~q − ~k + ~k′) · qx · z(ωm, ~q) . (4.157)
4. Lower-right neighbour
2t′
Nβ2
ei (~k−~k′)~xA ei~k′~x2
∫
dτ ei (νn−νn′ )τ ·∑
j
ei(~k−~k′)·~xj ·
[
−√2a z¯j∂yzj
]
= 2
√
2t′a i
Nβ3
ei (~k−~k′)~xA ei~k′~x2
∑
m,~q
z¯(ωm − νn + νn′ , ~q − ~k + ~k′) · qy · z(ωm, ~q) . (4.158)
In the end we have to sum up the terms which leads to
Σ(1,t
′)
A = −
4
√
2t′a
Nβ3
ei (~k−~k′)~xA
∑
m,~q
[
sin(
√
2 k′x a) · qx + sin(
√
2 k′y a) · qy
]
· z¯(ωm − νn + νn′ , ~q − ~k + ~k′)z(ωm, ~q) . (4.159)
91
4. Effective Field Theory for the t-J Model at Low Doping
Next we consider the terms proportional to γ− in (4.153) and call this contribution
Σ(1,t
′)
B . The sum over all neighbours eventually reads
Σ(1,t
′)
B =
4
√
2t′a
Nβ3
ei (~k−~k′)~xB
∑
m,~q
[
sin(
√
2 k′x a) · qx + sin(
√
2 k′y a) · qy
]
· z¯(ωm − νn + νn′ , ~q) · z(ωm, ~q + ~k − ~k′) . (4.160)
So in the end, we have to combine (4.159) and (4.160) and get
Result 14 (2nd-nearest neighbour t′ self-energy contribution in O(a))
K(t
′,1) =
∑
k,k′
χ∗(k)
(
Σ(1,t
′)
0 1+ Σ(1,t
′)
z σ
z
)
χ(k′) , (4.161)
where
Σ(1,t
′)
0 =
1
2
(
Σ(1,t
′)
A + Σ
(1,t′)
B
)
, (4.162)
Σ(1,t′)z =
1
2
(
Σ(1,t
′)
A − Σ(1,t
′)
B
)
. (4.163)
Second order in a: Here we look for the second order terms proportional to γ+ in
result (4.153) and call this contribution Σ(2,t
′)
A . The sum over upper-right, upper-left,
lower-left and lower-right neighbour gives
Σ(2,t
′)
A = −
4t′a2
Nβ3
ei (~k−~k′)~xA
∑
m,~q
([
cos(
√
2a k′x) q2x + cos(
√
2a k′y) q2y
]
· z¯(ωm + νn′ − νn, ~q + ~k′ − ~k) z(ωm, ~q)
−√2
[
sin(
√
2a k′x) qx + sin(
√
2a k′y) qy
][
z¯(ωm + νn′ − νn, ~q + ~k′ − ~k) ζ(ωm, ~q)
+ ζ¯(ωm + νn′ − νn, ~q + ~k′ − ~k) z(ωm, ~q)
])
. (4.164)
Then the second order terms proportional to γ− in (4.153)
Σ(2,t
′)
B = −
4t′a2
Nβ3
ei (~k−~k′)~xB
∑
m,~q
([
cos(
√
2a k′x)q2x + cos(
√
2a k′y)q2y
]
92
4.3. Rotating frame and staggered CP1-representation
· z¯(ωm, ~q) z(ωm + νn − νn′ , ~q + ~k − ~k′)
−√2
[
sin(
√
2a k′x)qx + sin(
√
2a k′y)qy
][
z¯(ωm, ~q) ζ(ωm + νn − νn′ , ~q + ~k − ~k′)
+ ζ¯(ωm, ~q) z(ωm + νn − νn′ , ~q + ~k − ~k′)
])
. (4.165)
And when we combine (4.164) and (4.165) we obtain
Result 15 (2nd-nearest neighbour t′ self-energy contribution in O(a2))
K(t
′,2) =
∑
k,k′
χ∗(k)
(
Σ(2,t
′)
0 1+ Σ(2,t
′)
z σ
z
)
χ(k′) , (4.166)
where Σ(2,t
′)
A/B is given by (4.164) and (4.165) and
Σ(2,t
′)
0 =
1
2
(
Σ(2,t
′)
A + Σ
(2,t′)
B
)
, (4.167)
Σ(2,t′)z =
1
2
(
Σ(2,t
′)
A − Σ(2,t
′)
B
)
. (4.168)
Third order in a: Finally, we extract the 3rd-order self-energy contribution Σ(3,t′)
from the 2nd-nearest neighbour hopping along the diagonal. We will see in Section 4.4
that for the 3rd-order terms, we only need to express the χ-fields in k-space. So first, we
look for the terms proportional to γ+ in result (4.153) and call this contribution Σ(3,t
′)
A .
With the definitions
ux/y :=
√
2
3 z¯j ∂
3
x/yzj +
√
2 ζ¯j ∂x/yζj −
√
2 ζ¯jζj z¯j ∂x/yzj ,
vx/y := −(z¯j ∂2x/yζj + ζ¯j ∂2x/yzj) , (4.169)
it reads
Σ(3,t
′)
A = −
4t′a3
Nβ2
ei (~k−~k′)~xA
∫
dτ ei (νn−νn′ )τ ·∑
j
ei(~k−~k′)·~xj ·
[
− cos(√2a k′x) · vx
+ i sin(
√
2a k′x) · ux − cos(
√
2a k′y) · vy + i sin(
√
2a k′y) · uy
]
. (4.170)
93
4. Effective Field Theory for the t-J Model at Low Doping
Then we look at the terms proportional to γ− in result (4.153) and call this contribution
Σ(3,t
′)
A . Here we get
Σ(3,t
′)
B = −
4t′a3
Nβ2
ei (~k−~k′)~xB
∫
dτ ei (νn−νn′ )τ ·∑
j
ei(~k−~k′)·~xj ·
[
cos(
√
2a k′x) · v∗x
+ i sin(
√
2a k′x) · u∗x + cos(
√
2a k′y) · v∗y + i sin(
√
2a k′y) · u∗y
]
. (4.171)
Finally, we have for the third order contribution
Result 16 (2nd-nearest neighbours t′ self-energy contribution in O(a3))
K(t
′,3) =
∑
k,k′
χ∗(k)
(
Σ(3,t
′)
0 1+ Σ(3,t
′)
z σ
z
)
χ(k′) , (4.172)
where
Σ(3,t
′)
0 =
1
2
(
Σ(3,t
′)
A + Σ
(3,t′)
B
)
, (4.173)
Σ(3,t′)z =
1
2
(
Σ(3,t
′)
A − Σ(3,t
′)
B
)
. (4.174)
2nd-nearest neighbours along the principal axis
In this case, the eight contribution from the 2nd-nearest neighbours are given in (4.100)-
(4.103) and (4.105)-(4.108). Once more, there are parts diagonal and off-diagonal in k
so we get contributions to the free Green’s function as well as the self-energy.
K(t
′′) =
∑
k,k′
χ∗(k)
(
2(~k)
β
δkk′1 + Σ(t
′′)
A (k, k′) γ+ + Σ
(t′′)
B (k, k′) γ−
)
χ(k′) , (4.175)
where we defined
2(~k) = 4t′′ cos(
√
2akx) cos(
√
2aky) . (4.176)
First order in a: We consider the first order terms proportional to γ+ in (4.175) and
call this contribution Σ(1,t
′′)
A . Summing over the contributions from right, upper, left and
lower neighbour, we get
94
4.3. Rotating frame and staggered CP1-representation
Σ(1,t
′′)
A = −
4
√
2t′′a
Nβ3
ei (~k−~k′)~xA
∑
m,~q
[
sin
[√
2a (k′x + k′y)
](
qx + qy
)
+ sin
[√
2a(k′x − k′y)
](
qx − qy
)]
· z¯(ωm − νn + νn′ , ~q − ~k + ~k′)z(ωm, ~q) . (4.177)
Then we look at the part proportional to γ− and call this contribution Σ(1,t
′′)
B .
Σ(1,t
′′)
B =
4
√
2t′′a
Nβ3
ei (~k−~k′)~xB
∑
m,~q
[
sin
[√
2a (k′x + k′y)
](
qx + qy
)
+ sin
[√
2a(k′x − k′y)
](
qx − qy
)]
· z¯(ωm − νn + νn′ , ~q)z(ωm, ~q + ~k − ~k′) . (4.178)
We combine (4.177) and (4.178) to obtain
Result 17 (2nd-nearest neighbours t′′ self-energy contribution in O(a))
K(t
′′,1) =
∑
k,k′
χ∗(k)
(
Σ(1,t
′′)
0 1 + Σ(1,t
′′)
z σ
z
)
χ(k′) , (4.179)
where
Σ(1,t
′′)
0 =
1
2
(
Σ(1,t
′′)
A + Σ
(1,t′′)
B
)
, (4.180)
Σ(1,t′′)z =
1
2
(
Σ(1,t
′′)
A − Σ(1,t
′′)
B
)
. (4.181)
Second order in a: The part proportional to γ+ in result (4.175) evaluates to
Σ(2,t
′′)
A = −
4t′′a2
Nβ3
ei (~k−~k′)~xA
∑
m,~q
([
cos(
√
2a [k′x + k′y]) (qx + qy)2
+ cos(
√
2a [k′x − k′y]) (qx − qy)2
]
· z¯(ωm + νn′ − νn, ~q + ~k′ − ~k) z(ωm, ~q)
−√2
[
sin(
√
2a [k′x + k′y]) (qx + qy) + sin(
√
2a [k′x − k′y]) (qx − qy)
]
·
[
z¯(ωm + νn′ − νn, ~q + ~k′ − ~k) ζ(ωm, ~q)
+ ζ¯(ωm + νn′ − νn, ~q + ~k′ − ~k) z(ωm, ~q)
])
, (4.182)
95
4. Effective Field Theory for the t-J Model at Low Doping
and the one proportional to γ− in (4.175) gives
Σ(2,t
′′)
B = −
4t′′a2
Nβ3
ei (~k−~k′)~xB
∑
m,~q
([
cos(
√
2a [k′x + k′y]) (qx + qy)2
+ cos(
√
2a [k′x − k′y]) (qx − qy)2
]
· z¯(ωm, ~q) z(ωm + νn − νn′ , ~q + ~k − ~k′)
−√2
[
sin(
√
2a [k′x + k′y]) (qx + qy) + sin(
√
2a [k′x − k′y]) (qx − qy)
]
·
[
z¯(ωm, ~q) ζ(ωm + νn − νn′ , ~q + ~k − ~k′)
+ ζ¯(ωm, ~q) z(ωm + νn − νn′ , ~q + ~k − ~k′)
])
. (4.183)
Such that when we combine (4.182) and (4.183) we obtain
Result 18 (2nd-nearest neighbours t′′ self-energy contribution in O(a2))
K(t
′′,2) =
∑
k,k′
χ∗(k)
(
Σ(2,t
′′)
0 1 + Σ(2,t
′′)
z σ
z
)
χ(k′) , (4.184)
where
Σ(2,t
′′)
0 =
1
2
(
Σ(2,t
′′)
A + Σ
(2,t′′)
B
)
, (4.185)
Σ(2,t′′)z =
1
2
(
Σ(2,t
′′)
A − Σ(2,t
′′)
B
)
. (4.186)
Third order in a: For the upcoming considerations, it is sufficient to express only the
χ-fields in k-space. We define
u± :=
√
2
3 z¯j (∂x ± ∂y)
3zj +
√
2 ζ¯j (∂x ± ∂y)ζj −
√
2 ζ¯jζj z¯j (∂x ± ∂y)zj ,
v± := −(z¯j (∂x ± ∂y)2ζj + ζ¯j (∂x ± ∂y)2zj) , (4.187)
such that the terms proportional to γ+ in result (4.175) are given by
Σ(3,t
′′)
A = −
4t′′a3
Nβ2
ei (~k−~k′)~xA
∫
dτ ei (νn−νn′ )τ ·∑
j
ei(~k−~k′)·~xj
96
4.3. Rotating frame and staggered CP1-representation
·
[
− cos(√2a [k′x + k′y]) · v+ + i sin(
√
2a [k′x + k′y]) · u+
− cos(√2a [k′x − k′y]) · v− + i sin(
√
2a [k′x − k′y]) · u−
]
, (4.188)
and the ones proportional to γ− read
Σ(3,t
′′)
B = −
4t′′a3
Nβ2
ei (~k−~k′)~xB
∫
dτ ei (νn−νn′ )τ ·∑
j
ei(~k−~k′)·~xj
·
[
cos(
√
2a [k′x + k′y]) · v∗+ + i sin(
√
2a [k′x + k′y]) · u∗+
+ cos(
√
2a [k′x − k′y]) · v∗− + i sin(
√
2a [k′x − k′y]) · u∗−
]
. (4.189)
So in the end, we get for the third order contribution
Result 19 (2nd-nearest neighbours t′′ self-energy contribution in O(a3))
K(t
′′,3) =
∑
k,k′
χ∗(k)
(
Σ(3,t
′′)
0 (k, k′) 1 + Σ(3,t
′′)
z (k, k′)σz
)
χ(k′) , (4.190)
where
Σ(3,t
′′)
0 =
1
2
(
Σ(3,t
′′)
A + Σ
(3,t′′)
B
)
, (4.191)
Σ(3,t′′)z =
1
2
(
Σ(3,t
′′)
A − Σ(3,t
′′)
B
)
. (4.192)
Fermion-dressed spin interaction term
The Fermion-dressed spin interaction term is given in (4.111). We separate contributions
diagonal in k from off-diagonal contributions, such that
F =
∑
k,k′
χ∗(k)
(
− J
β
· δk,k′1+ Σ(J)A (k, k′) γ+ + Σ(J)z (k, k′) γ−
)
χ(k′) . (4.193)
So let us sort the contributions in terms of powers of a.
First order in a: There is no contribution in O(a) as one can see from (4.59) and
(4.61).
97
4. Effective Field Theory for the t-J Model at Low Doping
Second order in a: The part proportional to γ+ reads
Σ(2,J)A (k, k′) = −
4Ja2
N2β5
ei(~k−~k′)~xA
∑
q,q′,q′′
2 ·G∗(q − q′ + q′′ − k + k′, q′)G(q′′, q)
+
(
2qxq′x − qxq′y − qyq′x + 2qyq′y
)
· F ∗(q − q′ + q′′ − k + k′, q′)F (q′′, q)
−√2i (q′x − q′y) · F ∗(q − q′ + q′′ − k + k′, q′)G(q′′, q)
+
√
2i (qx − qy) ·G∗(q − q′ + q′′ − k + k′, q′)F (q′′, q) , (4.194)
where
F (q, q′) = z1(ωm, ~q) z2(ωm′ , ~q′)− z2(ωm, ~q) z1(ωm′ , ~q′) , (4.195)
G(q, q′) = z1(ωm, ~q) ζ2(ωm′ , ~q′)− z2(ωm, ~q) ζ1(ωm′ , ~q′) . (4.196)
The expression for the odd lattice sites holds the same result, except for the exponential
factor which in the odd case reads ei(~k−~k′)~xB . So altogether, we have
Result 20 (Fermion-dressed spin interaction in O(a2))
F(2) =
∑
k,k′
χ∗(k)
(
Σ(2,J)0 1 + Σ(2,J)z σz
)
χ(k′) , (4.197)
where
Σ(2,J)0 =
1
2
(
ei(k−k′)~xA + ei(k−k′)~xB
)
Σ˜(2,J)(k, k′) , (4.198)
Σ(2,J)z =
1
2
(
ei(k−k′)~xA − ei(k−k′)~xB
)
Σ˜(2,J)(k, k′) , (4.199)
and
Σ˜(2,J)(k, k′) = − 4Ja
2
N2β5
∑
q,q′,q′′
2 ·G∗(q − q′ + q′′ − k + k′, q′)G(q′′, q)
+
(
2qxq′x − qxq′y − qyq′x + 2qyq′y
)
· F ∗(q − q′ + q′′ − k + k′, q′)F (q′′, q)
−√2i (q′x − q′y)F ∗(q − q′ + q′′ − k + k′, q′)G(q′′, q)
+
√
2i (qx − qy)G∗(q − q′ + q′′ − k + k′, q′)F (q′′, q) , (4.200)
where F (q, q′) and G(q, q′) are given by (4.195) and (4.196), respectively.
98
4.3. Rotating frame and staggered CP1-representation
Third order in a: Once again, we only express the χ-fields in k-space and obtain
Result 21 (Fermion-dressed spin interaction in O(a3))
F(3) =
∑
k,k′
χ∗(k)
(
Σ(3,J)0 1 + Σ(3,J)z σz
)
χ(k′) , (4.201)
where
Σ(3,J)0 =
1
2
(
ei(k−k′)~xA − ei(k−k′)~xB
)
Σ˜(3,J)(k, k′) , (4.202)
Σ(3,J)z =
1
2
(
ei(k−k′)~xA + ei(k−k′)~xB
)
Σ˜(3,J)(k, k′) , (4.203)
and
Σ˜(3,J)(k, k′) = − J
Nβ2
· (4.59)(3) , (4.204)
where by the superscript (3), we denote the third order contributions of (4.59).
Chemical Potential
The chemical potential term simply reads
Result 22 (Chemical potential)
µ
∫
dτ
∑
i,σ
ψ∗iσψiσ = µ
∑
k
1
β
χ∗(νn, ~k)χ(νn, ~k) , (4.205)
thus, it only contributes to the free fermionic Green’s function.
Summary of fermionic contributions
We can now summarize the results and write the fermionic part of the action as
99
4. Effective Field Theory for the t-J Model at Low Doping
Result 23 (Fermionic contributions to the action)
SF = −
∑
k, k′
χ(k)∗
[
G−10 (k, k′)− Σ(k, k′)
]
χ(k′) , (4.206)
with the inverse free Green’s function given by
G−10 (k, k′) =
1
β
(
iνn − [(~k) + J + µ]
)
δk,k′ (4.207)
and a dispersion which is the sum of the diagonal contributions in (4.153) and (4.175):
(~k) = 1(~k)+2(~k) = 2t′ ·
(
cos
√
2akx+cos
√
2aky
)
+4t′′ ·cos√2akx ·cos
√
2aky . (4.208)
The self-energy is given by
Σ(k, k′) = Σ(τ)(k, k′) + Σ(t)(k, k′) + Σ(t′)(k, k′) + Σ(t′′)(k, k′) + Σ(J)(k, k′) , (4.209)
where the different contributions decompose into
Σ(τ)(k, k′) =
(
Σ(1,τ)0 + Σ
(2,τ)
0 + Σ
(3,τ)
0
)
1 +
(
Σ(1,τ)z + Σ(2,τ)z + Σ(3,τ)z
)
σz ,
Σ(t)(k, k′) =
(
Σ(1,t)+ + Σ
(2,t)
+ + Σ
(3,t)
+
)
σ+ +
(
Σ(1,t)− + Σ(2,t)− + Σ(3,t)−
)
σ− ,
Σ(t′)(k, k′) =
(
Σ(1,t
′)
0 + Σ
(2,t′)
0 + Σ
(3,t′)
0
)
1 +
(
Σ(1,t′)z + Σ(2,t
′)
z + Σ(3,t
′)
z
)
σz ,
Σ(t′′)(k, k′) =
(
Σ(1,t
′′)
0 + Σ
(2,t′′)
0 + Σ
(3,t′′)
0
)
1 +
(
Σ(1,t′′)z + Σ(2,t
′′)
z + Σ(3,t
′′)
z
)
σz ,
Σ(J)(k, k′) =
(
Σ(2,J)0 + Σ
(3,J)
0
)
1 +
(
Σ(2,J)z + Σ(3,J)z
)
σz . (4.210)
We will see in the following section that among the third order terms only the ones
proportional to 1 remain. This is reassuring, since contributions from Σ(3,t)+ and Σ(3,t)−
would spoil our analysis, as they contain terms ∼ ζ3 which we would not know how to
integrate out.
Note however that most of the first and second order terms will also contribute in
third order, since the integration over fermionic degrees of freedom and the subsequent
expansion of the logarithm essentially recombine the respective terms.
100
4.4. Integrating out the fermions
4.4. Integrating out the fermions
Up to this point, the partition function reads
Z =
∫
D(z¯, z)D(ζ¯ , ζ) D(χ∗, χ) J2a
(
1+ 32a
2ζ¯ζ
)
δ
(
z¯z−1
)
δ
(
z¯ζ+ ζ¯z
)
e−(SB+SF ) . (4.211)
The fermionic part of the action SF is given in (4.206). After integrating over the
fermionic degrees of freedom [79], it will assume the form
SF → Tr ln
(
G−10 (k, k′)− Σ(k, k′)
)
= Tr lnG−10 + Tr ln
(
1−G0(k, k′′)Σ(k′′, k′)
)
. (4.212)
4.4.1. Evaluating the trace
Tr lnG−10 may be absorbed in the normalization of the path integral, since it contains
no dynamical variables. After expanding the logarithm, we arrive at
SF → −Tr
(
G0Σ +
1
2G0ΣG0Σ +
1
3G0ΣG0ΣG0Σ
)
+O(a4) , (4.213)
where
G0(k, k′) = β · 1
iνn −
(
(~k) + J + µ
) δk,k′ (4.214)
and the selfenergy Σ(k, k′) as given in (4.209). Sorted in terms of powers of a we have
Σ(k, k′) = Σ(1)(k, k′) + Σ(2)(k, k′) + Σ(3)(k, k′) +O(a4) ,
(4.215)
where the different summands are given by
Σ(1)(k, k′) = Σ(1,τ)(k, k′) + Σ(1,t)(k, k′) + Σ(1,t′)(k, k′) + Σ(1,t′′)(k, k′)
= Σ(1)0 (k, k′) 1 + Σ(1)z (k, k′)σz + Σ
(1)
+ (k, k′)σ+ + Σ
(1)
− (k, k′)σ− , (4.216)
Σ(2)(k, k′) = Σ(2,τ)(k, k′) + Σ(2,t)(k, k′) + Σ(2,t′)(k, k′) + Σ(2,t′′)(k, k′) + Σ(2,J)(k, k′)
= Σ(2)0 (k, k′) 1 + Σ(2)z (k, k′)σz + Σ
(2)
+ (k, k′)σ+ + Σ
(2)
− (k, k′)σ− , (4.217)
Σ(3)(k, k′) = Σ(3,τ)(k, k′) + Σ(3,t)(k, k′) + Σ(3,t′)(k, k′) + Σ(3,t′′)(k, k′) + Σ(3,J)(k, k′)
101
4. Effective Field Theory for the t-J Model at Low Doping
= Σ(3)0 (k, k′) 1 + Σ(3)z (k, k′)σz + Σ
(3)
+ (k, k′)σ+ + Σ
(3)
− (k, k′)σ− . (4.218)
If we plug this into the (4.213) and sort with respect to powers of a, we obtain
Result 24 (Contributions to the trace ordered in powers of a)
• 1st order in a
S
(1)
F → Tr
(
G0Σ
)(1)
= 2 · Tr
(
G0Σ(1)0
)
. (4.219)
• 2nd order in a
S
(2)
F → Tr
(
G0Σ +
1
2G0ΣG0Σ
)(2)
= Tr
(
2G0Σ(2)0 +G0Σ
(1)
0 G0Σ
(1)
0 +G0Σ(1)z G0Σ(1)z +G0Σ
(1)
+ G0Σ
(1)
−
)
. (4.220)
• 3rd order in a
S
(3)
F → Tr
(
G0Σ +
1
2G0ΣG0Σ +
1
3G0ΣG0ΣG0Σ
)(3)
= Tr
[
2G0Σ(3)0 + 2
(
G0Σ(1)0 G0Σ
(2)
0 +G0Σ(1)z G0Σ(2)z
)
+
(
G0Σ(1)+ G0Σ
(2)
− +G0Σ(1)− G0Σ(2)+
)
+ 23
(
G0Σ(1)0 G0Σ
(1)
0 G0Σ
(1)
0 + 3G0Σ
(1)
0 G0Σ(1)z G0Σ(1)z
)
+G0Σ(1)0
(
G0Σ(1)+ G0Σ
(1)
− +G0Σ(1)− G0Σ(1)+
)
+G0Σ(1)z
(
G0Σ(1)+ G0Σ
(1)
− −G0Σ(1)− G0Σ(1)+
)]
, (4.221)
with
Σ(1)0 =
(
Σ(1,τ)0 + Σ
(1,t′)
0 + Σ
(1,t′′)
0
)
, Σ(1)z =
(
Σ(1,τ)z + Σ(1,t
′)
z + Σ(1,t
′′)
z
)
,
Σ(2)0 =
(
Σ(2,τ)0 + Σ
(2,t′)
0 + Σ
(2,t′′)
0 + Σ
(2,J)
0
)
, Σ(2)z =
(
Σ(2,τ)z + Σ(2,t
′)
z + Σ(2,t
′′)
z + Σ(2,J)z
)
,
Σ(3)0 =
(
Σ(3,τ)0 + Σ
(3,t′)
0 + Σ
(3,t′′)
0 + Σ
(3,J)
0
)
,
Σ(1)+ = Σ
(1,t)
+ , Σ
(1)
− = Σ(1,t)− ,
Σ(2)+ = Σ
(2,t)
+ , Σ
(2)
− = Σ(2,t)− .
102
4.4. Integrating out the fermions
4.4.2. Contribution in O(a)
As we have seen in the previous paragraph, for the fermionic contribution in O(a) we
get:
(4.219) = 2 · Tr
[
G0
(
Σ(1,τ) + t′ · Σ(1,t′) + t′′ · Σ(1,t′′)
)]
, (4.222)
where the factor of 2 originates from the trace over 1, see (4.216).
1. Σ(1,τ)0 : As can be seen from (4.128) the trace over Σ
(1,τ)
0 gives a vanishing contribution
due to the exponential factors.
2. Σ(1,t
′)
0 : The trace over (4.162) disappears as well.
3. Σ(1,t
′′)
0 : and the same with the next term, as can be seen from (4.180).
Note that G0 is diagonal, so if Σ has vanishing diagonal elements we also have Tr(G0Σ) =
0. So in the end, there is no contribution to the selfenergy in O(a).
4.4.3. Contribution in O(a2)
Contribution with one Green’s function
We first look at the simple contributions to (4.220), the summands with one Green’s
function:
2 · Tr
(
G0Σ(2)0
)
= 2 · Tr
(
G0
(
Σ(2,τ)0 + Σ
(2,t′)
0 + Σ
(2,t′′)
0 + Σ
(2,J)
0
))
. (4.223)
1. For the first summand we have:
Tr
[
G0Σ(2,τ)0
]
=
∑
n,~k
n′, ~k′
G0(~k,~k′)Σ(2,τ)0 (~k′, ~k)
= β
∑
n,~k
n′, ~k′
1
iνn −
(
(~k) + J + µ
) δk,k′δn,n′Σ(2,τ)0 (~k′, ~k)
= − 2a
Nβ
∑
n,~k
1
iνn −
(
(~k) + J + µ
) ∫ dτ∑
j
(
z¯j(τ) ∂τζj(τ) + ζ¯j(τ) ∂τzj(τ)
)
,(4.224)
where for Σ(2,τ)0 (~k′, ~k), we plugged in the real-space representation from (4.136).
103
4. Effective Field Theory for the t-J Model at Low Doping
The sum over Matsubara frequencies νn can be performed (see Appendix A.3). It
yields the Fermi-Dirac distribution nF (~k) for the holes, which we sum over all wave
vectors ~k to obtain ρ˜, the density of holes:
ρ˜ = 1
N
∑
~k
nF (~k) =
1
Nβ
∑
n,~k
1
iνn −
(
(~k) + J + µ
) . (4.225)
We finally obtain
Tr
[
G0Σ(2,τ)0
]
= −2aρ˜
∫
dτ
∑
j
(
z¯j∂τζj + ζ¯j∂τzj
)
= −2ρ˜
a
∫
dτ d2x
(
z¯∂τζ + ζ¯∂τz
)
, (4.226)
where we picked up a factor of 1/a2 from going to the continuum. Thus, we get
Result 25 (Tr G0Σ(2,τ)0 )
Tr
[
G0Σ(2,τ)0
]
= −2ρ˜
a
∫
dτ d2x
(
z¯∂τζ + ζ¯∂τz
)
. (4.227)
2. The second summand contains the 2nd-nearest neighbour contribution proportional
to t′:
Tr
[
G0Σ(2,t
′)
0
]
=
∑
n,~k
n′, ~k′
β δk,k′
iνn −
(
(~k) + J + µ
) · Σ(2,t′)0 (k′, k) , (4.228)
where Σ(2,t
′)
0 is given by (4.167). We plug in Σ
(2,t′)
0 (k′, k) from (4.167), with z and ζ
in their real-space representation, and get
= 2t
′a2
Nβ
∫
dτ
∑
n,~k
1
iνn −
(
(~k) + J + µ
)[ cos(√2akx)∑
j
(
z¯j∂
2
xzj + zj∂2xz¯j
)
+ cos(
√
2aky)
∑
j
(
z¯j∂
2
yzj + zj∂2y z¯j
)
+
√
2i sin(
√
2akx)
∑
j
(
z¯j∂xζj + ζ¯j∂xzj − zj∂xζ¯j − ζj∂xz¯j
)
+
√
2i sin(
√
2aky)
∑
j
(
z¯j∂yζj + ζ¯j∂yzj − zj∂y ζ¯j + ζj∂yz¯j
)]
. (4.229)
104
4.4. Integrating out the fermions
Since (~k) is even in ~k, the last two summands in (4.229) are odd functions of ~k. The
index ~k runs over the whole Brillouin zone, so all odd terms cancel. Furthermore, we
perform the sum over ν (see Appendix A.3) after which we obtain
= 2t
′a2
N
∑
~k
nF (~k)
∫
dτ
∑
j
(
cos(
√
2akx)
(
z¯j∂
2
xzj + zj∂2xz¯j
)
+ cos(
√
2aky)
(
z¯j∂
2
yzj + zj∂2y z¯j
))
. (4.230)
Also note that on the square lattice the sum over cos(
√
2akx) gives the same result
as the sum over cos(
√
2aky) so that we can define
ρ˜1 =
1
N
∑
~k
nF (~k) cos(
√
2akx) =
1
N
∑
~k
nF (~k) cos(
√
2aky) . (4.231)
With that we have
Tr
[
G0Σ(2,t
′)
0
]
= 2t′a2 ρ˜1
∫
dτ
∑
j
(
z¯j∂
2
xzj + zj∂2xz¯j + z¯j∂2yzj + zj∂2y z¯j
)
. (4.232)
Finally, using the constraint 2 ∂µz¯∂µz + z¯ ∂2µz + z ∂2µz¯ = 0, we end up with
Result 26 (Tr G0Σ(2,t
′)
0 )
Tr
[
G0Σ(2,t
′)
0
]
= −4t′ρ˜1
∫
dτ d2x
(
∂xz¯ ∂xz + ∂yz¯ ∂yz
)
. (4.233)
3. The third summand gives the 2nd-nearest neighbour contribution proportional to t′′:
Tr
[
G0Σ(2,t
′′)
0
]
=
∑
n,~k
n′, ~k′
β δ~k,~k′δn,n′
iνn −
(
(~k) + J + µ
) · Σ(2,t′′)0 , (4.234)
where Σ(2,t
′′)
0 is given by (4.185). Repeating the steps from above, we obtain
Tr
[
G0Σ(2,t
′′)
0
]
= 4t
′′a2
Nβ
∫
dτ
∑
n,~k
1
iνn −
(
(~k) + J + µ
)
105
4. Effective Field Theory for the t-J Model at Low Doping
· cos(√2akx) cos(
√
2aky)
∑
j
(
z¯j∂
2
xzj + z¯j∂2yzj + zj∂2xz¯j + zj∂2y z¯j
)
. (4.235)
We perform then the sum over νn and define
ρ˜2 =
1
N
∑
~k
nF (~k) cos(
√
2akx) cos(
√
2aky) , (4.236)
so that, using the constraint 2 ∂µz¯∂µz + z¯ ∂2µz + z ∂2µz¯ = 0, we finally obtain
Result 27 (Tr G0Σ(2,t
′′)
0 )
Tr
[
G0Σ(2,t
′′)
0
]
= −8t′′ρ2
∫
dτ d2x
(
∂xz¯ ∂xz + ∂yz¯ ∂yz
)
. (4.237)
4. The last summand is the contribution from the fermion-dressed spin interaction:
Tr
[
G0Σ(2,J)0
]
=
∑
n,~k
n′, ~k′
β δ~k,~k′δn,n′
iνn −
(
(~k) + J + µ
) · Σ(2,J)0 (k′, k) , (4.238)
where Σ(2,J)0 is given in (4.198). We obtain the result:
Result 28 (Tr G0Σ(2,J)0 )
Tr
[
G0Σ(2,J)0
]
= −2Jρ˜
∫
dτ d2x
[
2
(
G∗G+ F ∗xFx + F ∗yFy
)
−
(
F ∗xFy + F ∗yFx
)
−√2
(
[F ∗x − F ∗y ]G+G∗[Fx − Fy]
)]
. (4.239)
Contribution with two Green’s function
The next pair of terms in (4.220) features products of first order contributions
2a2 · Tr
[
G0Σ(1)0 G0Σ
(1)
0 +G0Σ(1)z G0Σ(1)z
]
= 2a2 ·
(
Tr
[
G0Σ(1,τ)0 G0Σ
(1,τ)
0 +G0Σ(1,τ)z G0Σ(1,τ)z
]
106
4.4. Integrating out the fermions
+ 2 Tr
[
G0Σ(1,τ)0 G0Σ
(1,t′)
0 +G0Σ(1,τ)z G0Σ(1,t
′)
z
]
+ 2 Tr
[
G0Σ(1,τ)0 G0Σ
(1,t′′)
0 +G0Σ(1,τ)z G0Σ(1,t
′′)
z
]
+ Tr
[
G0Σ(1,t
′)
0 G0Σ
(1,t′)
0 +G0Σ(1,t
′)
z G0Σ(1,t
′)
z
]
+ 2 Tr
[
G0Σ(1,t
′)
0 G0Σ
(1,t′′)
0 +G0Σ(1,t
′)
z G0Σ(1,t
′′)
z
]
+ Tr
[
G0Σ(1,t
′′)
0 G0Σ
(1,t′′)
0 +G0Σ(1,t
′′)
z G0Σ(1,t
′′)
z
])
. (4.240)
We will consider these summands one by one.
1. So first we investigate Tr
[
G0Σ(1,τ)0 G0Σ
(1,τ)
0 +G0Σ(1,τ)z G0Σ(1,τ)z
]
:
We plug in the result from (4.128)
Tr
[
. . .
]
=
∑
nn′
~k~k′
β
iνn −
(
(~k) + J + µ
) β
iνn′ −
(
(~k′) + J + µ
) · Σ˜(1,τ)(k, k′)Σ˜(1,τ)(k′, k)
= − 4
N2β4
∑
nn′
~k~k′
1
iνn −
(
(~k) + J + µ
) 1
iνn′ −
(
(~k′) + J + µ
)
· ∑
mm′
~q ~q′
[(
νn − νn′ + ωm
)
z¯(~q, ωm)z(~k − ~k′ + ~q, νn − νn′ + ωm)
]
·
[(
νn′ − νn + ωm′
)
z¯(~q′, ωm′)z(~k′ − ~k + ~q′, νn′ − νn + ωm′)
]
. (4.241)
Now we use the fact that the z-field is slowly varying, so that it has only sizable
components for long wavelength and small energies. In particular, ωm and νn− νn′ +
ωm can be considered small so that the double sum over n and n′ will contribute
only for small values of νn − νn′ . In order to make this clear, we introduce a change
of variables:
νn − νn′ + ωm = ωm∗ , ~k − ~k′ + ~q = ~q∗ (4.242)
and get
= − 4
N2β4
∑
nm∗mm′
~k ~q∗~q ~q′
1
iνn −
(
(~k) + J + µ
)
· 1
i(νn + ωm − ωm∗)−
(
(~k + ~q − ~q∗) + J + µ
) · ω∗m(ωm + ωm′ − ωm∗)
107
4. Effective Field Theory for the t-J Model at Low Doping
·
[
z¯(~q, ωm)z(~q∗, ωm∗)
][
z¯(~q′, ωm′)z(~q + ~q′ − ~q∗, ωm + ωm′ − ωm∗)
]
. (4.243)
Thus we replaced the double sum over fermionic frequencies νn and νn′ by a sum
over νn and the bosonic frequency ωm∗ . Now we are able to evaluate the sum over
νn via a contour integral, as is done in Appendix A.3. Introducing the Fermi-Dirac
distributions
nF (~k) =
1
β
∑
n
1
iνn −
(
(~k) + J + µ
) ,
nF (~k + ~q − ~q∗) = 1
β
∑
n
1
i(νn + ωm − ωm∗)−
(
(~k + ~q − ~q∗) + J + µ
) , (4.244)
we obtain
= − 4
N2β3
∑
m∗mm′
~q∗~q ~q′
∑
k
nF (~k)− nF (~k + ~˜q)
iω˜m −
(
(~k + ~˜q)− (~k)
) · ωm∗(ωm + ωm′ − ωm∗)
·
[
z¯(~q, ωm)z(~q∗, ωm∗)
][
z¯(~q′, ωm′)z(~q − ~q∗ + ~q′, ωm − ωm∗ + ωm′)
]
, (4.245)
with the susceptibility
χ0(ω˜m, ~˜q) =
2
N
∑
k
nF (~k)− nF (~k + ~˜q)
iω˜m −
(
(~k + ~˜q)− (~k)
) , (4.246)
where we defined ω˜m = ωm − ωm∗ and ~˜q = ~q − ~q∗. Since ~q and ~q′ can be considered
small, we may evaluate the susceptibility in the static limit, expanding it to first
order in ~˜q around ~k while setting ω˜m to zero, see Appendix A.3. We obtain
χ0(ω˜m, ~˜q) ' − 2
N
∂
∂µ
∑
k
nF (~k) = −2ρ˜κ , (4.247)
where κ denotes the electronic compressibility, defined by κ = 1
ρ˜
∂ρ˜
∂µ
. So we get
= 4ρ˜κ
Nβ3
∑
m∗mm′
~q∗~q ~q′
ωm∗(ωm + ωm′ − ωm∗)
· z¯(~q, ωm)z(~q∗, ωm∗) · z¯(~q′, ωm′)z(~q − ~q∗ + ~q′, ωm − ωm∗ + ωm′) . (4.248)
108
4.4. Integrating out the fermions
We plug in the z-field’s real space representation and obtain in the continuum limit
Result 29 (Tr G0Σ(1,τ)0 G0Σ
(1,τ)
0 +G0Σ(1,τ)z G0Σ(1,τ)z )
Tr
[
G0Σ(1,τ)G0Σ(1,τ)
]
= −2ρ˜κ
a2
∫
dτ d2x
(
z¯ ∂τz
)2
. (4.249)
2. Next we look at Tr
[
G0Σ(1,τ)0 G0Σ
(1,t′)
0 +G0Σ(1,τ)z G0Σ(1,t
′)
z
]
Tr
[
G0Σ(1,τ)0 G0Σ
(1,t′)
0 +G0Σ(1,τ)z G0Σ(1,t
′)
z
]
= 4
√
2t′ai
N2β4
∑
nn′
~k~k′
1
iνn −
(
(~k) + J + µ
) 1
iνn′ −
(
(~k′) + J + µ
)
· ∑
mm′
~q ~q′
[(
ωm + νn − νn′
)
z¯(ωm, ~q)z(ωm + νn − νn′ , ~q + ~k − ~k′)
]
·
[
sin
√
2akx · q′x + sin
√
2aky · q′y
]
·
[
z¯(ωm′ + νn − νn′ , ~q′ + ~k − ~k′)z(ωm′ , ~q′)
+ z¯(ωm′ + νn − νn′ , ~q′)z(ωm′ , ~q′ − ~k + ~k′)
]
. (4.250)
Again we apply a change of variables
νn − νn′ + ωm = ωm∗ , ~k − ~k′ + ~q = ~q∗ , (4.251)
so that we are able to perform the Matsubara sum and expand the susceptibility
around ~k. We obtain
= −4
√
2t′ai
N2β3
∑
m∗mm′
~q∗~q ~q′
∂
∂µ
∑
~k
nF (~k) ·
(
sin
√
2akx · q′x + sin
√
2aky · q′y
)
· ω∗m · z¯(ωm, ~q)z(ωm∗ , ~q∗) ·
(
z¯(ωm′ − ωm + ωm∗, ~q′ − ~q + ~q′∗)z(ωm′ , ~q′)
+ z¯(ωm′ − ωm + ωm∗, ~q′)z(ωm′ , ~q′ + ~q − ~q∗)
)
, (4.252)
where now we see that the k-sum vanishes since all summand are odd functions in ~k.
109
4. Effective Field Theory for the t-J Model at Low Doping
3. Next we look at Tr
[
G0Σ(1,τ)0 G0Σ
(1,t′′)
0 +G0Σ(1,τ)z G0Σ(1,t
′′)
z
]
Repeating the steps we did for the previous contribution, we obtain a similar term
where all summands are either odd functions of kx or of ky. Thus the contribution
vanishes as well
Tr
[
G0Σ(1,τ)0 G0Σ
(1,t′′)
0 +G0Σ(1,τ)z G0Σ(1,t
′′)
z
]
= 0 . (4.253)
4. Then we consider Tr
[
G0Σ(1,t
′)
0 G0Σ
(1,t′)
0 +G0Σ(1,t
′)
z G0Σ(1,t
′)
z
]
= 16t
′2a2
N2β4
∑
nn′
~k~k′
1
iνn −
(
(~k) + J + µ
) 1
iνn′ −
(
(~k′) + J + µ
)
·∑
mm′
~q ~q′
[
sin
√
2ak′x · qx + sin
√
2ak′y · qy
][
sin
√
2akx · q′x + sin
√
2aky · q′y
]
·
[
z¯(ωm − νn + νn′ , ~q − ~k + ~k′)z(ωm, ~q) z¯(ωm′ + νn − νn′ , ~q′ + ~k − ~k′)z(ωm′ , ~q′)
+ z¯(ωm − νn + νn′ , ~q)z(ωm, ~q + ~k − ~k′) z¯(ωm′ + νn − νn′ , ~q′)z(ωm′ , ~q′ − ~k + ~k′)
]
.
(4.254)
Here we have to use two different substitutions. In both summands we take:
ωm∗ = ωm − νn + νn′ , (4.255)
but for the momentum parts we use
~q∗ = ~q − ~k + ~k′ in the first product and (4.256)
~q∗ = ~q + ~k − ~k′ in the second one. (4.257)
In the first product this leads to a susceptibility
2
N
∑
~k
nF (~k)− nF (~k − ~˜q)
−iω˜m −
(
(~k − ~˜q)− (~k)
) , (4.258)
110
4.4. Integrating out the fermions
whereas we have in the second product
2
N
∑
~k
nF (~k)− nF (~k + ~˜q)
−iω˜m −
(
(~k + ~˜q)− (~k)
) . (4.259)
But in the static limit, expanding ~˜q around ~k to 1st order, both terms are equal and
we obtain
= −16t
′2a2
N2β3
∑
mm′m∗
~q ~q′~q∗
∂
∂µ
∑
~k
nF (~k) ·
(
sin2
√
2akx cos
√
2aq˜x · qx · q′x
+ sin2
√
2aky cos
√
2aq˜y · qy · q′y
)
·
(
z¯(ωm∗ , ~q∗)z(ωm, ~q) z¯(ωm + ωm′ − ωm∗ , ~q + ~q′ − ~q′∗)z(ωm′ , ~q′)
+ z¯(ωm∗ , ~q)z(ωm, ~q∗) z¯(ωm + ωm′ − ωm∗ , ~q′)z(ωm′ , ~q + ~q′ − ~q′∗)
)
, (4.260)
where again q˜α = qα − q∗α. We have cos
√
2aq˜α ≈ 1 in O(a3) (since the term already
is a O(a2)-contribution). Then, after introducing the electronic compressibility
κ3 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) sin2
√
2akx =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) sin2
√
2aky , (4.261)
we go back to the z-field’s real space representation to arrive at
Result 30 (Tr G0Σ(1,t
′)
0 G0Σ
(1,t′)
0 +G0Σ(1,t
′)
z G0Σ(1,t
′)
z )
Tr
[
G0Σ(1,t
′)G0Σ(1,t
′)
]
= 16t′2ρ˜κ3
∫
dτ d2x
[(
z¯ ∂xz
)2
+
(
z¯ ∂yz
)2]
. (4.262)
5. Next we look at Tr
[
G0Σ(1,t
′)
0 G0Σ
(1,t′′)
0 +G0Σ(1,t
′)
z G0Σ(1,t
′′)
z
]
. Here we have
Tr
[
G0Σ(1,t
′)
0 G0Σ
(1,t′′)
0 +G0Σ(1,t
′)
z G0Σ(1,t
′′)
z
]
= 16t
′t′′a2
N2β4
∑
nn′
~k~k′
1
iνn −
(
(~k) + J + µ
) 1
iνn′ −
(
(~k′) + J + µ
)
·∑
mm′
~q ~q′
[
sin
√
2ak′x · qx + sin
√
2ak′y · qy
]
111
4. Effective Field Theory for the t-J Model at Low Doping
·
[
sin
√
2a(kx + ky) · (q′x + q′y) + sin
√
2akx − ky · (q′x − q′y)
]
·
[
z¯(ωm − νn + νn′ , ~q − ~k + ~k′)z(ωm, ~q)
· z¯(ωm′ + νn − νn′ , ~q′ + ~k − ~k′)z(ωm′ , ~q′)
+ z¯(ωm − νn + νn′ , ~q)z(ωm, ~q + ~k − ~k′)
· z¯(ωm′ + νn − νn′ , ~q′)z(ωm′ , ~q′ − ~k + ~k′)
]
. (4.263)
Performing a similar analysis as for the last term and introducing the compressibility
κ6 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) sin2
√
2akx cos
√
2aky , (4.264)
we eventually obtain
Result 31 (Tr G0Σ(1,t
′)
0 G0Σ
(1,t′′)
0 +G0Σ(1,t
′)
z G0Σ(1,t
′′)
z )
Tr
[
G0Σ(1,t
′)G0Σ(1,t
′′)
]
= 32t′t′′ρ˜κ6
∫
dτ d2x
[(
z¯ ∂xz
)2
+
(
z¯ ∂yz
)2]
. (4.265)
6. Then consider Tr
[
G0Σ(1,t
′′)
0 G0Σ
(1,t′′)
0 +G0Σ(1,t
′′)
z G0Σ(1,t
′′)
z
]
. We have
Tr
[
G0Σ(1,t
′′)
0 G0Σ
(1,t′′)
0 +G0Σ(1,t
′′)
z G0Σ(1,t
′′)
z
]
= 16t
′′2a2
N2β4
∑
nn′
~k~k′
1
iνn −
(
(~k) + J + µ
) 1
iνn′ −
(
(~k′) + J + µ
)
·∑
mm′
~q ~q′
[
sin
√
2a(k′x + k′y) · (qx + qy) + sin
√
2a(k′x − k′y) · (qx − qy)
]
·
[
sin
√
2a(kx + ky) · (q′x + q′y) + sin
√
2akx − ky · (q′x − q′y)
]
·
[
z¯(ωm − νn + νn′ , ~q − ~k + ~k′)z(ωm, ~q)
z¯(ωm′ + νn − νn′ , ~q′ + ~k − ~k′)z(ωm′ , ~q′)
+ z¯(ωm − νn + νn′ , ~q)z(ωm, ~q + ~k − ~k′)
z¯(ωm′ + νn − νn′ , ~q′)z(ωm′ , ~q′ − ~k + ~k′)
]
. (4.266)
112
4.4. Integrating out the fermions
Performing once more the steps we did for the last term and introducing the com-
pressibility
κ4 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) sin2
√
2akx cos2
√
2aky , (4.267)
we get
Result 32 (Tr G0Σ(1,t
′′)
0 G0Σ
(1,t′′)
0 +G0Σ(1,t
′′)
z G0Σ(1,t
′′)
z )
Tr
[
G0Σ(1,t
′′)G0Σ(1,t
′′)
]
= 64t′′2ρ˜κ4
∫
dτ d2x
[(
z¯ ∂xz
)2
+
(
z¯ ∂yz
)2]
. (4.268)
Finally we examine the last summand in (4.220). Unfortunately, the term
Tr
[
G0Σ(1,t)+ G0Σ
(1,t)
−
]
(4.269)
is too large to fit on a single page. For its derivation we refer to the master document [96].
The steps are basically the same that we used for the previous contributions. There is,
however, one special feature. We obtain trigonometric functions in the bosonic wave
vectors that have a non-vanishing first order expansion. Consequently, (4.269) which
originally is of 2nd order in a, gives a 3rd order contribution as well. We obtain
Result 33 (12TrG0Σ
(1,t)
+ G0Σ
(1,t)
− + 12TrG0Σ
(1,t)
− G0Σ(1,t)+ )
for the O(a2) contribution:
Tr
[
G0Σ(1,t)+ G0Σ
(1,t)
−
](2)
= −16t2ρ˜
∫
dτ d2x
(
κ1
[
2G∗G−√2G∗(Fx − Fy)−
√
2 (F ∗x − F ∗y )G
− F ∗xFy − F ∗yFx
]
+ κ2
[
F ∗xFx + F ∗yFy)
])
, (4.270)
and for the O(a3) contribution:
113
4. Effective Field Theory for the t-J Model at Low Doping
Tr
[
G0Σ(1,t)+ G0Σ
(1,t)
−
](3)
= −8at2ρ˜
∫
dτ d2x
(
κ1
[(
∂xG
∗ − ∂yG∗
)(√
2G− Fx + Fy
)
−
(√
2G∗ − F ∗x + F ∗y
)(
∂xG− ∂yG
)
−
(
F ∗xx − F ∗xy + F ∗yy
)(
2G−√2Fx +
√
2Fy
)
+
(
2G∗ −√2F ∗x +
√
2F ∗y
)(
Fxx − Fxy + Fyy
)]
− κ5
[(
∂xG
∗ +
√
2F ∗xy −
√
2F ∗xx
)
Fx − F ∗x
(
∂xG+
√
2Fxy −
√
2Fxx
)
+
(
∂yG
∗ −√2F ∗xy +
√
2F ∗yy
)
Fy − F ∗y
(
∂yG−
√
2Fxy +
√
2Fyy
)])
. (4.271)
Effective action up to O(a2) after integrating out the fermions
We summarize all contributions up to O(a2), coming from the spin part, see (4.49) and
(4.62), and the fermionic part, see (4.220).
For the fermionic part, we collect the results from (4.227), (4.233), (4.237), (4.239),
(4.249), (4.262), (4.265), (4.268) and (4.270) to obtain
S
(2)
F −→
∫
dτ d2x
[
4ρ˜
a
(
z¯∂τζ + ζ¯∂τz
)
+ 8t′ρ˜1
(
∂xz¯ ∂xz + ∂yz¯ ∂yz
)
+ 16t′′ρ˜2
(
∂xz¯ ∂xz + ∂yz¯ ∂yz
)
+ 4Jρ˜
[
2
(
G∗G+ F ∗xFx + F ∗yFy
)
−
(
F ∗xFy + F ∗yFx
)
−√2
(
G∗Fx −G∗Fy
)
−√2
(
F ∗xG− F ∗yG
))]
+ 2ρ˜κ
a2
(
z¯∂τz
)2 − 16t′2ρ˜κ3 [(z¯∂xz)2 + (z¯∂yz)2]− 64t′t′′ρ˜κ6 [(z¯∂xz)2 + (z¯∂yz)2]
− 64t′′2ρ˜κ4
[(
z¯∂xz
)2
+
(
z¯∂yz
)2]
+ 16t2ρ˜
[
κ1
(
2G∗G−√2(G∗Fx −G∗Fy)
−√2(F ∗xG− F ∗yG)− (F ∗xFy + F ∗yFx)
)
+ κ2(F ∗xFx + F ∗yFy)
]]
. (4.272)
And for the spin part, we get from (4.49) and (4.62)
114
4.4. Integrating out the fermions
SS =
∫
dτ d2x
(
2
a
(
z¯∂τζ + ζ¯∂τz
)
+ J
[
2
(
G∗G+ F ∗xFx + F ∗yFy
)
−
(
F ∗xFy + F ∗yFx
)
−√2
(
F ∗x − F ∗y
)
Gj −
√
2G∗j (Fx − Fy)
])
. (4.273)
Both, the fermionic- (4.272) and the spin-contribution (4.273) contain a part with only
F -fields, namely
(
2J + 8Jρ˜+ 16t2ρ˜κ2
)(
F ∗xFx + F ∗yFy
)
−
(
J + 4Jρ˜+ 16t2ρ˜κ1
)(
F ∗xFy + F ∗yFx
)
, (4.274)
which we now evaluate further. We see from the identities in Appendix (B.2) that
F ∗αFβ = ∂αz¯ ∂βz + (z¯ ∂αz)(z¯ ∂βz) . (4.275)
Note that via integration by parts in the first summand, we can show that
F ∗αFβ = ∂αz¯ ∂βz + (z¯ ∂αz)(z¯ ∂βz) = ∂β z¯ ∂αz + (z¯ ∂βz)(z¯ ∂αz) = F ∗βFα , (4.276)
so that we have(
2J + 8Jρ˜+ 16t2ρ˜κ2
)(
F ∗xFx + F ∗yFy
)
−
(
J + 4Jρ˜+ 16t2ρ˜κ1
)(
F ∗xFy + F ∗yFx
)
=
(
2J + 8Jρ˜+ 16t2ρ˜κ2
)(
∂xz¯ ∂xz + (z¯ ∂xz)2 + ∂yz¯ ∂yz + (z¯ ∂yz)2
)
− 2
(
J + 4Jρ˜+ 16t2ρ˜κ1
)(
∂xz¯ ∂yz + (z¯ ∂xz)(z¯ ∂yz)
)
. (4.277)
Thus, collecting the results from (4.272), (4.273) and (4.277) we finally get
Result 34 (Effective action up to O(a2) after integrating out the fermions)
S(2) →
∫
dτ d2x
(
c1
(
∂xz¯ ∂xz + ∂yz¯ ∂yz
)
+ c2
(
∂xz¯ ∂yz
)
+ c3
(
z¯ ∂xz
)2
+ c3
(
z¯ ∂yz
)2
+ c2
(
z¯ ∂xz
)(
z¯ ∂yz
)
+ c5
(
z¯ ∂τz
)2
+ c6
(
z¯ ∂τζ + ζ¯ ∂τz
)
+ c7
(
2G∗G−√2(G∗Fx −G∗Fy)−
√
2(F ∗xG− F ∗yG)
))
, (4.278)
115
4. Effective Field Theory for the t-J Model at Low Doping
where
c1 = 2J + 8 Jρ˜+ 16 t2ρ˜κ2 + 8 t′ρ˜1 + 16 t′′ρ˜2 ,
c2 = −2J − 8 Jρ˜− 32 t2ρ˜κ1 ,
c3 = 2J + 8 Jρ˜+ 16 t2ρ˜κ2 − 16 t′2ρ˜κ3 − 64t′t′′ρ˜κ6 − 64t′′2ρ˜κ4 ,
c5 =
2ρ˜κ
a2
, c6 =
2
a
+ 4ρ˜
a
,
c7 = J + 4Jρ˜+ 16t2ρ˜ κ1 . (4.279)
The densities and compressibilities are summarized in Appendix B.1.
Note that most of the results summarized in Subsection 4.4.2 and 4.4.3 were already
derived in [10]. However, we are able to give some corrections to the coupling con-
stants that were obtained there and we carefully checked for higher order contributions
originating from the expansion of trigonometric functions, see for instance (4.271).
4.4.4. Contribution in O(a3)
Contribution with one Green’s function
We first look at the simple contributions to (4.221), the summands with one Green’s
function
Tr
(
2 ·G0Σ(3)0
)
= 2 · Tr
(
G0Σ(3,τ)0 +G0Σ
(3,t′)
0 +G0Σ
(3,t′′)
0 +G0Σ
(3,J)
0
)
. (4.280)
1. For the first summand we have
Tr
[
G0Σ(3,τ)0
]
=
∑
k,k′
G0(k, k′) Σ(3,τ)0 (k′, k) =
∑
k
β
iνn −
[
(~k) + J + µ
]Σ(3,τ)0 (k, k) ,
(4.281)
where Σ(3,τ)0 (k, k′) is given by (4.138). We easily see that the term vanishes for k = k′.
2. The second summand reads
Tr
[
G0Σ(3,t
′)
0
]
=
∑
k
β
iνn −
[
(~k) + J + µ
]Σ(3,t′)0 (k, k) , (4.282)
116
4.4. Integrating out the fermions
where Σ(3,t
′)
0 (k, k′) is given by (4.173), so that
Tr
[
G0Σ(3,t
′)
0
]
= 12
∑
k
β
iνn −
[
(~k) + J + µ
](Σ˜(3,t′)A (k, k) + Σ˜(3,t′)B (k, k))
= −2t
′a3
Nβ
∑
k
1
iνn −
[
(~k) + J + µ
]
·
∫
dτ
∑
j
(
cos
√
2a kx · (v∗x − vx) + cos
√
2a ky · (v∗y − vy)
)
. (4.283)
We plug in (4.187) and ρ˜1, defined in (4.231), so that we finally obtain
Result 35 (Tr G0Σ(3,t
′)
0 )
Tr
[
G0Σ(3,t
′)
0
]
= −2t′aρ˜1
∫
dτ d2x
(
z¯j(∂2x + ∂2y)ζj + ζ¯j(∂2x + ∂2y)zj
− zj(∂2x + ∂2y)ζ¯j − ζj(∂2x + ∂2y)z¯j
)
. (4.284)
3. The third term
Tr
[
G0Σ(3,t
′′)
0
]
=
∑
k
β
iνn −
[
(~k) + J + µ
]Σ(3,t′′)0 (k, k) , (4.285)
where Σ(3,t
′′)
0 (k, k′) is given by (4.191). We perform a similar analysis as for the
previous term and plug in ρ˜2, defined in (4.236). Finally, we obtain
Result 36 (Tr G0Σ(3,t
′′)
0 )
Tr
[
G0Σ(3,t
′′)
0
]
= −4t′′aρ˜2
∫
dτ d2x
(
z¯(∂2x + ∂2y)ζ + ζ¯(∂2x + ∂2y)z
− z(∂2x + ∂2y)ζ¯ − ζ(∂2x + ∂2y)z¯
)
. (4.286)
117
4. Effective Field Theory for the t-J Model at Low Doping
4. The last summand in (4.280) reads
Tr
[
G0Σ(3,J)0
]
=
∑
k,k′
G0(k, k′) Σ(3,J)0 (k′, k) =
∑
k
β
iνn −
[
(~k) + J + µ
]Σ(3,J)0 (k, k) ,
(4.287)
where Σ(3,J)0 (k, k′) is given in (4.204). We see that it vanishes for k = k′.
Contribution with two Green’s function
The next terms in O(a3) are given by
Tr
[
2 ·
(
G0Σ(1)0 G0Σ
(2)
0 +G0Σ(1)z G0Σ(2)z
)]
= 2 · Tr
[
G0
(
Σ(1,τ)0 + Σ
(1,t′)
0 + Σ
(1,t′′)
0
)
G0
(
Σ(2,τ)0 + Σ
(2,t′)
0 + Σ
(2,t′′)
0 + Σ
(2,J)
0
)
+G0
(
Σ(1,τ)z + Σ(1,t
′)
z + Σ(1,t
′′)
z
)
G0
(
Σ(2,τ)z + Σ(2,t
′)
z + Σ(2,t
′′)
z + Σ(2,J)z
)]
= 2 · Tr
[
G0Σ(1,τ)0 G0Σ
(2,τ)
0 +G0Σ
(1,τ)
0 G0Σ
(2,t′)
0 +G0Σ
(1,τ)
0 G0Σ
(2,t′′)
0
+G0Σ(1,τ)0 G0Σ
(2,J)
0 +G0Σ
(1,t′)
0 G0Σ
(2,τ)
0 +G0Σ
(1,t′)
0 G0Σ
(2,t′)
0
+G0Σ(1,t
′)
0 G0Σ
(2,t′′)
0 +G0Σ
(1,t′)
0 G0Σ
(2,J)
0 +G0Σ
(1,t′′)
0 G0Σ
(2,τ)
0
+G0Σ(1,t
′′)
0 G0Σ
(2,t′)
0 +G0Σ
(1,t′′)
0 G0Σ
(2,t′′)
0 +G0Σ
(1,t′′)
0 G0Σ
(2,J)
0 + Σz-terms
]
. (4.288)
All of these terms were analyzed, though only a few give contributions to the effective
action. These are summarized in the following.
1. There is a contribution from
Tr
[
G0Σ(1,τ)0 G0Σ
(2,t′)
0 +G0Σ(1,τ)z G0Σ(2,t
′)
z
]
=
∑
k,k′
β
iνn −
[
(~k) + J + µ
] β
iνn′ −
[
(~k′) + J + µ
]
·
[
Σ(1,τ)0 (k, k′)Σ
(2,t′)
0 (k′, k) + Σ(1,τ)z (k, k′)Σ(2,t
′)
z (k′, k)
]
= 12
∑
k,k′
β
iνn −
[
(~k) + J + µ
] β
iνn′ −
[
(~k′) + J + µ
]
· Σ˜(1,τ)(k, k′) ·
[
Σ˜(2,t
′)
A (k′, k)− Σ˜(2,t
′)
B (k′, k)
]
, (4.289)
118
4.4. Integrating out the fermions
where we used (4.128). Now we plug in (4.167) and (4.168) so that, after the change
of variables
νn − νn′ + ωm = ωm∗ , ~k − ~k′ + ~q = ~q∗ , (4.290)
we obtain
= −4t
′a2 i
N2β3
∑
q,q′,q∗
∂
∂µ
∑
~k
nF (~k) cos
√
2a kx ·
(
q′2x + q′2y
)
ωm∗ · z¯(ωm, ~q)z(ωm∗ , ~q∗)
·
[
z¯(−ωm + ωm′ + ωm∗ ,−~q + ~q′ + ~q∗)z(ωm′ , ~q′)
− z¯(ωm′ , ~q′)z(ωm + ωm′ − ωm∗ , ~q + ~q′ − ~q∗)
]
, (4.291)
where we already discarded terms antisymmetric in ~k. We introduce
κ7 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) cos
√
2a kx (4.292)
and transform the fields back to real space to obtain
Result 37 (Tr G0Σ(1,τ)0 G0Σ
(2,t′)
0 +G0Σ(1,τ)z G0Σ(2,t
′)
z )
Tr
[
G0Σ(1,τ)0 G0Σ
(2,t′)
0 +G0Σ(1,τ)z G0Σ(2,t
′)
z
]
= −4t′ρ˜κ7
∫
dτ d2x
[
z¯ ∂τz ·
(
∂xz¯ ∂xz + ∂yz¯ ∂yz
)
+ z¯ ∂τz ·
(
z¯ ∂ 2x z + z¯ ∂ 2y z
)]
. (4.293)
2. Then
Tr
[
G0Σ(1,τ)0 G0Σ
(2,t′′)
0 +G0Σ(1,τ)z G0Σ(2,t
′′)
z
]
=
∑
k,k′
β
iνn −
[
(~k) + J + µ
] β
iνn′ −
[
(~k′) + J + µ
]
·
[
Σ(1,τ)0 (k, k′)Σ
(2,t′′)
0 (k′, k) + Σ(1,τ)z (k, k′)Σ(2,t
′′)
z (k′, k)
]
, (4.294)
where we perform a similar analysis as for the last term. After introducing
κ8 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) cos
√
2a kx cos
√
2a ky , (4.295)
119
4. Effective Field Theory for the t-J Model at Low Doping
we obtain
Result 38 (Tr G0Σ(1,τ)0 G0Σ
(2,t′′)
0 +G0Σ(1,τ)z G0Σ(2,t
′′)
z )
Tr
[
G0Σ(1,τ)0 G0Σ
(2,t′′)
0 +G0Σ(1,τ)z G0Σ(2,t
′′)
z
]
= −8t′′ρ˜κ8
∫
dτ d2x
[
z¯ ∂τz ·
(
∂xz¯ ∂xz + ∂yz¯ ∂yz
)
+ z¯ ∂τz ·
(
z¯ ∂ 2x z + z¯ ∂ 2y z
)]
. (4.296)
Finally, we evaluate the next neighbour term Tr(G0Σ(1)+ G0Σ
(2)
− + G0Σ(1)− G0Σ(2)+ ). How-
ever, the term is too big to fit on a single page. For its derivation, we refer again to
Reference [96]. Here we only state the result
Result 39 (Tr G0Σ(1,t)+ G0Σ(2,t)− +G0Σ(1,t)− G0Σ(2,t)+ )
Tr
[
G0Σ(1,t)+ G0Σ
(2,t)
− +G0Σ(1,t)− G0Σ(2,t)+
]
= 8at2ρ˜
∫
dτ
∫
d2x
(
2κ1
[
G∗(Fxx − Fxy + Fyy)− (F ∗xx − F ∗xy + F ∗yy)G
]
−√2κ1
[
(F ∗xFyy − F ∗yFxx)− (F ∗yyFx − F ∗xxFy)
]
+
√
2κ2
[
F ∗x (Fxy − Fxx)− F ∗y (Fxy − Fyy)− (F ∗xy − F ∗xx)Fx + (F ∗xy − F ∗yy)Fy
])
. (4.297)
Contribution with three Green’s functions
The terms in (4.221) involving a product of three fermionic Green’s functions are given
by
Tr
(1
3 G0ΣG0ΣG0Σ
)(3)
= Tr
[
2
3
(
G0Σ(1)0 G0Σ
(1)
0 G0Σ
(1)
0 + 3G0Σ
(1)
0 G0Σ(1)z G0Σ(1)z
)
+
(
G0Σ(1)0
(
G0Σ(1)+ G0Σ
(1)
− +G0Σ(1)− G0Σ(1)+
)
120
4.4. Integrating out the fermions
+G0Σ(1)z
(
G0Σ(1)+ G0Σ
(1)
− −G0Σ(1)− G0Σ(1)+
))]
. (4.298)
Each contribution will have a prefactor
S =
∑
n
1
iνn −
[
(~k) + J + µ
] 1
iνn′ −
[
(~k′) + J + µ
] 1
iνn′′ −
[
(~k′′) + J + µ
] , (4.299)
which after the appropriate change of variables
νn′ → νn + ω˜m , ~k′ → ~k + ~˜q ,
νn′′ → νn + ˜˜ωm , ~k′′ → ~k + ˜˜~q , (4.300)
where ω˜m, ˜˜ωm, ~˜q, ˜˜~q are considered small, can be expanded (see Appendix A.3) to hold in
lowest order
2
N
∑
~k
1
β
S = 2
N
1
2
∂2
∂µ2
∑
~k
nF (~k) =
∂2ρ˜
∂µ2
. (4.301)
1. We start by considering the first two summands in (4.298)
2
3 Tr
(
G0Σ(1)0 G0Σ
(1)
0 G0Σ
(1)
0 + 3G0Σ
(1)
0 G0Σ(1)z G0Σ(1)z
)
, (4.302)
where
Σ(1)0 = Σ
(1,τ)
0 + Σ
(1,t′)
0 + Σ
(1,t′′)
0 ,
Σ(1)z = Σ(1,τ)z + Σ(1,t
′)
z + Σ(1,t
′′)
z , (4.303)
so that
2
3 Tr
(
G0Σ(1)0 G0Σ
(1)
0 G0Σ
(1)
0
)
= 23 Tr
(
G0Σ(1,τ)0 G0Σ
(1,τ)
0 G0Σ
(1,τ)
0
+ 3G0Σ(1,τ)0 G0Σ
(1,τ)
0 G0Σ
(1,t′)
0 + 3G0Σ
(1,τ)
0 G0Σ
(1,τ)
0 G0Σ
(1,t′′)
0
+G0Σ(1,t
′)
0 G0Σ
(1,t′)
0 G0Σ
(1,t′)
0
+ 3G0Σ(1,t
′)
0 G0Σ
(1,t′)
0 G0Σ
(1,τ)
0 + 3G0Σ
(1,t′)
0 G0Σ
(1,t′)
0 G0Σ
(1,t′′)
0
+G0Σ(1,t
′′)
0 G0Σ
(1,t′′)
0 G0Σ
(1,t′′)
0
+ 3G0Σ(1,t
′′)
0 G0Σ
(1,t′′)
0 G0Σ
(1,τ)
0 + 3G0Σ
(1,t′′)
0 G0Σ
(1,t′′)
0 G0Σ
(1,t′)
0
121
4. Effective Field Theory for the t-J Model at Low Doping
+ 3G0Σ(1,τ)0 G0Σ
(1,t′)
0 G0Σ
(1,t′′)
0 + 3G0Σ
(1,τ)
0 G0Σ
(1,t′′)
0 G0Σ
(1,t′)
0
)
(4.304)
and
2
3 Tr
(
3G0Σ(1)0 G0Σ(1)z G0Σ(1)z
)
= 23 Tr
(
3G0Σ(1,τ)0 G0Σ(1,τ)z G0Σ(1,τ)z + 3G0Σ
(1,τ)
0 G0Σ(1,τ)z G0Σ(1,t
′)
z
+ 3G0Σ(1,τ)0 G0Σ(1,t
′)
z G0Σ(1,τ)z + 3G0Σ
(1,t′)
0 G0Σ(1,τ)z G0Σ(1,τ)z
+ 3G0Σ(1,τ)0 G0Σ(1,τ)z G0Σ(1,t
′′)
z + 3G0Σ
(1,τ)
0 G0Σ(1,t
′′)
z G0Σ(1,τ)z
+ 3G0Σ(1,t
′′)
0 G0Σ(1,τ)z G0Σ(1,τ)z + 3G0Σ
(1,t′)
0 G0Σ(1,t
′)
z G0Σ(1,t
′)
z
+ 3G0Σ(1,t
′)
0 G0Σ(1,t
′)
z G0Σ(1,τ)z + 3G0Σ
(1,t′)
0 G0Σ(1,τ)z G0Σ(1,t
′)
z
+ 3G0Σ(1,τ)0 G0Σ(1,t
′)
z G0Σ(1,t
′)
z + 3G0Σ
(1,t′)
0 G0Σ(1,t
′)
z G0Σ(1,t
′′)
z
+ 3G0Σ(1,t
′)
0 G0Σ(1,t
′′)
z G0Σ(1,t
′)
z + 3G0Σ
(1,t′′)
0 G0Σ(1,t
′)
z G0Σ(1,t
′)
z
+ 3G0Σ(1,t
′′)
0 G0Σ(1,t
′′)
z G0Σ(1,t
′′)
z + 3G0Σ
(1,t′′)
0 G0Σ(1,t
′′)
z G0Σ(1,τ)z
+ 3G0Σ(1,t
′′)
0 G0Σ(1,τ)z G0Σ(1,t
′′)
z + 3G0Σ
(1,τ)
0 G0Σ(1,t
′′)
z G0Σ(1,t
′′)
z
+ 3G0Σ(1,t
′′)
0 G0Σ(1,t
′′)
z G0Σ(1,t
′)
z + 3G0Σ
(1,t′′)
0 G0Σ(1,t
′)
z G0Σ(1,t
′′)
z
+ 3G0Σ(1,t
′)
0 G0Σ(1,t
′′)
z G0Σ(1,t
′′)
z + 3G0Σ
(1,τ)
0 G0Σ(1,t
′)
z G0Σ(1,t
′′)
z
+ 3G0Σ(1,t
′)
0 G0Σ(1,t
′′)
z G0Σ(1,τ)z + 3G0Σ
(1,τ)
0 G0Σ(1,t
′)
z G0Σ(1,t
′′)
z
+ 3G0Σ(1,τ)0 G0Σ(1,t
′′)
z G0Σ(1,t
′)
z + 3G0Σ
(1,t′′)
0 G0Σ(1,t
′)
z G0Σ(1,τ)z
+ 3G0Σ(1,τ)0 G0Σ(1,t
′′)
z G0Σ(1,t
′)
z
)
. (4.305)
All these terms were carefully analyzed but turned out to give vanishing results. For
details on the calculation, we refer again to the master document [96].
2. We now turn to the last two summands in (4.298)
Tr
(
G0Σ(1)0
(
G0Σ(1)+ G0Σ
(1)
− +G0Σ(1)− G0Σ(1)+
)
+G0Σ(1)z
(
G0Σ(1)+ G0Σ
(1)
− −G0Σ(1)− G0Σ(1)+
))
, (4.306)
where
Σ(1)0 = Σ
(1,τ)
0 + Σ
(1,t′)
0 + Σ
(1,t′′)
0 ,
122
4.4. Integrating out the fermions
Σ(1)+ = Σ
(1,t)
+ , Σ
(1)
− = Σ(1,t)− , (4.307)
so that
Tr
(
G0Σ(1)0
(
G0Σ(1)+ G0Σ
(1)
− +G0Σ(1)− G0Σ(1)+
)
+G0Σ(1)z
(
G0Σ(1)+ G0Σ
(1)
− −G0Σ(1)− G0Σ(1)+
))
= Tr
(
G0Σ(1,τ)0
(
G0Σ(1,t)+ G0Σ
(1,t)
− +G0Σ(1,t)− G0Σ(1,t)+
)
+G0Σ(1,t
′)
0
(
G0Σ(1,t)+ G0Σ
(1,t)
− +G0Σ(1,t)− G0Σ(1,t)+
)
+G0Σ(1,t
′′)
0
(
G0Σ(1,t)+ G0Σ
(1,t)
− +G0Σ(1,t)− G0Σ(1,t)+
)
+G0Σ(1,τ)z
(
G0Σ(1,t)+ G0Σ
(1,t)
− −G0Σ(1,t)− G0Σ(1,t)+
)
+G0Σ(1,t
′)
z
(
G0Σ(1,t)+ G0Σ
(1,t)
− −G0Σ(1,t)− G0Σ(1,t)+
)
+G0Σ(1,t
′′)
z
(
G0Σ(1,t)+ G0Σ
(1,t)
− −G0Σ(1,t)− G0Σ(1,t)+
))
. (4.308)
This time, we get non-vanishing results from two summands, namely
Tr
(
G0Σ(1,τ)z G0Σ
(1,t)
+ G0Σ
(1,t)
− −G0Σ(1,t)z G0Σ(1,t)− G0Σ(1,t)+
)
. (4.309)
Only those will be derived in the following and we refer again to [96] for the remain-
ing terms.
From (4.214), (4.128), (4.147) and (4.148) we have
Tr
(
G0Σ(1,τ)z G0Σ
(1,t)
+ G0Σ
(1,t)
−
)
=
∑
k,k′,k′′
G0(k)Σ(1,τ)z (k, k′)G0(k′)Σ
(1,t)
+ (k′, k′′)G0(k′′)Σ
(1,t)
− (k′′, k)
= i a
2 t2
N3β9
∑
k,k′,k′′
β
iνn − (~k)− J − µ
β
iνn′ − (~k′)− J − µ
β
iνn′′ − (~k′′)− J − µ
· ∑
q,q′,q′′
(
ei(~k−~k′)~xA + ei(~k−~k′)~xB
)
(νn − νn′ + ωm)
· z¯(ωm, ~q)z(νn − νn′ + ωm, ~k − ~k′ + ~q) e−i
√
2a
2 [(kx−k′′x)−(ky−k′′y )]e(~k−~k′)~xA
·
[
16 cos
√
2ak′′x
2 cos
√
2ak′′y
2 G
∗(ωm′ , ~q′;−νn′ + νn′′ − ωm′ ,−~k′ + ~k′′ − ~q′)
123
4. Effective Field Theory for the t-J Model at Low Doping
− 4√2i
(
ei
√
2a
2 k
′′
x cos
√
2ak′′y
2 (−k
′
x + k′′x − q′x)
− e−i
√
2a
2 k
′′
y cos
√
2ak′′x
2 (−k
′
y + k′′y − q′y)
)
· F ∗(ωm′ , ~q′;−νn′ + νn′′ − ωm′ ,−~k′ + ~k′′ − ~q′)
]
·
[
16 cos
√
2akx
2 cos
√
2aky
2 G(ωm
′′ , ~q′′; νn′′ − νn − ωm′′ , ~k′′ − ~k − ~q′′)
+ 4
√
2i
(
e−i
√
2a
2 kx cos
√
2aky
2 (k
′′
x − kx − q′′x)
− ei
√
2a
2 ky cos
√
2akx
2 (k
′′
y − ky − q′′y)
)
· F (ωm′′ , ~q′′; νn′′ − νn − ωm′′ , ~k′′ − ~k − ~q′′)
]
. (4.310)
We perform a change of variables,
νn′ → νn + ωm − ωm∗ , ~k′ → ~k + ~q − ~q∗ ,
νn′′ → νn + ωm′′ + ωm∗∗ , ~k′′ → ~k + ~q′′ + ~q∗∗ , (4.311)
and sum over Matsubara frequencies νn. Then we expand all trigonometric functions
with respect to bosonic wave vectors and obtain
= i a
2 t2
N3β5
∂2
∂µ2
∑
~k
nF (~k)
∑
q, q′,q′′
q∗,q∗∗
ω∗m · z¯(ωm, ~q)z(ωm∗ , ~q∗)
·
[
8 cos
√
2akx
2 cos
√
2aky
2
·G∗(ωm′ , ~q′;−ωm − ωm′ + ωm′′ + ωm∗ + ωm∗∗ ,−~q − ~q′ + ~q′′ + ~q∗ + ~q∗∗)
− 4√2i
(
ei
√
2a
2 kx cos
√
2aky
2 (−qx − q
′
x + q′′x + q∗x + q∗∗x )
− e−i
√
2a
2 ky cos
√
2akx
2 (−qy − q
′
y + q′′y + q∗y + q∗∗y )
)
· F ∗(ωm′ , ~q′;−ωm − ωm′ + ωm′′ + ωm∗ + ωm∗∗ ,−~q − ~q′ + ~q′′ + ~q∗ + ~q∗∗)
]
·
[
8 cos
√
2akx
2 cos
√
2aky
2 ·G(ωm′′ , ~q
′′;ωm∗∗ , ~q∗∗)
124
4.4. Integrating out the fermions
+ 4
√
2i
(
e−i
√
2a
2 kx cos
√
2aky
2 q
∗∗
x − ei
√
2a
2 ky cos
√
2akx
2 q
∗∗
y
)
· F (ωm′′ , ~q′′;ωm∗∗ , ~q∗∗)
]
. (4.312)
Only summands symmetric in ~k contribute to the ~k-sum. We introduce
λ4 =
1
N
∂2
∂µ2
∑
~k
nF (~k) cos2
√
2a
2 kx cos
2
√
2a
2 ky , (4.313)
λ5 =
1
N
∂2
∂µ2
∑
~k
nF (~k) cos2
√
2a
2 kx sin
2
√
2a
2 ky (4.314)
and transform the fields back to realspace, so that finally
Tr
(
G0Σ(1,τ)z G0Σ
(1,t)
+ G0Σ
(1,t)
−
)
= 8 a2t2ρ˜
∫
dτ
∑
j
(z¯ ∂τz)
(
λ4
[
2G∗G−√2G∗(Fx − Fy)−
√
2(F ∗x − F ∗y )G
+ (F ∗x − F ∗y )(Fx − Fy)
]
+ λ5(F ∗xFx + F ∗yFy)
)
. (4.315)
We get the same result via a similar calculation for
Tr
(
G0Σ(1,τ)z G0Σ
(1,t)
− G0Σ(1,t)+
)
=
∑
k,k′,k′′
G0(k)Σ(1,τ)z (k, k′)G0(k′)Σ
(1,t)
− (k′, k′′)G0(k′′)Σ(1,t)+ (k′′, k) , (4.316)
so that in summary, after going to the continuum, we end up with
Result 40 (TrG0Σ(1,τ)z G0Σ
(1,t)
+ G0Σ
(1,t)
− − TrG0Σ(1,τ)z G0Σ(1,t)− G0Σ(1,t)+ )
Tr
(
G0Σ(1,τ)z G0Σ
(1,t)
+ G0Σ
(1,t)
− −G0Σ(1,τ)z G0Σ(1,t)− G0Σ(1,t)+
)
= 16 t2ρ˜
∫
dτ d2x (z¯ ∂τz)
(
λ4
[
2G∗G−√2G∗(Fx − Fy)
−√2(F ∗x − F ∗y )G+ (F ∗x − F ∗y )(Fx − Fy)
]
+ λ5(F ∗xFx + F ∗yFy)
)
. (4.317)
125
4. Effective Field Theory for the t-J Model at Low Doping
4.4.5. Effective action after integrating out the fermions
Let us finally summarize all contributions to the effective action up to O(a3). We collect
the results from the spin part and the 2nd-order fermionic contribution, given in (4.278).
The total contribution to O(a3) comes from the fermionic part of the action
S
(3)
F = S
(3,std)
F + S
(3,exp)
F , (4.318)
where S(3,std)F are the ones we calculated in Section 4.4.4
S
(3,std)
F = −Tr
(
G0Σ +
1
2G0ΣG0Σ +
1
3G0ΣG0ΣG0Σ
)(3)
= −2 Tr
(
G0Σ(3)0
)
− Tr
[
2
(
G0Σ(1)0 G0Σ
(2)
0 +G0Σ(1)z G0Σ(2)z
)
+
(
G0Σ(1)+ G0Σ
(2)
− +G0Σ(1)− G0Σ(2)+
)]
− Tr
[2
3
(
G0Σ(1)0 G0Σ
(1)
0 G0Σ
(1)
0 + 3G0Σ
(1)
0 G0Σ(1)z G0Σ(1)z
)
+G0Σ(1)0
(
G0Σ(1)+ G0Σ
(1)
− +G0Σ(1)− G0Σ(1)+
)
+G0Σ(1)z
(
G0Σ(1)+ G0Σ
(1)
− −G0Σ(1)− G0Σ(1)+
)]
, (4.319)
and S(3,exp)F are 3rd-order contributions coming from 2nd-order terms where we expanded
functions of the bosonic wave vectors
S
(3,exp)
F = −Tr
(
G0Σ +
1
2G0ΣG0Σ +
1
3G0ΣG0ΣG0Σ
)(2→3)
= −Tr
[
2G0Σ(2)0 +G0Σ
(1)
0 G0Σ
(1)
0 +G0Σ(1)z G0Σ(1)z
+ 12
(
G0Σ(1)+ G0Σ
(1)
− +G0Σ(1)− G0Σ(1)+
)](3,exp)
. (4.320)
We separate all summands that contain fluctuations ζ, from those containing only z-
fields. In the fluctuation-free part, we rewrite F in terms of the z-fields by means of
the identities given in Appendix B.2. After some algebra, we obtain the effective action
that results from the integration over fermionic degrees of freedom:
126
4.4.
Integrating
out
the
ferm
ions
Summary 2 (Effective action up to O(a3) after integrating out the fermions)
S =
∫
dτ d2x
[
Aαβ1 (∂αz¯ ∂βz) + Aαβ2 (z¯ ∂αz)(z¯ ∂βz) +Bαβτ (z¯ ∂τz)
(
∂αz¯ ∂βz + (z¯ ∂αz)(z¯ ∂βz)
)
+Dαβτ (z¯ ∂τz)(z¯ ∂α∂βz− z ∂α∂β z¯) + γ1
(
z¯∂τζ + ζ¯∂τz
)
+ γ2
(
2G∗G−√2(G∗Fx−G∗Fy)−
√
2(F ∗xG−F ∗yG)
)
+ γ3
(
2
[
(∂xG∗)G− (∂yG∗)G
]
+
√
2
[
(∂xG∗)Fy + (∂yG∗)Fx
]
− c.c.
)
− γ4
(√
2
[
(∂xG∗)Fx + (∂yG∗)Fy
]
− c.c.
)
− γ5
(
z¯ ∂τz
)(
2G∗G−√2
(
G∗Fx −G∗Fy
)
−√2
(
F ∗xG− F ∗yG
))
+ γ6
(
z¯j(∂2x + ∂2y)ζj + ζ¯j(∂2x + ∂2y)zj − c.c.
)]
, (4.321)
Aαβ1 =

0 0 0
0 a1 a2
0 a2 a1
 , Aαβ2 =

a3 0 0
0 a4 a2
0 a2 a4
 , Bαβτ =

0 0 0
0 b1 b2
0 b2 b1
 , Dαβτ =

0 0 0
0 d1 0
0 0 d1
 , (4.322)
a1 = 2J
(
1 + 4ρ˜
)
+ 16 t2ρ˜κ2 + 8 t′ρ˜1 + 16 t′′ρ˜2 , a2 = −J
(
1 + 4ρ˜
)
− 16 t2ρ˜κ1 ,
a3 =
2ρ˜κ
a2
, a4 = 2J
(
1 + 4ρ˜
)
+ 16 t2ρ˜κ2 − 16 t′2ρ˜κ3 − 64t′t′′ρ˜κ6 − 64t′′2ρ˜κ4 , (4.323)
b1 = −16 t2 ρ˜ λ8 , b2 = 16 t2 ρ˜ λ4 , d1 = 4 ρ˜ (t′κ7 + 2 t′′κ8) , (4.324)
γ1 =
2
a
(
1 + 2ρ˜
)
, γ2 = J + 4Jρ˜+ 16t2ρ˜ κ1 , γ3 = 4
√
2 a t2ρ˜ κ1 ,
γ4 = 4
√
2 a t2ρ˜ κ2 , γ5 = 16 t2 ρ˜ λ4 , γ6 = 4 a(t′ρ˜1 + 2t′′ρ˜2) . (4.325)127
4. Effective Field Theory for the t-J Model at Low Doping
4.5. Integrating out small bosonic fluctuations
After integrating out the fermionic degrees of freedom, we are left with an action con-
taining fields in the staggered CP1 representation. We divide it in a part containing only
z-fields and another one with ζ-fields.
S = Sz + Sζ . (4.326)
ζ-part of the effective action
The remaining variable to be integrated out is ζ, representing small fluctuations. From
(4.321) we have
Sζ =
∫
dτ d2x
[
γ1
(
z¯∂τζ + ζ¯∂τz
)
+ γ2
(
2G∗G−√2(G∗Fx −G∗Fy)
−√2(F ∗xG− F ∗yG)
)
+ γ3
(
2
[
(∂xG∗)G− (∂yG∗)G
]
+
√
2
[
(∂xG∗)Fy + (∂yG∗)Fx
]
− c.c.
)
− γ4
(√
2
[
(∂xG∗)Fx + (∂yG∗)Fy
]
− c.c.
)
− γ5
(
z¯ ∂τz
)(
2G∗G−√2
(
G∗Fx −G∗Fy
)
−√2
(
F ∗xG− F ∗yG
))
+ γ6
(
z¯(∂2x + ∂2y)ζ + ζ¯(∂2x + ∂2y)z − c.c.
)]
, (4.327)
and the γ’s were defined in (4.325). The first two lines of (4.327) constitute the O(a2)-
contribution which consists of the Berry phase and a part with spatial derivatives which
is real. The O(a3)-term in the third to sixth line is purely imaginary. We use integration
by parts to simplify the expression.
Result 41 (ζ-part of the effective action after integrating by parts)
Sζ =
∫
dτ d2x
[
γ1
(
ζ¯ ∂τz − ζ ∂τ z¯
)
+ γ2
(
2G∗G−√2G∗(Fx − Fy)−
√
2 (F ∗x − F ∗y )G
)
+ 2 γ3
(
(∂xG∗)G− (∂yG∗)G− c.c.
)
+
√
2
(
G∗
(
γ4Fxx + 2 γ3Fxy + γ4Fyy
)
− c.c.
)
− γ5
(
z¯ ∂τz
)(
2G∗G−√2G∗
(
Fx − Fy
)
−√2
(
F ∗x − F ∗y
)
G
)]
. (4.328)
128
4.5. Integrating out small bosonic fluctuations
Gauge fixing and the constraint for ζ
It is clear from our discussion in Section 2.4 that CP1-variables are defined only up
to gauge transformations. However in Section 4.3, we decomposed the original CP 1-
variables into slowly varying fields and fast fluctuations, z → (z, a ζ). Consequently,
there is a gauge degree of freedom for ζ as well and in order to integrate only over
physical degrees of freedom, we need to fix the gauge for the ζ-fields.
As discussed in Appendix A.4, as well as in [10], the ζ-fields must be constrained to
have the same gauge transformation as the z-fields and we may choose the global phases
of z and ζ to differ by pi2 , which implies the following gauge fixing condition for the ζ-fields
ζ1
|ζ1|
ζ2
|ζ2| = −
z1
|z1|
z2
|z2| , (4.329)
such that
ζ1 = − z1|z1|
z2
|z2|
|ζ2|
ζ2
|ζ1| = − ζ
∗
2
z∗1z∗2
ζ∗1ζ1 . (4.330)
Note that this condition only fixes the gauge for ζ, while retaining the freedom of choos-
ing a gauge for the z-fields. Using (4.330), we get for
z¯ζ + ζ¯z = − z
∗
1
|z1|
z∗2
|z2|
|ζ2|
ζ∗2
|ζ1| z1 + ζ∗2z2 − z∗1
z1
|z1|
z2
|z2|
|ζ2|
ζ2
|ζ1|+ z∗2ζ2
= (z∗2ζ2 + ζ∗2z2)
(
1− |z1||ζ1||z2||ζ2|
)
, (4.331)
so that the constraint z¯ ζ + ζ¯ z = 0 translates into one for |ζ1|:
δ
(
z¯ζ + ζ¯z
)
= δ
[
(z∗2ζ2 + ζ∗2z2)
(
1− |z1||ζ1||z2||ζ2|
)]
= 1
z∗2ζ2 + ζ∗2z2
δ
(
1− |z1||ζ1||z2||ζ2|
)
= |z2||ζ2||z1|(z∗2ζ2 + ζ∗2z2)
δ
( |z2||ζ2|
|z1| − |ζ1|
)
. (4.332)
And when we combine (4.330) and (4.332), we obtain
ζ1 = − z1z2|z1|2
|ζ2|2
ζ2
= −z2
z∗1
ζ∗2 , (4.333)
such that we can now eliminate Re ζ1 and Im ζ1 from the calculation.
129
4. Effective Field Theory for the t-J Model at Low Doping
Consequences for the measure
From (4.41) we have for the partition function
Z =
∫
D(z¯, z)D(ζ¯ , ζ) 12a
(
1 + 32a
2ζ¯ζ
)
J δ
(
z¯z − 1
)
δ
(
z¯ζ + ζ¯z
)
e−S . (4.334)
After fixing the gauge for the ζ-fields, (4.330), the independent variables are ζ2, ζ∗2 and
|ζ1|. Using (4.332) we get
Z =
∫
D(z¯, z)D(ζ∗2 , ζ2)D|ζ1|
1
2a
(
1 + 32a
2ζ¯ζ
)
J
· |z2||ζ2||z1|(z∗2ζ2 + ζ∗2z2)
δ
( |z2||ζ2|
|z1| − |ζ1|
)
δ
(
z¯z − 1
)
e−S . (4.335)
And after eliminating ζ1 from all terms, we can trivially integrate over |ζ1| to obtain
Z =
∫
D(z¯, z)D(ζ∗2 , ζ2)
1
2a
(
1 + 32a
2 |ζ2|2
|z1|2
)
J |z2||ζ2||z1|(z∗2ζ2 + ζ∗2z2)
δ
(
z¯z − 1
)
e−S . (4.336)
At this point, we need to evaluate the Jacobian J that results from the transformation
(4.31). In order to do so, we have to fix the gauge for the z-fields. We use a local gauge
fixing condition z∗1 + z1 = 0, as was done in [10]. Since the terms are rather lengthy, we
only sketch the procedure:
• Expand the transformation (4.31) up to third order in a.
• Apply the gauge fixing conditions z∗1 + z1 = 0 and (4.330).
• Calculate the Jacobi matrix J =
(
∂ΩA1 ...∂ΩB3
∂Im z1...Im ζ2
)
and its determinant.
• Integrate over the constraint (4.332).
• Expand the result with respect to powers of a.
The result finally reads
J =
∣∣∣∣∣det
(
∂ΩA1 . . . ∂ΩB3
∂Im z1 . . . Im ζ2
)∣∣∣∣∣
= a3 64 z1
|z1|2 − |z2|2
|z1|2
|z1|(ζ∗2z2 + z∗2ζ2)
|z2||ζ2|
(
1− 32a
2 |ζ2|2
|z1|2 +O(a
4)
)
. (4.337)
130
4.5. Integrating out small bosonic fluctuations
So we get for the integral measure (4.336)
∫
D(z¯, z)D(ζ∗2 , ζ2) 16a2(z1 − z∗1)
|z1|2 − |z2|2
|z1|2
(
1 +O(a4)
)
δ
(
z¯z − 1
)
. (4.338)
Consequences for the effective action
Let us examine the effect of (4.333) on the terms appearing in the effective action. We
have
• in O(a2)
ζ¯ ∂τz − ζ ∂τ z¯ = z2
(
z1∂τz
∗
1
|z1|2 +
z∗2∂τz2
|z2|2
)
ζ∗2 − z∗2
(
z∗1∂τz1
|z1|2 +
z2∂τz
∗
2
|z2|2
)
ζ2
G∗G = 4
( |z1|4 + |z2|4
|z1|2 ζ
∗
2ζ2 + z22 ζ∗2ζ∗2 + z∗22 ζ2ζ2
)
G∗ Fα + F ∗α G = 2 z2
(
z∗1z
∗
2
|z2|2Fα +
z1z2
|z1|2F
∗
α
)
ζ∗2 + 2 z∗2
(
z∗1z
∗
2
|z1|2Fα +
z1z2
|z2|2F
∗
α
)
ζ2 , (4.339)
• In O(a3)
There is a term with derivatives of ζ,
∂αG
∗G−G∗∂αG = 8 z22
(
z1∂αz
∗
1
|z1|2 −
z∗2∂αz2
|z2|2
)
ζ∗2ζ
∗
2 − 8 z∗2
(
z∗1∂αz1
|z1|2 −
z2∂αz
∗
2
|z2|2
)
ζ2ζ2
+ 4
( |z1|4 + |z2|4
|z1|4 (z1∂αz
∗
1 − z∗1∂αz1) + 2
|z2|2
|z1|2 (z2∂αz
∗
2 − z∗2∂αz2)
)
ζ∗2ζ2
+ 4 |z2|
4 − |z1|4
|z1|2
(
ζ∗2 (∂αζ2)− (∂αζ∗2 )ζ2
)
, (4.340)
and we have
G∗ Fαβ − F ∗αβ G = 2 z2
(
z∗1z
∗
2
|z2|2Fαβ −
z1z2
|z1|2F
∗
αβ
)
ζ∗2
+ 2 z∗2
(
z∗1z
∗
2
|z1|2Fαβ −
z1z2
|z2|2F
∗
αβ
)
ζ2 . (4.341)
131
4. Effective Field Theory for the t-J Model at Low Doping
4.5.1. Path integral over ζ
From our considerations in the last paragraph and (4.328), we obtain the following path
integral over ζ:
Zζ =
∫
D(ζ∗2 , ζ2) exp
(
−
∫
dτ d2x
[
ζ∗2 A1 ζ2 + ζ∗2 A2 (∂x − ∂y)ζ2 − ζ2A2 (∂x − ∂y)ζ∗2
+B ζ∗2ζ∗2 + C ζ2ζ2 +D ζ∗2 + E ζ2
])
, (4.342)
where we defined
A1 = ∆(0) + ∆(1) , A2 = Π(0) ,
B = Λ(0) + Λ(1) , C = Λ∗(0) − Λ∗(1) ,
D = Ξ(1) − Γ(1) + Γ(2) , E = −Ξ∗(1) − Γ∗(1) − Γ∗(2) (4.343)
and
∆(0) = 8 γ2
|z1|4 + |z2|4
|z1|2 ,
∆(1) = 8γ3
( |z1|4 + |z2|4
|z1|4
(
z1(∂x − ∂y)z∗1 − z∗1(∂x − ∂y)z1
)
+2 |z2|
2
|z1|2
(
z2(∂x − ∂y)z∗2 − z∗2(∂x − ∂y)z2
))
− 8 γ5 |z1|
4 + |z2|4
|z1|2 (z¯ ∂τz) ,
Π(0) = 8 γ3
|z2|4 − |z1|4
|z1|2 ,
Λ(0) = 8 γ2 z22 ,
Λ(1) = 16 γ3 z22
(
z1(∂x − ∂y)z∗1
|z1|2 −
z∗2(∂x − ∂y)z2
|z2|2
)
− 8 γ5 z22
(
z¯ ∂τz
)
,
Ξ(1) = γ1 z2
(
z1∂τz
∗
1
|z1|2 +
z∗2∂τz2
|z2|2
)
,
Γ(1) = 2
√
2 γ2 z2
(
z∗1z
∗
2
|z2|2 (Fx − Fy) +
z1z2
|z1|2 (F
∗
x − F ∗y )
)
,
Γ(2) = 2
√
2γ5 z2
(
z¯ ∂τz
)(z∗1z∗2
|z2|2 (Fx − Fy) +
z1z2
|z1|2 (F
∗
x − F ∗y )
)
+2
√
2 z2
(
z∗1z
∗
2
|z2|2
(
γ4Fxx − 2 γ3Fxy + γ4Fyy
)
+ z1z2|z1|2
(
γ4F
∗
xx − 2 γ3F ∗xy + γ4F ∗yy
))
. (4.344)
132
4.5. Integrating out small bosonic fluctuations
Since the effective action is not in the usual form of a complex Gaussian integral, we
need to decompose ζ2 = η+ i ξ and integrate explicitly over the real- and imaginary part
of ζ2. We get
ζ∗2ζ2 = η2 + ξ2 ,
ζ∗2 ∂αζ2 = η ∂αη + ξ ∂αξ + i
(
η ∂αξ − ξ ∂αη
)
,
−ζ2 ∂αζ∗2 = −η ∂αη − ξ ∂αξ + i
(
η ∂αξ − ξ ∂αη
)
,
ζ∗2ζ
∗
2 = η2 − ξ2 − 2i η ξ ,
ζ2ζ2 = η2 − ξ2 + 2i η ξ . (4.345)
and the path integral now reads
Zζ =
∫
DηDξ exp
(
−
∫
dτ d2x
[
η
(
A1 +B + C
)
η + ξ
(
A1 −B − C
)
ξ
+ 2i η
(
−B + A2∂α
)
ξ + 2i ξ
(
C − A2∂α
)
η +
(
D + E
)
η + i
(
−D + E
)
ξ
])
. (4.346)
The exponent contains derivatives of η and ξ, so we need to evaluate the integral in
Fourier space, where
η(~x) =
√
2
N
∑
~q
e−i~q~x η(~q) and ξ(~x) =
√
2
N
∑
~q
e−i~q~x ξ(~q) . (4.347)
Note that the Fourier Transform of a real function in general is not real-valued, but
satisfies
η∗(~q) = η(−~q) . (4.348)
Thus, a quadratic form can be expressed as
∑
~q,~q′
η(~q)A(~q, ~q ′)η(~q ′) =
∑
~q,~q′
η(−~q)A(−~q, ~q ′)η(~q ′) = ∑
~q,~q′
η∗(~q)A(−~q, ~q ′)η(~q ′) . (4.349)
In the same way, we see that
∑
~q
d(~q)η(~q) =
∑
~q
d(−~q)η(−~q) = ∑
~q
d(−~q)η∗(~q) . (4.350)
133
4. Effective Field Theory for the t-J Model at Low Doping
So we arrive at
Result 42 (Functional integral over fluctuations)
Zζ =
∫
DηDξ exp
(
−∑
~q,~q′
[
η∗(~q)A(~q − ~q ′)η(~q ′) + ξ∗(~q)B(~q − ~q ′)ξ(~q ′)
+ ξ∗(~q) C(~q, ~q ′)η(~q ′)
]
−∑
~q
[
d(−~q)η(~q) + e(−~q)ξ(~q)
])
, (4.351)
with
A(~q − ~q ′) = 2
N
∫
d2x ei(~q−~q′)~xA(~x)
= 2
N
∫
d2x ei(~q−~q′)~x
(
∆(0) +
(
Λ(0) + Λ∗(0)
)
+ ∆(1) +
(
Λ(1) − Λ∗(1)
))
, (4.352)
B(~q − ~q ′) = 2
N
∫
d2x ei(~q−~q′)~x B(~x)
= 2
N
∫
d2x ei(~q−~q′)~x
(
∆(0) −
(
Λ(0) + Λ∗(0)
)
+ ∆(1) −
(
Λ(1) − Λ∗(1)
))
, (4.353)
C(~q, ~q ′) = C1(−~q ′,−~q) + C2(~q, ~q ′)
= 2
N
∫
d2x ei(~q−~q′)~x
(
− 2i
(
Λ(0) − Λ∗(0)
)
− 2i
(
Λ(1) + Λ∗(1)
)
−2
(
qα + q′α
)
Π(0)
)
, (4.354)
d(−~q) =
√
2
N
∫
d2x e−i~q~x d(~x)
=
√
2
N
∫
d2x e−i~q~x
((
Ξ(1) − Ξ∗(1)
)
−
(
Γ(1) + Γ∗(1)
)
+
(
Γ(2) − Γ∗(2)
))
, (4.355)
e(−~q) =
√
2
N
∫
d2x e−i~q~x e(~x)
=
√
2
N
∫
d2x e−i~q~x
(
− i
(
Ξ(1) + Ξ∗(1)
)
+ i
(
Γ(1) − Γ∗(1)
)
−i
(
Γ(2) + Γ∗(2)
))
. (4.356)
Note that we wrote qα and q′α as a short-hand for qx− qy and q′x− q′y and suppressed the
τ -dependencies as well as the integral over τ , since they are trivial.
134
4.5. Integrating out small bosonic fluctuations
Evaluating the path integral
In general, a Gaussian functional integral over η, where η is the Fourier-transform of a
real-valued field, is evaluated as in (A.19). First we look at the η-part of (4.351) where
we have
Zη =
∫
D(η, η†) exp
(
−∑
~q,~q′
η∗(~q)A(~q − ~q ′) η(~q ′)
−∑
~q,~q′
ξ∗(~q)C(~q, ~q ′)η(~q ′)−∑
~q
d(−~q) η(~q)
)
, (4.357)
and the term linear in η can be written as
∑
~q,~q′
ξ∗(~q)C(~q, ~q ′)η(~q ′) +∑
~q
d(−~q)η(~q)
=
∑
~q
1
2
[∑
~q′
ξ∗(~q ′)C(~q ′, ~q) + d(−~q)
︸ ︷︷ ︸
v∗(~q)
]
η(~q)
+
∑
~q
η∗(~q) 12
[∑
~q′
ξ(~q ′)C(−~q ′,−~q) + d(~q)
︸ ︷︷ ︸
v′(~q)
]
, (4.358)
so that
Zη =
∫
D(η, η†) exp
(
−
[
η†A η+ v†η+ η†v′
])
= exp
(
− 12Tr lnA+ v
†A−1v′
)
, (4.359)
where
v†A−1v′ = ∑
~q,~q′
v∗(~q)A−1(~q, ~q ′) v′(~q ′)
= 14
∑
~q,~q′
[∑
~q′′
ξ∗(~q ′′)C(~q ′′, ~q) + d(−~q)
]
A−1(~q, ~q ′)
[∑
~q′′′
ξ(~q ′′′)C(−~q ′′′,−~q ′) + d(~q ′)
]
.
(4.360)
A similar reasoning can be applied for the ξ-part of the integral so that we finally obtain
135
4. Effective Field Theory for the t-J Model at Low Doping
Result 43 (Result of the ζ-integration)
Zζ = exp
(
− 12Tr ln
[
AB′
]
+ 116dA
−1CB′−1CA−1d
− 18dA
−1CB′−1e− 18eB
′−1CA−1d+ 14eB
′−1e+ 14dA
−1d
)
= exp
(
− 12Tr ln
[∑
~q′′
A(~q, ~q ′′)B′(~q ′′, ~q ′)
]
+ 116
∑
~q,~q′
∑
~q′′,~q′′′
∑
~qIV,~qV
d(−~q)A−1(~q, ~q ′)C(−~q ′′,−~q ′)B′−1(~q ′′, ~q ′′′)
· C(~q ′′′, ~qIV )A−1(−~qV ,−~qIV )d(~qV )
− 18
∑
~q,~q′
∑
~q′′,~q′′′
d(−~q)A−1(~q, ~q ′)C(−~q ′′,−~q ′)B′−1(~q ′′, ~q ′′′)e(~q ′′′)
− 18
∑
~q,~q′
∑
~q′′,~q′′′
e(−~q)B′−1(~q, ~q ′)C(~q ′, ~q ′′)A−1(−~q ′′′,−~q ′′)d(~q ′′′)
+ 14
∑
~q,~q′
e(−~q)B′−1(~q, ~q ′) e(~q ′) + 14
∑
~q,~q′
d(−~q)A−1(~q, ~q ′) d(~q ′)
)
, (4.361)
with
B′ = B − 14 C A
−1C
= B(~q, ~q ′′′)− 14
∑
~q′,~q′′
C(~q, ~q ′)A−1(~q ′, ~q ′′) C(−~q ′′′,−~q ′′) . (4.362)
Let us sort all terms with respect to powers of a. From (4.352)-(4.356) we get
A(~q, ~q ′) = A(0)(~q, ~q ′) +A(1)(~q, ~q ′) ,
B(~q, ~q ′) = B(0)(~q, ~q ′) + B(1)(~q, ~q ′) ,
C(~q, ~q ′) = C(0)(~q, ~q ′) + C(1a)(~q, ~q ′) + (qα + q′α)C(1b)(~q, ~q ′) ,
d(~q) = d(1)(~q) + d(2)(~q) ,
e(~q) = e(1)(~q) + e(2)(~q) . (4.363)
The subindices (0), (1) and (2) refer to the number of derivatives in the respective terms
136
4.5. Integrating out small bosonic fluctuations
and we have
{A(0),B(0), C(0), d(1), e(1)} ∼ O(a2) , {A(1),B(1), C(1a), C(1b), d(2), e(2)} ∼ O(a3) , (4.364)
which can be made explicit by writing the integrals as sums and taking into account the
factors of a from (4.325). In the following, we will make successive use of expansions
with respect to a which will allow us to evaluate the terms in the effective action.
The inverse of A
The inverse of an operator A, which is diagonal in ~q − ~q ′, is easily obtained in terms of
its real space-representation
A(~q, ~q ′) = 2
N
∫
d2x ei(~q−~q′)~xf(~x) , A−1(~q, ~q ′) = 2
N
∫
d2x ei(~q−~q′)~x 1
f(~x) , (4.365)
since
(
AA−1
)
(~q, ~q ′′) = 4
N2
∑
~q′
A(~q, ~q ′)A−1(~q ′, ~q ′′) (4.366)
= 4
N2
∑
~q′
∫
d2x d2x′ ei(~q−~q′)~xei(~q′−~q′′)~x′ f(~x)
f(~x′)
= 2
N
∫
d2x d2x′ δ(~x− ~x′) ei~q~xe−i~q′′~x′ f(~x)
f(~x′)
= δ(~q − ~q ′′) . (4.367)
However, when A consists of an O(a2) and an O(a3) part
A(~q, ~q ′) = A(0)(~q, ~q ′) +A(1)(~q, ~q ′) , (4.368)
we need to expand A−1 up to higher orders in a
A−1 =
(
A(0) +A(1)
)−1
=
[(
1+A(1)A −1(0)
)
A(0)
]−1
= A −1(0)
(
1+A(1)A −1(0)
)−1
= A −1(0) −A −1(0)A(1)A −1(0) +A −1(0)A(1)A −1(0)A(1)A −1(0)
−A −1(0)A(1)A −1(0)A(1)A −1(0)A(1)A −1(0) + . . . (4.369)
Thus, A−1 can be expressed in terms of A(1) and A−1(0), which is obtained as in (4.365).
137
4. Effective Field Theory for the t-J Model at Low Doping
Result 44 (Inverse of A)
A−1(~q, ~q ′) = A−1(0)(~q, ~q ′) +A−1(1)(~q, ~q ′) + . . . (4.370)
where
A−1(0)(~q, ~q ′) =
2
N
∫
d2x ei(~q−~q′)~x 1A(0)(~x) , (4.371)
A−1(n)(~q, ~q ′) =
(
− 1
)n 2
N
∫
d2x ei(~q−~q′)~x A(1)(~x)
n
A(0)(~x)n+1 . (4.372)
The operator B′
For B′ we have
B′(~q, ~q ′) = B(~q, ~q ′)− 14
∑
~q′′,~q′′′
C(~q, ~q ′′)A−1(~q ′′, ~q ′′′) C(−~q ′,−~q ′′′)
=
[
B(0)(~q, ~q ′) + B(1)(~q, ~q ′)
]
− 14
∑
~q′′,~q′′′
[
C(0)(~q, ~q ′′) + C(1)(~q, ~q ′′)
]
·
[
A−1(0)(~q ′′, ~q ′′′) +A−1(1)(~q ′′, ~q ′′′)
][
C(0)(−~q ′,−~q ′′′)− C(1)(−~q ′,−~q ′′′)
]
. (4.373)
So up to third order in a
B′(~q, ~q ′) = B′(0)(~q, ~q ′) + B′(1)(~q, ~q ′) , (4.374)
where
B′(0)(~q, ~q ′) = B(0)(~q, ~q ′)−
1
4
∑
~q′′,~q′′′
C(0)(~q, ~q ′′)A−1(0)(~q ′′, ~q ′′′)C(0)(−~q ′,−~q ′′′)
= 2
N
∫
d2x ei(~q−~q′)~x 4A(0)(~x)B(0)(~x)− C(0)(~x)C(0)(~x)4A(0)(~x) , (4.375)
B′(1)(~q, ~q ′) = B(1)(~q, ~q ′)−
1
4
∑
~q′′,~q′′′
[
C(1)(~q, ~q ′′)A−1(0)(~q ′′, ~q ′′′)C(0)(−~q ′,−~q ′′′)
138
4.5. Integrating out small bosonic fluctuations
+ C(0)(~q, ~q ′′)A−1(1)(~q ′′, ~q ′′′)C(0)(−~q ′,−~q ′′′)
+ C(0)(~q, ~q ′′)A−1(0)(~q ′′, ~q ′′′)C(1)(−~q ′,−~q ′′′)
]
. (4.376)
For the last three summands we have
∑
~q′′,~q′′′
C(1)(~q, ~q ′′)A−1(0)(~q ′′, ~q ′′′)C(0)(−~q ′,−~q ′′′)
= 2
N
∫
d2x ei(~q−~q′)~x
(C(0)(~x)C(1a)(~x)
A(0)(~x) + 2 qα
C(0)(~x)C(1b)(~x)
A(0)(~x) − i
C(0)(~x) ∂∂xαC(1b)(~x)
A(0)(~x)
)
,
(4.377)
∑
~q′′,~q′′′
C(0)(~q, ~q ′′)A−1(1)(~q ′′, ~q ′′′)C(0)(−~q ′,−~q ′′′) = −
2
N
∫
d2x ei(~q−~q′)~xC(0)(~x)C(0)(~x)A(1)(~x)A(0)(~x)A(0)(~x) ,
(4.378)
1
4
∑
~q′′,~q′′′
C(0)(~q, ~q ′′)A−1(0)(~q ′′, ~q ′′′)C(1)(−~q ′,−~q ′′′)
= 2
N
∫
d2x ei(~q−~q′)~x
(C(0)(~x)C(1a)(~x)
A(0)(~x) − 2 q
′
α
C(0)(~x)C(1b)(~x)
A(0)(~x) − i
C(0)(~x) ∂∂xαC(1b)(~x)
A(0)(~x)
)
,
(4.379)
and we finally get
Result 45 (Operator B′ up to O(a3))
B′(~q, ~q ′) = B′(0)(~q, ~q ′) + B′(1)(~q, ~q ′) , (4.380)
where
B′(0)(~q, ~q ′) =
2
N
∫
d2x ei(~q−~q′)~x 4A(0)B(0) − C(0)C(0)4A(0) ,
B′(1)(~q, ~q ′) =
2
N
∫
d2x ei(~q−~q′)~x
(
B(1) − C(0)C(1a) + i ∂αC(0) C(1b)2A(0)
+ A(1)C(0)C(0) + 2i ∂αA(0) C(0)C(1b)4A(0)A(0)
)
. (4.381)
139
4. Effective Field Theory for the t-J Model at Low Doping
The inverse of B′
We evaluate the inverse up to third order in a
B′−1(~q, ~q ′) = B′−1(0) (~q, ~q ′) + B′−1(1) (~q, ~q ′) . (4.382)
We have
B′ −1(0) (~q, ~q ′) =
2
N
∫
d2x ei(~q−~q′)x 4A(0)(~x)4A(0)(~x)B(0)(~x)− C(0)(~x)C(0)(~x) , (4.383)
and
B′ −1(1) (~q, ~q ′) = −
∑
~q′′,~q′′′
B′ −1(0) (~q, ~q ′′)B′(1)(~q ′′, ~q ′′′)B′ −1(0) (~q ′′′, ~q ′)
= − 2
N
∫
d2x ei(~q−~q′)~x
[
4A(0)(x)(~x)
4A(0)(~x)B(0)(~x)− C(0)(~x)C(0)(~x)
]2
B′(1)(~x)
= 2
N
∫
d2x ei(~q−~q′)~x
(−16A(0)A(0) B(1) + 8A(0)C(0) C(1a) − 4A(1)C(0) C(0)(
4A(0)B(0) − C(0) C(0)
)2
+ 8iA(0)∂αC(0) C(1b) − 8i ∂αA(0)C(0) C(1b)(
4A(0)B(0) − C(0) C(0)
)2
)
. (4.384)
Result 46 (Inverse of B′ up to second leading order)
B′−1(~q, ~q ′) = B′−1(0) (~q, ~q ′) + B′−1(1) (~q, ~q ′) , (4.385)
where
B′ −1(0) (~q, ~q ′) =
2
N
∫
d2x ei(~q−~q′)~x 4A(0)4A(0)B(0) − C(0)C(0) , (4.386)
B′ −1(1) (~q, ~q ′) =
2
N
∫
d2x ei(~q−~q′)~x
(−16A(0)A(0) B(1) + 8A(0)C(0) C(1a) − 4A(1)C(0) C(0)(
4A(0)B(0) − C(0) C(0)
)2
+ 8iA(0)∂αC(0) C(1b) − 8i ∂αA(0)C(0) C(1b)(
4A(0)B(0) − C(0) C(0)
)2
)
. (4.387)
140
4.5. Integrating out small bosonic fluctuations
4.5.2. Contribution in O(a2)
Evaluating the result of the ζ-integration
Let us now evaluate the O(a2)-contribution in (4.361). With the separation introduced
in (4.363), we get
dAd-term1: Using (4.370) we get
1
4
∑
~q,~q′
d(1)(−~q)A−1(0)(~q, ~q ′) d(1)(~q ′)
= 1
N2
∑
~q,~q′
∫
d2x d2x′ d2x′′ e−i ~q~xd(1)(~x) ei(~q−~q
′)~x′ 1
A(0)(~x′) e
i ~q′~x′′d(1)(~x′′)
=
∫
d2x
d(1)(~x)d(1)(~x)
4A(0)(~x) . (4.388)
eBe-term: Using (4.385) this contribution reads
1
4
∑
~q,~q′
e(1)(−~q)B′−1(0) (~q, ~q ′) e(1)(~q ′) =
∫
d2x
A(0)e(1)e(1)
4A(0)B(0) − C(0)C(0) . (4.389)
eBCAd-term: Together with (4.370) and (4.385) we obtain
1
8
∑
~q,~q′
∑
~q′′,~q′′′
e(1)(−~q)B′−1(0) (~q, ~q ′)C(0)(~q ′, ~q ′′)A−1(0)(~q ′′, ~q ′′′)d(1)(~q ′′′)
= 2
N4
∑
~q,~q′
∑
~q′′,~q′′′
∫
d2x d2x′ d2x′′ d2x′′′ d2xIV e−i ~q~xe(1)(~x) ei (~q−~q
′)~x′B′−1(0) (~x′)
·ei (~q′−~q′′)~x′′C(0)(~x′′)ei (~q′′−~q′′′)~x′′′A−1(0)(~x′′′)ei ~q
′′′~xIV d(1)(~xIV )
= 12
∫
d2x
C(0)d(1)e(1)
4A(0)B(0) − C(0)C(0) . (4.390)
dACBe-term: Here we get
1
8
∑
~q,~q′
∑
~q′′,~q′′′
d(1)(−~q)A−1(0)(~q, ~q ′)C(0)(−~q ′′,−~q ′)B′−1(0) (~q ′′, ~q ′′′)e(1)(~q ′′′)
= 12
∫
d2x
C(0)d(1)e(1)
4A(0)B(0) − C(0)C(0) . (4.391)
1the notation refers to the names of variables in (4.361)
141
4. Effective Field Theory for the t-J Model at Low Doping
dACBCAd-term: Using again (4.370) and (4.385) this reads
1
16
∑
~q,~q′
∑
~q′′,~q′′′
∑
~qIV,~qV
d(1)(−~q)A−1(0)(~q, ~q ′)C(0)(−~q ′′,−~q ′)B′−1(0) (~q ′′, ~q ′′′)
· C(0)(~q ′′′, ~qIV )A−1(0)(−~qV ,−~qIV )e(1)(~qV )
= 14
∫
d2x
C(0)C(0)d(1)d(1)
A(0)(4A(0)B(0) − C(0)C(0)) . (4.392)
So when we sum up all contributions in O(a2) we obtain
Result 47 (O(a2)-contribution from the integral over fluctuations)
S
(2)
ζ =
∫
d2x
(
− B(0)d(1)d(1) − C(0)d(1)e(1) +A(0)e(1)e(1)4A(0)B(0) − C(0)C(0)
)
. (4.393)
Final manipulations
Let us now plug in (4.352)-(4.355). For the denominator we obtain
4A(0)B(0) − C(0)C(0) = 4
[
∆2(0) − 4 Λ(0)Λ∗(0)
]
, (4.394)
and when we insert the identities from (4.344) we have
∆2(0) − 4 Λ(0)Λ∗(0) = 64 γ22
( |z1|2 − |z2|2
|z1|2
)2
. (4.395)
The tricky part is to get the numerator into a form where the denominator cancels. The
numerator reads
−
(
B(0)d(1)d(1) − C(0)d(1)e(1) +A(0)e(1)e(1)
)
= 4 Λ(0)Ξ∗2(0) + 4 Λ∗(0)Ξ2(0) + 4 ∆(0)Ξ∗(0)Ξ(0)
+ 4 ∆(0)
(
Ξ∗(1)Γ(1) − Ξ(1)Γ∗(1)
)
+ 8 Λ∗(0)Γ(1)Ξ(1) − 8 Λ(0)Γ∗(1)Ξ∗(1)
+ 4 ∆(0)Γ∗(1)Γ(1) − 4 Λ∗(0)Γ2(1) − 4 Λ(0)Γ∗2(1) . (4.396)
The different summands may be grouped into three contributions. The first, involving
142
4.5. Integrating out small bosonic fluctuations
two time derivatives. The second with only one time derivative and the third, containing
derivatives with respect to the x and y only. These three groups will lead to distinct
contributions to the effective action, so we expect the denominator to cancel for each
group. After some lengthy algebra, see [96], we finally obtain
∆(0)Ξ∗(0)Ξ(0) + Λ∗(0)Ξ2(0) + Λ(0)Ξ∗2(0)
∆2(0) − 4 Λ(0)Λ∗(0)
= γ
2
1
8 γ2
[
(z¯ ∂τz)2 + ∂τ z¯ ∂τz
]
,
∆(0)
(
Ξ∗(1)Γ(1) − Ξ(1)Γ∗(1)
)
+ 2 Λ∗(0)Γ(1)Ξ(1) − 2 Λ(0)Γ∗(1)Ξ∗(1)
∆2(0) − 4 Λ(0)Λ∗(0)
= 0 ,
∆(0)Γ∗(1)Γ(1) − Λ∗(0)Γ(1)Γ(1) − Λ(0)Γ∗(1)Γ∗(1)
∆2(0) − 4 Λ(0)Λ∗(0)
= γ24
[
(∂x − ∂y)z¯ (∂x − ∂y)z +
(
z¯ (∂x − ∂y)z
)2]
. (4.397)
Total contribution from the ζ-part of the effective action in O(a2)
Finally, we summarize all O(a2)-contributions from (4.397). We plug in γ1 and γ2 and
arrive at
Result 48 (O(a2)-contribution from the integral over fluctuations)
S
(2)
ζ =
∫
dτ d2x
(
(1 + 2ρ˜)2
2a2 (J + 4Jρ˜+ 16t2ρ˜κ1)
[(
z¯ ∂τz
)2
+ ∂τ z¯ ∂τz
]
+ 14(J + 4Jρ˜+ 16t
2ρ˜κ1)
[
(∂x − ∂y)z¯ (∂x − ∂y)z +
(
z¯ (∂x − ∂y)z
)2])
. (4.398)
4.5.3. Contribution in O(a3)
Let us now evaluate the O(a3)-contribution from (4.361). We have
dAd-term:(
1
4
∑
~q,~q′
d(−~q)A−1(~q, ~q ′) d(~q ′)
)
(3)
143
4. Effective Field Theory for the t-J Model at Low Doping
=
∫
d2x
[
d(1)(~x)d(2)(~x)
2A(0)(~x) −
d(1)(~x)d(1)(~x)A(1)(~x)
4A(0)(~x)A(0)(~x)
]
. (4.399)
eBe-term:
(
1
4
∑
~q,~q′
e(−~q)B′−1(~q, ~q ′) e(~q ′)
)
(3)
=
∫
d2x
[
2A(0)e(1)e(2)
4A(0)B(0) − C(0)C(0)
+
(
− 4A(0)A(0) B(1) + 2A(0)C(0) C(1a) −A(1)C(0) C(0)
)
e(1)e(1)(
4A(0)B(0) − C(0) C(0)
)2
+
(
2iA(0)∂αC(0) C(1b) − 2i ∂αA(0)C(0) C(1b)
)
e(1)e(1)(
4A(0)B(0) − C(0) C(0)
)2
]
. (4.400)
eBCAd-term:
(
1
8
∑
~q,~q′
∑
~q′′,~q′′′
e(−~q)B′−1(~q, ~q ′)C(~q ′, ~q ′′)A−1(−~q ′′′,−~q ′′)d(~q ′′′)
)
(3)
= 18
∑
~q,~q′
∑
~q′′,~q′′′
(
e(2)(−~q)B′−1(0) (~q, ~q ′)C(0)(~q ′, ~q ′′)A−1(0)(~q ′′, ~q ′′′)d(1)(~q ′′′)
+ e(1)(−~q)B′−1(1) (~q, ~q ′)C(0)(~q ′, ~q ′′)A−1(0)(~q ′′, ~q ′′′)d(1)(~q ′′′)
+ e(1)(−~q)B′−1(0) (~q, ~q ′)C(1)(~q ′, ~q ′′)A−1(0)(~q ′′, ~q ′′′)d(1)(~q ′′′)
+ e(1)(−~q)B′−1(0) (~q, ~q ′)C(0)(~q ′, ~q ′′)A−1(1)(~q ′′, ~q ′′′)d(1)(~q ′′′)
+ e(1)(−~q)B′−1(0) (~q, ~q ′)C(0)(~q ′, ~q ′′)A−1(0)(~q ′′, ~q ′′′)d(2)(~q ′′′) , (4.401)
where we used A−1(−~q ′,−~q) = A−1(~q, ~q ′). We examine the summands one by one. The
first reads
1
8
∑
~q,~q′
∑
~q′′,~q′′′
e(2)(−~q)B′−1(0) (~q, ~q ′)C(0)(~q ′, ~q ′′)A−1(0)(~q ′′, ~q ′′′)d(1)(~q ′′′)
= 12
∫
d2x
C(0)d(1)e(2)
4A(0)B(0) − C(0)C(0) . (4.402)
For the second we get
144
4.5. Integrating out small bosonic fluctuations
1
8
∑
~q,~q′
∑
~q′′,~q′′′
e(1)(−~q)B′−1(1) (~q, ~q ′)C(0)(~q ′, ~q ′′)A−1(0)(~q ′′, ~q ′′′)d(1)(~q ′′′)
= 18
∫
d2x
C(0)d(1)e(1)
A(0)
(−16A(0)A(0) B(1) + 8A(0)C(0) C(1a) − 4A(1)C(0) C(0)(
4A(0)B(0) − C(0) C(0)
)2
+ 8iA(0)∂αC(0) C(1b) − 8i ∂αA(0)C(0) C(1b)(
4A(0)B(0) − C(0) C(0)
)2
)
. (4.403)
The third one features terms proportional to q′α and q′′α
1
8
∑
~q,~q′
∑
~q′′,~q′′′
e(1)(−~q)B′−1(0) (~q, ~q ′)C(1)(~q ′, ~q ′′)A−1(0)(~q ′′, ~q ′′′)d(1)(~q ′′′)
= 18
∑
~q,~q′
∑
~q′′,~q′′′
e(1)(−~q)B′−1(0) (~q, ~q ′)
(
C(1a)(~q ′, ~q ′′)
+ (q′α + q′′α)C(1b)(~q ′, ~q ′′)
)
A−1(0)(~q ′′, ~q ′′′)d(1)(~q ′′′)
= 12
∫
d2x
C(1a)d(1)e(1)
4A(0)B(0) − C(0)C(0)
− i8
∫
d2x
[
∂α
(
e(1)
B′(0)
)C(1b)d(1)
A(0) −
e(1)C(1b)
B′(0)
∂α
(
d(1)
A(0)
)]
. (4.404)
And the fourth,
1
8
∑
~q,~q′
∑
~q′′,~q′′′
e(1)(−~q)B′−1(0) (~q, ~q ′)C(0)(~q ′, ~q ′′)A−1(1)(~q ′′, ~q ′′′)d(1)(~q ′′′)
= 12
∫
d2x
−A(1)C(0)d(1)e(1)
A(0)(4A(0)B(0) − C(0)C(0)) . (4.405)
Finally, the fifth summand reads
1
8
∑
~q,~q′
∑
~q′′,~q′′′
e(1)(−~q)B′−1(0) (~q, ~q ′)C(0)(~q ′, ~q ′′)A−1(0)(~q ′′, ~q ′′′)d(2)(~q ′′′)
= 12
∫
d2x
C(0)d(2)e(1)
4A(0)B(0) − C(0)C(0) . (4.406)
dACBe-term: This term gives the same contribution as eBCAd.
145
4. Effective Field Theory for the t-J Model at Low Doping
dACBe + eBCAd: After some algebra, we obtain the form
2
(
1
8
∑
~q,~q′
∑
~q′′,~q′′′
e(−~q)B′−1(~q, ~q ′)C(~q ′, ~q ′′)A−1(−~q ′′′,−~q ′′)d(~q ′′′)
)
(3)
=
∫
d2x
(C(0)d(1)e(2) + C(1a)d(1)e(1) + C(0)d(2)e(1)
4A(0)B(0) − C(0)C(0)
+
(
− 4A(0)B(1) + 2 C(0)C(1a) − 4A(1)B(0)
)
C(0)d(1)e(1)(
4A(0)B(0) − C(0) C(0)
)2
− iC(1b)d(1)∂αe(1) − C(1b)∂αd(1) e(1)4A(0)B(0) − C(0)C(0)
− 4i
(
∂αA(0) B(0) −A(0)∂αB(0)
)
C(1b)d(1)e(1)(
4A(0)B(0) − C(0)C(0)
)2
)
. (4.407)
dACBCAd-term:
(
1
16
∑
~q,~q′
∑
~q′′,~q′′′
∑
~qIV,~qV
d(−~q)A−1(~q, ~q ′)C(−~q ′′,−~q ′)B′−1(~q ′′, ~q ′′′)
· C(~q ′′′, ~qIV )A−1(−~qV ,−~qIV )d(~qV )
)
(3)
, (4.408)
which decomposes into seven summands. The first one reads
1
16
∑
~q,~q′
∑
~q′′,~q′′′
∑
~qIV,~qV
d(2)(−~q)A−1(0)(~q, ~q ′)C(0)(−~q ′′,−~q ′)B′−1(0) (~q ′′, ~q ′′′)
· C(0)(~q ′′′, ~qIV )A−1(0)(−~qV ,−~qIV )d(1)(~qV )
= 14
∫
d2x
C(0)C(0)d(1)d(2)
A(0)(4A(0)B(0) − C(0)C(0)) . (4.409)
The second summand is given by
1
16
∑
~q,~q′
∑
~q′′,~q′′′
∑
~qIV,~qV
d(1)(−~q)A−1(1)(~q, ~q ′)C(0)(−~q ′′,−~q ′)B′−1(0) (~q ′′, ~q ′′′)
· C(0)(~q ′′′, ~qIV )A−1(0)(−~qV ,−~qIV )d(1)(~qV )
= 14
∫
d2x
−A(1)C(0)C(0)d(1)d(1)
A(0)A(0)(4A(0)B(0) − C(0)C(0)) . (4.410)
146
4.5. Integrating out small bosonic fluctuations
Then the third
1
16
∑
~q,~q′
∑
~q′′,~q′′′
∑
~qIV,~qV
d(1)(−~q)A−1(0)(~q, ~q ′)C(1)(−~q ′′,−~q ′)B′−1(0) (~q ′′, ~q ′′′)
· C(0)(~q ′′′, ~qIV )A−1(0)(−~qV ,−~qIV )d(1)(~qV )
= 14
∫
d2x
C(0)C(1a)d(1)d(1)
A(0)(4A(0)B(0) − C(0)C(0))
+ i16
2
N
∑
~q,~q′
∫
d2x e−i~q~xe−i~q′~x(−iqα + iq′α)
[
d(1)
A(0)
]
(~q) C(1b)(~x)
[ C(0)d(1)
A(0)B′(0)
]
(~q ′)
= 14
∫
d2x
C(0)C(1a)d(1)d(1)
A(0)(4A(0)B(0) − C(0)C(0))
+ i16
∫
d2x
[
∂α
(
d(1)
A(0)
)C(0)C(1b)d(1)
A(0)B′(0)
− C(1b)d(1)A(0) ∂α
( C(0)d(1)
A(0)B′(0)
)]
= 14
∫
d2x
d(1)C(0)C(1a)d(1)
A(0)
(
4A(0)B(0) − C(0)C(0)
)
+ i4
∫
d2x
(
− d(1)∂αC(0) C(1b)d(1)A(0)
(
4A(0)B(0) − C(0)C(0)
) + 4 d(1)C(0)∂αB′(0)C(1b)d(1)(
4A(0)B(0) − C(0)C(0)
)2
)
. (4.411)
Next is the fourth summand
1
16
∑
~q,~q′
∑
~q′′,~q′′′
∑
~qIV,~qV
d(1)(−~q)A−1(0)(~q, ~q ′)C(0)(−~q ′′,−~q ′)B′−1(1) (~q ′′, ~q ′′′)
· C(0)(~q ′′′, ~qIV )A−1(0)(−~qV ,−~qIV )d(1)(~qV )
= 14
∫
d2x
(−4A(0) B(1)C(0)C(0)d(1)d(1) + 2 C(0)C(0)C(0) C(1a)d(1)d(1)
A(0)
(
4A(0)B(0) − C(0) C(0)
)2
+ 2i C(0)C(0)∂αC(0) C(1b)d(1)d(1)
A(0)
(
4A(0)B(0) − C(0) C(0)
)2
− A(1)C(0)C(0)C(0) C(0)d(1)d(1) + 2i ∂αA(0)C(0)C(0)C(0) C(1b)d(1)d(1)
A(0)A(0)
(
4A(0)B(0) − C(0) C(0)
)2
)
, (4.412)
and the fifth, which reads
1
16
∑
~q,~q′
∑
~q′′,~q′′′
∑
~qIV,~qV
d(1)(−~q)A−1(0)(~q, ~q ′)C(0)(−~q ′′,−~q ′)B′−1(0) (~q ′′, ~q ′′′)
· C(1)(~q ′′′, ~qIV )A−1(0)(−~qV ,−~qIV )d(1)(~qV )
147
4. Effective Field Theory for the t-J Model at Low Doping
= 14
∫
d2x
C(0)C(1a)d(1)d(1)
A(0)(4A(0)B(0) − C(0)C(0))
− i16
2
N
∑
~q,~q′
∫
d2x e−i~q~xe−i~q′~x(−iqα + iq′α)
[ C(0)d(1)
A(0)B′(0)
]
(~q) C(1b)(~x)
[
d(1)
A(0)
]
(~q ′)
= 14
∫
d2x
d(1)C(0)C(1a)d(1)
A(0)
(
4A(0)B(0) − C(0)C(0)
)
+ i4
∫
d2x
(
− d(1)∂αC(0) C(1b)d
∗
(1)
A(0)
(
4A(0)B(0) − C(0)C(0)
) + 4 d(1)C(0)∂αB′(0)C(1b)d(1)(
4A(0)B(0) − C(0)C(0)
)2
)
. (4.413)
Then the sixth summand,
1
16
∑
~q,~q′
∑
~q′′,~q′′′
∑
~qIV,~qV
d(1)(−~q)A−1(0)(~q, ~q ′)C(0)(−~q ′′,−~q ′)B′−1(0) (~q ′′, ~q ′′′)
· C(0)(~q ′′′, ~qIV )A−1(1)(−~qV ,−~qIV )d(1)(~qV )
= 14
∫
d2x
−A(1)C(0)C(0)d(1)d(1)
A(0)A(0)(4A(0)B(0) − C(0)C(0)) . (4.414)
And finally, the seventh summand
1
16
∑
~q,~q′
∑
~q′′,~q′′′
∑
~qIV,~qV
d(1)(−~q)A−1(0)(~q, ~q ′)C(0)(−~q ′′,−~q ′)B′−1(0) (~q ′′, ~q ′′′)
· C(0)(~q ′′′, ~qIV )A−1(0)(−~qV ,−~qIV )d(2)(~qV )
= 14
∫
d2x
C(0)C(0)d(1)d(2)
A(0)(4A(0)B(0) − C(0)C(0)) . (4.415)
When we now sum up all contributions to the dACBCAd-term, we see that the first and
the last, the 2nd and 6th, the 3rd and 5th are all equal. After some algebra, it simplifies
to
(
1
16
∑
~q,~q′
∑
~q′′,~q′′′
∑
~qIV,~qV
d(−~q)A−1(~q, ~q ′)C(−~q ′′,−~q ′)B′−1(~q ′′, ~q ′′′)
· C(~q ′′′, ~qIV )A−1(−~qV ,−~qIV )d(~qV )
)
(3)
= 14
∫
d2x
(
2 C(0)C(0)d(1)d(2)
A(0)
(
4A(0)B(0) − C(0)C(0)
)
+ 8B(0)C(0)C(A1)d(1)d(1) − 4B(1)C(0)C(0)d(1)d(1)(
4A(0)B(0) − C(0)C(0)
)2
148
4.5. Integrating out small bosonic fluctuations
− 8A(1)B(0)C(0)C(0)d(1)d(1)
A(0)
(
4A(0)B(0) − C(0)C(0)
)2 + A(1)C(0)C(0)C(0)C(0)d(1)d(1)A(0)A(0)(4A(0)B(0) − C(0)C(0))2
+
8i
(
∂αB(0)C(0) − B(0)∂αC(0)
)
C(1b)d(1)d(1)(
4A(0)B(0) − C(0)C(0)
)2
)
. (4.416)
Summarizing the O(a3) contribution to the ζ-part of the effective action
The total O(a3)-contribution from integrating out fast fluctuations, see (4.361), is given
by
S
(3)
ζ =
∫
d2x
(
− 14dAd−
1
4eBe +
1
8eBCAd +
1
8dACBe−
1
16dACBCAd
)
, (4.417)
where the summands were derived in (4.399), (4.400), (4.407) and (4.416). Some further
algebra leads to
Result 49 (O(a3)-contribution from integrating out fluctuations)
S
(3)
ζ =
∫
d2x
(
− 2B(0)d(1)d(2) − C(0)
(
d(1)e(2) + d(2)e(1)
)
+ 2A(0)e(1)e(2)
4A(0)B(0) − C(0)C(0)
− A(1)e(1)e(1) − C(1a)d(1)e(1) + B(1)d(1)d(1)4A(0)B(0) − C(0)C(0)
+ 2
(
2A(1)B(0) − C(0)C(1a) + 2A(0)B(1)
)(
A(0)e(1)e(1) − C(0)d(1)e(1) + B(0)d(1)d(1)
)
(
4A(0)B(0) − C(0)C(0)
)2
− i C(1b)
(
d(1)∂αe(1) − ∂αd(1)e(1)
)
4A(0)B(0) − C(0)C(0) + 2i
(
B(0)∂αC(0) − ∂αB(0)C(0)
)
C(1b)d(1)d(1)(
4A(0)B(0) − C(0)C(0)
)2
+4i
(
A(0)∂αB(0) − ∂αA(0)B(0)
)
C(1b)d(1)e(1)(
4A(0)B(0) − C(0)C(0)
)2 −2i
(
A(0)∂αC(0) − ∂αA(0)C(0)
)
C(1b)e(1)e(1)(
4A(0)B(0) − C(0) C(0)
)2
)
.
(4.418)
Final manipulations
Now let us plug in (4.352)-(4.356).
149
4. Effective Field Theory for the t-J Model at Low Doping
Denominators: From (4.394), we get
(
4A(0)B(0) − C(0)C(0)
)2
= 16
[
∆2(0) − 4 Λ(0)Λ∗(0)
]2
. (4.419)
Terms proportional to d(2) and e(2):
d(1)d(2) =
[(
Ξ(1) − Ξ∗(1)
)
−
(
Γ(1) + Γ∗(1)
)][
Γ(2) − Γ∗(2)
]
,
e(1)e(2) = −
[(
Ξ(1) + Ξ∗(1)
)
−
(
Γ(1) − Γ∗(1)
)][(
Γ(2) + Γ∗(2)
)]
,
d(1)e(2) =
[(
Ξ(1) − Ξ∗(1)
)
−
(
Γ(1) + Γ∗(1)
)][
− i
(
Γ(2) + Γ∗(2)
)]
,
e(1)d(2) =
[
− i
(
Ξ(1) + Ξ∗(1)
)
+ i
(
Γ(1) − Γ∗(1)
)][(
Γ(2) − Γ∗(2)
)]
. (4.420)
Then, there is
2B(0)
4A(0)B(0) − C(0)C(0) =
∆2(0) − (Λ(0) + Λ∗(0))
2
(
∆(0) − 4 Λ∗(0)Λ(0)
) ,
2A(0)
4A(0)B(0) − C(0)C(0) =
∆2(0) + (Λ(0) + Λ∗(0))
2
(
∆(0) − 4 Λ∗(0)Λ(0)
) ,
−C(0)
4A(0)B(0) − C(0)C(0) =
i (Λ(0) − Λ∗(0))
2
(
∆(0) − 4 Λ∗(0)Λ(0)
) , (4.421)
such that we obtain
2B(0)d(1)d(2) − C(0)
(
d(1)e(2) + d(2)e(1)
)
+ 2A(0)e(1)e(2)
4A(0)B(0) − C(0)C(0)
= − ∆(0)∆(0) − 4 Λ∗(0)Λ(0)
[(
Ξ∗(1) + Γ∗(1)
)
Γ(2) +
(
Ξ(1) − Γ(1)
)
Γ∗(2)
]
− 2 Λ(0)∆(0) − 4 Λ∗(0)Λ(0)
(
Ξ∗(1) + Γ∗(1)
)
Γ∗(2) − 2
Λ∗(0)
∆(0) − 4 Λ∗(0)Λ(0)
(
Ξ(1) − Γ(1)
)
Γ(2) . (4.422)
Terms proportional to d(1) and e(1): Here we get
− A(1)e(1)e(1) − C(1a)d(1)e(1) + B(1)d(1)d(1)4A(0)B(0) − C(0)C(0)
150
4.5. Integrating out small bosonic fluctuations
= 12
∆(1)Θ(2a) + 2 Λ∗(1)Θ(2b) − 2 Λ(1)Θ(2c)
∆2(0) − 4 Λ(0)Λ∗(0)
(4.423)
and
2
(
2A(1)B(0) − C(0)C(1a) + 2A(0)B(1)
)(
A(0)e(1)e(1) − C(0)d(1)e(1) + B(0)d(1)d(1)
)
(
4A(0)B(0) − C(0)C(0)
)2
=
(
∆(0)∆(1) + 2 Λ∗(1)Λ(0) − 2 Λ∗(0)Λ(1)
)(
∆(0)Θ(2a) − 2 Λ∗(0)Θ(2b) − 2 Λ(0)Θ(2c)
)
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 , (4.424)
where we defined
Θ(2a) = −2
(
Ξ∗(1) + Γ∗(1)
)(
Ξ(1) − Γ(1)
)
,
Θ(2b) =
(
Ξ(1) − Γ(1)
)2
,
Θ(2c) = Ξ∗2(1) + Γ∗2(1) + 2 Ξ∗(1)Γ∗(1) . (4.425)
Terms with additional derivatives: The d(1)d(1)-term reads
2i
(
B(0)∂αC(0) − ∂αB(0)C(0)
)
C(1b)(
4A(0)B(0) − C(0)C(0)
)2 d(1)d(1)
= −
(
∆(0) − (Λ(0) + Λ∗(0)
) (
∂αΛ(0) − ∂αΛ∗(0)
)
Π(0)
2
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (Θ(2a) + Θ(2b) + Θ(2c))
+
(
∂α∆(0) − (∂αΛ(0) + ∂αΛ∗(0))
)(
Λ(0) − Λ∗(0)
)
Π(0)
2
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (Θ(2a) + Θ(2b) + Θ(2c)) . (4.426)
Then we have the e(1)e(1)-term
− 2i
(
A(0)∂αC(0) − ∂αA(0)C(0)
)
C(1b)(
4A(0)B(0) − C(0)C(0)
)2 d(1)d(1)
=
(
∆(0) + (Λ(0) + Λ∗(0)
) (
∂αΛ(0) − ∂αΛ∗(0)
)
Π(0)
2
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (Θ(2a) −Θ(2b) −Θ(2c))
151
4. Effective Field Theory for the t-J Model at Low Doping
−
(
∂α∆(0) + (∂αΛ(0) + ∂αΛ∗(0))
)(
Λ(0) − Λ∗(0)
)
Π(0)
2
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (Θ(2a) −Θ(2b) −Θ(2c)) . (4.427)
And the d(1)e(1)-contribution
4i
(
A(0)∂αB(0) − ∂αA(0)B(0)
)
C(1b)(
4A(0)B(0) − C(0) C(0)
)2 d(1)e(1)
= −
(
∆(0) + (Λ(0) + Λ∗(0)
) (
∂α∆(0) − (∂αΛ(0) + ∂αΛ∗(0))
)
Π(0)
2
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (Θ(2b) −Θ(2c))
+
(
∂α∆(0) + (∂αΛ(0) + ∂αΛ∗(0))
)(
∆(0) − (Λ(0) + Λ∗(0))
)
Π(0)
2
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (Θ(2b) −Θ(2c)) . (4.428)
And when we sum up the three contributions we get
4
Π(0)
(
Λ∗(0)∂αΛ(0) − Λ(0)∂αΛ∗(0)
)
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (Ξ∗(1) + Γ∗(1))(Ξ(1) − Γ(1))
+ 2
Π(0)
(
∆(0)∂αΛ∗(0) − Λ∗(0)∂α∆(0)
)
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (Ξ(1) − Γ(1))2
− 2 Π(0)
(
∆(0)∂αΛ(0) − Λ(0)∂α∆(0)
)
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (Ξ∗(1) + Γ∗(1))2 . (4.429)
Now let us look at the part with derivatives of d(1) and e(1):
− i C(1b) ∂αd(1)e(1)4A(0)B(0) − C(0)C(0) + i
C(1b) d(1)∂αe(1)
4A(0)B(0) − C(0)C(0)
= Π(0)∆2(0) − 4 Λ(0)Λ∗(0)
((
Ξ(1) − Γ(1)
)
∂α
(
Ξ∗(1) + Γ∗(1)
)
−
(
Ξ∗(1) + Γ∗(1)
)
∂α
(
Ξ(1) − Γ(1)
))
. (4.430)
Finally, we summarize all O(a3)-contributions to the ζ-part of the action (4.418), which
were derived in (4.422), (4.423), (4.424), (4.429), and (4.430)
152
4.5. Integrating out small bosonic fluctuations
Result 50 (O(a3)-contributions to the ζ-part in terms of ∆, . . . ,Γ)
S
(3)
ζ =
∫
d2x
 1
∆2(0) − 4 Λ∗(0)Λ(0)
(
∆(0)
[(
Ξ∗(1) + Γ∗(1)
)
Γ(2) +
(
Ξ(1) − Γ(1)
)
Γ∗(2)
]
+ 2 Λ∗(0)
(
Ξ(1) − Γ(1)
)
Γ(2) + 2 Λ(0)
(
Ξ∗(1) + Γ∗(1)
)
Γ∗(2)
)
+ 1∆2(0) − 4 Λ(0)Λ∗(0)
(
∆(1)
(
Ξ∗(1) + Γ∗(1)
)(
Ξ(1) − Γ(1)
)
− Λ∗(1)
(
Ξ(1) − Γ(1)
)2
+ Λ(1)
(
Ξ∗(1) + Γ∗(1)
)2)
− 2 ∆(0)∆(1) + 2
(
Λ∗(1)Λ(0) − Λ∗(0)Λ(1)
)
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2
(
∆(0)
(
Ξ∗(1) + Γ∗(1)
)(
Ξ(1) − Γ(1)
)
+ Λ∗(0)
(
Ξ(1) − Γ(1)
)2
+ Λ(0)
(
Ξ∗(1) + Γ∗(1)
)2)
− 2 Π(0)(
∆2(0) − 4 Λ(0)Λ∗(0)
)2
(
2
[
Λ∗(0)∂αΛ(0) − Λ(0)∂αΛ∗(0)
](
Ξ∗(1) + Γ∗(1)
)(
Ξ(1) − Γ(1)
)
−
[
∆(0)∂αΛ∗(0) − Λ∗(0)∂α∆(0)
](
Ξ(1) − Γ(1)
)2
+
[
∆(0)∂αΛ(0) − Λ(0)∂α∆(0)
] (
Ξ∗(1) + Γ∗(1)
)2)
+ Π(0)∆2(0) − 4 Λ(0)Λ∗(0)
((
Ξ(1) − Γ(1)
)
∂α
(
Ξ∗(1) + Γ∗(1)
)
−
(
Ξ∗(1) + Γ∗(1)
)
∂α
(
Ξ(1) − Γ(1)
)) . (4.431)
This is still a fairly complicated expression, so we split it into four summands
S
(3)
ζ = S
(3,τττ)
ζ + S
(3,ττα)
ζ + S
(3,ταα)
ζ + S
(3,ααα)
ζ , (4.432)
characterized by the number of time derivatives τ and spatial derivatives α. Each of
these summands will lead to a separate term in the effective action so we can be sure
that the denominators in (4.431) cancel for each one of them. Note that besides Ξ(1),
the term with only one time derivative, there are time derivatives included in ∆(1), Λ(1)
153
4. Effective Field Theory for the t-J Model at Low Doping
and Γ(1) as well. So we separate
∆(1) = ∆(1,α) −∆(1,τ) ,
Λ(1) = Λ(1,α) − Λ(1,τ) ,
Γ(2) = Γ(2,αα) + Γ(2,τα) . (4.433)
where
∆(1,α) = 8γ3
( |z1|4 + |z2|4
|z1|4
(
z1(∂x − ∂y)z∗1 − z∗1(∂x − ∂y)z1
)
+2 |z2|
2
|z1|2
(
z2(∂x − ∂y)z∗2 − z∗2(∂x − ∂y)z2
))
,
∆(1,τ) = 8 γ5
|z1|4 + |z2|4
|z1|2 (z¯ ∂τz) ,
Λ(1,α) = 16 γ3 z22
(
z1(∂x − ∂y)z∗1
|z1|2 −
z∗2(∂x − ∂y)z2
|z2|2
)
,
Λ(1,τ) = 8 γ5 z22
(
z¯ ∂τz
)
,
Γ(2,αα) = 2
√
2 γ3 z2
(
z∗1z
∗
2
|z2|2
(
γ4Fxx − 2 γ3Fxy + γ4Fyy
)
+ z1z2|z1|2
(
γ4F
∗
xx − 2 γ3F ∗xy + γ4F ∗yy
))
,
Γ(2,τα) = 2
√
2γ5 z2
(
z¯ ∂τz
)(z∗1z∗2
|z2|2 (Fx − Fy) +
z1z2
|z1|2 (F
∗
x − F ∗y )
)
. (4.434)
With these definitions we arrive at contributions with:
• three time derivatives:
S
(3,τττ)
ζ =
∫
d2x
[
− ∆(1,τ)Ξ
∗
(1)Ξ(1) − Λ∗(1,τ)Ξ2(1) + Λ(1,α)Ξ∗2(1)
∆2(0) − 4 Λ(0)Λ∗(0)
+ 2
∆(0)∆(1,τ) + 2
(
Λ∗(1,τ)Λ(0) − Λ∗(0)Λ(1,τ)
)
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (∆(0)Ξ∗(1)Ξ(1) + Λ∗(0)Ξ2(1) + Λ(0)Ξ∗2(1))
]
,
(4.435)
• two time derivatives:
154
4.5. Integrating out small bosonic fluctuations
S
(3,ττα,A)
ζ =
∫
d2x
[∆(1,α)Ξ∗(1)Ξ(1) − Λ∗(1,α)Ξ2(1) + Λ(1,α)Ξ∗2(1)
∆2(0) − 4 Λ(0)Λ∗(0)
− 2 ∆(0)∆(1,α) + 2
(
Λ∗(1,α)Λ(0) − Λ∗(0)Λ(1,α)
)
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (∆(0)Ξ∗(1)Ξ(1) + Λ∗(0)Ξ2(1) + Λ(0)Ξ∗2(1))
− 2 Π(0)(
∆2(0) − 4 Λ(0)Λ∗(0)
)2(2 (Λ∗(0)∂αΛ(0) − Λ(0)∂αΛ∗(0))Ξ∗(1)Ξ(1)
−
(
∆(0)∂αΛ∗(0) − Λ∗(0)∂α∆(0)
)
Ξ2(1) +
(
∆(0)∂αΛ(0) − Λ(0)∂α∆(0)
)
Ξ∗2(1)
)
− Π(0)∆2(0) − 4 Λ(0)Λ∗(0)
(
Ξ∗(1)∂αΞ(1) − Ξ(1)∂αΞ∗(1)
)]
, (4.436)
S
(3,ττα,B)
ζ =
∫
d2x
[∆(0)(Ξ∗(1)Γ(2,τα) + Ξ(1)Γ∗(2,τα))+ 2 (Λ∗(0)Ξ(1)Γ(2,τα) + Λ(0)Ξ∗(1)Γ∗(2,τα))
∆2(0) − 4 Λ(0)Λ∗(0)
+
∆(1,τ)
(
Ξ∗(1)Γ(1) − Ξ(1)Γ∗(1)
)
− 2
(
Λ∗(1,τ)Ξ(1)Γ(1) + Λ(1,τ)Ξ∗(1)Γ∗(1)
)
∆2(0) − 4 Λ(0)Λ∗(0)
− 2 ∆(0)∆(1,τ) + 2
(
Λ∗(1,τ)Λ(0) − Λ∗(0)Λ(1,τ)
)
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (∆(0)(Ξ∗(1)Γ(1) − Γ∗(1)Ξ(1))
+ 2
(
Λ∗(0)Ξ(1)Γ(1) − Λ(0)Ξ∗(1)Γ∗(1)
))]
, (4.437)
• one time derivative:
S
(3,ταα,A)
ζ =
∫
d2x
[∆(0)(Ξ∗(1)Γ(2,αα) + Ξ(1)Γ∗(2,αα))+ 2 (Λ∗(0)Ξ(1)Γ(2,αα) + Λ(0)Ξ∗(1)Γ∗(2,αα))
∆2(0) − 4 Λ(0)Λ∗(0)
− ∆(1,α)
(
Ξ∗(1)Γ(1) − Ξ(1)Γ∗(1)
)
− 2
(
Λ∗(1,α)Ξ(1)Γ(1) + Λ(1,α)Ξ∗(1)Γ∗(1)
)
∆2(0) − 4 Λ(0)Λ∗(0)
+ 2
∆(0)∆(1,α) + 2
(
Λ∗(1,α)Λ(0) − Λ∗(0)Λ(1,α)
)
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (∆(0)(Ξ∗(1)Γ(1) − Γ∗(1)Ξ(1))
+ 2
(
Λ∗(0)Ξ(1)Γ(1) − Λ(0)Ξ∗(1)Γ∗(1)
))
+ 4 Π(0)(
∆2(0) − 4 Λ(0)Λ∗(0)
)2((Λ∗(0)∂αΛ(0) − Λ(0)∂αΛ∗(0))(Ξ∗(1)Γ(1) − Γ∗(1)Ξ(1))
155
4. Effective Field Theory for the t-J Model at Low Doping
−
(
∆(0)∂αΛ∗(0) − Λ∗(0)∂α∆(0)
)
Ξ(1)Γ(1) +
(
∆(0)∂αΛ(0) − Λ(0)∂α∆(0)
)
Ξ∗(1)Γ∗(1)
)
+ Π(0)∆2(0) − 4 Λ(0)Λ∗(0)
(
Ξ∗(1)∂αΓ(1) − Γ∗(1)∂αΞ(1) + Ξ(1)∂αΓ∗(1) − Γ(1)∂αΞ∗(1)
)]
, (4.438)
S
(3,ταα,B)
ζ =
∫
d2x
[∆(0)(Γ∗(1)Γ(2,τα) − Γ(1)Γ∗(2,τα))− 2 (Λ∗(0)Γ(1)Γ(2,τα) − Λ(0)Γ∗(1)Γ∗(2,τα))
∆2(0) − 4 Λ(0)Λ∗(0)
+
∆(1,τ)Γ∗(1)Γ(1) + Λ∗(1,τ)Γ(1)Γ(1) − Λ(1,τ)Γ∗(1)Γ∗(1)
∆2(0) − 4 Λ(0)Λ∗(0)
− 2 ∆(0)∆(1,τ) + 2
(
Λ∗(1,τ)Λ(0) − Λ∗(0)Λ(1,τ)
)
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (∆(0)Γ∗(1)Γ(1)
− Λ∗(0)Γ(1)Γ(1) − Λ(0)Γ∗(1)Γ∗(1)
)]
, (4.439)
• no time derivatives:
S
(3,ααα)
ζ =
∫
d2x
[∆(0)(Γ∗(1)Γ(2,αα) − Γ(1)Γ∗(2,αα))− 2 (Λ∗(0)Γ(1)Γ(2,αα) + Λ(0)Γ∗(1)Γ∗(2,αα))
∆2(0) − 4 Λ(0)Λ∗(0)
− ∆(1,α)Γ
∗
(1)Γ(1) + Λ∗(1,α)Γ2(1) − Λ(1,α)Γ∗2(1)
∆2(0) − 4 Λ(0)Λ∗(0)
+ 2
∆(0)∆(1,α) + 2
(
Λ∗(1,α)Λ(0) − Λ∗(0)Λ(1,α)
)
(
∆2(0) − 4 Λ(0)Λ∗(0)
)2 (∆(0)Γ∗(1)Γ(1) − (Λ∗(0)Γ2(1) + Λ(0)Γ∗2(1)))
+ 2 Π(0)(
∆2(0) − 4 Λ(0)Λ∗(0)
)2(2 (Λ∗(0)∂αΛ(0) − Λ(0)∂αΛ∗(0))Γ∗(1)Γ(1)
+
(
∆(0)∂αΛ∗(0) − Λ∗(0)∂α∆(0)
)
Γ2(1) −
(
∆(0)∂αΛ(0) − Λ(0)∂α∆(0)
)
Γ∗2(1)
)
+ Π(0)∆2(0) − 4 Λ(0)Λ∗(0)
(
Γ∗(1)∂αΓ(1) − Γ(1)∂αΓ∗(1)
)]
. (4.440)
Final manipulations
Finally, we plug in (4.344) and (4.434). After some extraordinarily long algebra, see [96],
we obtain
156
4.5. Integrating out small bosonic fluctuations
• The contribution with three time derivatives
S
(3,τττ)
ζ =
1
8
γ21γ5
γ22
∫
dτ d2x (z¯ ∂τz)
(
∂τ z¯ ∂τz + (z¯ ∂τz)2
)
, (4.441)
where
γ21γ5
γ22
= 64 t
2ρ˜ λ4 (2ρ˜+ 1)2
a2
(
4ρ˜(J + 4 t2κ1) + J
)2 . (4.442)
• The Contribution with two time derivatives
S
(3,ττα)
ζ = −
γ21γ3
4γ22
∫
dτ d2x
[(
(z¯ ∂τz)
(
∂xz¯ ∂τz + ∂xz ∂τ z¯
)
− (z¯ ∂τz)
(
∂yz¯ ∂τz + ∂yz ∂τ z¯
)
− (z¯ ∂xz)(∂τ z¯ ∂τz)
+ (z¯ ∂yz)(∂τ z¯ ∂τz)− ∂τ z¯ ∂τ∂xz + ∂τ z¯ ∂τ∂yz
)]
, (4.443)
where
γ21γ3
γ22
= 16
√
2 t2ρ˜ κ1 (2ρ˜+ 1)2
a
(
4ρ˜(J + 4 t2κ1) + J
)2 . (4.444)
• The contribution with one time derivative
S
(3,ταα)
ζ =
∫
dτ d2x
[√
2
4
γ1γ4
γ2
(
(z¯ ∂τz)(z¯ ∂x∂xz − z ∂x∂xz¯)
+ (z¯ ∂τz)(z¯ ∂y∂yz − z ∂y∂yz¯)
)
−
√
2
2
γ1γ3
γ2
(
(z¯ ∂τz)(z¯ ∂x∂xz − z ∂x∂xz¯) + (z¯ ∂τz)(z¯ ∂y∂yz − z ∂y∂yz¯)
− (z¯ ∂τz)(z¯ ∂x∂yz − z ∂x∂yz¯)
)
+ γ5
(
(z¯ ∂τz)
(
∂xz¯ ∂xz + (z¯ ∂xz)2
)
+ (z¯ ∂τz)
(
∂yz¯ ∂yz + (z¯ ∂yz)2
)
− (z¯ ∂τz)
(
∂xz¯ ∂yz + (z¯ ∂xz)(z¯ ∂yz) + ∂yz ∂xz¯ + (z¯ ∂xz)(z¯ ∂yz)
))]
, (4.445)
with
γ1γ3
γ2
= 8
√
2 t2ρ˜ κ1 (2ρ˜+ 1)
4ρ˜(J + 4 t2κ1) + J
,
γ1γ4
γ2
= 8
√
2 t2ρ˜ κ2 (2ρ˜+ 1)
4ρ˜(J + 4 t2κ1) + J
, γ5 = 16t2ρ˜λ4 .
(4.446)
157
4. Effective Field Theory for the t-J Model at Low Doping
• Contribution with only spatial derivatives
S
(3,ααα)
ζ = −2 (γ3 − γ4)
∫
dτ d2x
(
(∂xz¯ − ∂yz¯)(∂x∂xz + ∂y∂yz)
− (z¯ ∂xz − z¯ ∂yz)(∂xz¯ ∂xz + ∂yz¯ ∂yz)
)
, (4.447)
where
γ3 = 4
√
2 a t2ρ˜ κ1 , γ4 = 4
√
2 a t2ρ˜ κ2 . (4.448)
Total O(a3)-contribution from the ζ-part of the effective action
Combining all O(a3)-contributions from (4.441), (4.443), (4.445) and (4.447) we obtain
Result 51 (O(a3)-contribution from the integral over fluctuations)
S
(3)
ζ = S
(3,τττ)
ζ + S
(3,ττα)
ζ + S
(3,ταα)
ζ + S
(3,ααα)
ζ
=
∫
dτ d2x
[
1
8
γ21γ5
γ22
(z¯ ∂τz)
(
∂τ z¯ ∂τz + (z¯ ∂τz)2
)
− γ
2
1γ3
4γ22
(
(z¯ ∂τz)
(
∂xz¯ ∂τz + ∂xz ∂τ z¯
)
− (z¯ ∂τz)
(
∂yz¯ ∂τz + ∂yz ∂τ z¯
)
− (z¯ ∂xz)(∂τ z¯ ∂τz) + (z¯ ∂yz)(∂τ z¯ ∂τz)− ∂τ z¯ ∂τ∂xz + ∂τ z¯ ∂τ∂yz
)
+
√
2
4
γ1γ4
γ2
(
(z¯ ∂τz)(z¯ ∂x∂xz − z ∂x∂xz¯) + (z¯ ∂τz)(z¯ ∂y∂yz − z ∂y∂yz¯)
)
−
√
2
2
γ1γ3
γ2
(
(z¯ ∂τz)(z¯ ∂x∂xz − z ∂x∂xz¯) + (z¯ ∂τz)(z¯ ∂y∂yz − z ∂y∂yz¯)
− (z¯ ∂τz)(z¯ ∂x∂yz − z ∂x∂yz¯)
)
+ γ5
(
(z¯ ∂τz)
(
∂xz¯ ∂xz + (z¯ ∂xz)2
)
+ (z¯ ∂τz)
(
∂yz¯ ∂yz + (z¯ ∂yz)2
)
− (z¯ ∂τz)
(
∂xz¯ ∂yz + (z¯ ∂xz)(z¯ ∂yz) + ∂yz ∂xz¯ + (z¯ ∂xz)(z¯ ∂yz)
))]
. (4.449)
158
4.5. Integrating out small bosonic fluctuations
4.5.4. Logarithmic contribution
The ln-part in the effective action will affect the measure of the path integral but even-
tually holds contribution to the effective action as well. We have
S
(log)
ζ =
1
2 Tr lnAB
′ = 12 Tr ln
(
AB − 14 AC A
−1C︸ ︷︷ ︸
M
)
. (4.450)
We want to treat this part of the calculation with special care. So in order to keep track
of the different orders of a, let us write the ζ-part of the effective action (4.328) in its
discretized form, replacing again
∫
d2x → a2∑
j
, (4.451)
so that we have the orders of a clearly separated:
Sζ =
∫
dτ
∑
j
[
a2 γ1
(
ζ¯j ∂τzj − ζj ∂τ z¯j
)
+ a2 γ2
(
2G∗jGj −
√
2G∗j(Fjx − Fjy)−
√
2 (F ∗jx − F ∗jy)Gj
)
+ 2 a3 γ3
(
(∂xG∗j)Gj − (∂yG∗j)Gj − c.c.
)
+
√
2 a3
(
G∗j
(
γ4Fjxx + 2 γ3Fjxy + γ4Fjyy
)
− c.c.
)
− a3 γ5
(
z¯j ∂τzj
)(
2G∗jGj −
√
2G∗j
(
Fjx − Fjy
)
−√2
(
F ∗jx − F ∗jy
)
Gj
)]
, (4.452)
where we redefined γ3 and γ4 from (4.325), so that now
γ1 =
2
a
(
1 + 2ρ˜
)
, γ2 = J + 4Jρ˜+ 16t2ρ˜ κ1 ,
γ3 = 4
√
2 t2ρ˜ κ1 , γ4 = 4
√
2 t2ρ˜ κ2 ,
γ5 = 16 t2 ρ˜ λ4 . (4.453)
After implementing the Fourier representation of ζ2, or rather of η and ξ, we rewrite this
as in (4.351)
Sζ =
∑
~q,~q′
[
η∗(~q)A(~q − ~q ′)η(~q ′) + ξ∗(~q)B(~q − ~q ′)ξ(~q ′)
159
4. Effective Field Theory for the t-J Model at Low Doping
+ ξ∗(~q) C(~q, ~q ′)η(~q ′)
]
+
∑
~q
[
d(−~q)η(~q) + e(−~q)ξ(~q)
])
, (4.454)
where we defined A, B and C just as in (4.352)-(4.354) but with the powers of a now
written explicitly:
A(~q, ~q ′) = a2A(0)(~q − ~q ′) + a3A(1)(~q − ~q ′)
= 2
N
∑
j
ei(~q−~q′)~xj
(
a2A(0)(~xj) + a3A(1)(~xj)
)
,
B(~q, ~q ′) = a2 B(0)(~q − ~q ′) + a3 B(1)(~q − ~q ′)
= 2
N
∑
j
ei(~q−~q′)~xj
(
a2 B(0)(~xj) + a3 B(1)(~xj)
)
,
C(~q, ~q ′) = a2 C(0)(~q − ~q ′) + a3 C(1)(~q, ~q ′)
= a2 C(0)(~q − ~q ′) + a3
(
C(1a)(~q − ~q ′) + (qα + q′α)C(1b)(~q − ~q ′)
)
= 2
N
∑
j
ei(~q−~q′)~xj
(
a2 C(0)(~xj) + a3
(
C(1a)(~xj) + (qα + q′α)C(1b)(~xj)
))
. (4.455)
and the inverse of A, derived in (4.370), now reads
A−1(~q, ~q ′) = 1
a2
(
A−1(0)(~q − ~q ′) + aA−1(1)(~q − ~q ′) + a2A−1(2)(~q − ~q ′) + . . .
)
= 1
a2
N
2
∑
j
ei(~q−~q′)~xj
(
1
A(0)(~xj) − a
A(1)(~xj)
A(0)(~xj)2 + a
2A(1)(~xj)2
A(0)(~xj)3 + . . .
)
. (4.456)
So we get for the logarithmic contribution
S
(log)
ζ =
1
2 Tr ln
(
a4M(0) + a5M(1) + a6M(2) + a7M(3) + . . .
)
, (4.457)
where
M(0) = A(0)B(0) − 14A(0)C(0)A
−1
(0)C(0) ,
M(1) = A(1)B(0) +A(0)B(1) − 14
(
A(1)C(0)A−1(0)C(0) +A(0)C(1)A−1(0)C(0)
+A(0)C(0)A−1(1)C(0) +A(0)C(0)A−1(0)C(1)
)
,
M(2) = . . . (4.458)
160
4.5. Integrating out small bosonic fluctuations
Let us now separate the lowest order contribution from the higher ones
S
(log)
ζ =
1
2 Tr ln
(
a4M(0)
)
+ 12 Tr ln
(
1 + aM−1(0)M(1) + a2M−1(0)M(2) + . . .
)
(4.459)
and expand the logarithm up to third order in a via:
ln
(
1 +X
)
' X − 12X
2 + 13X
3 . (4.460)
We finally get
Result 52 (Logarithmic part of the effective action)
S
(log)
ζ ' S(0,log)ζ + aS(1,log)ζ + a2S(2,log)ζ + a3S(3,log)ζ , (4.461)
where
S
(0,log)
ζ =
1
2 Tr ln
(
a4M(0)
)
,
S
(1,log)
ζ =
1
2 Tr
(
M −1(0) M(1)
)
,
S
(2,log)
ζ =
1
2 Tr
(
M −1(0) M(2) −
1
2M
−1
(0) M(1)M −1(0) M(1)
)
,
S
(3,log)
ζ =
1
2 Tr
(
M −1(0) M(3) −
1
2M
−1
(0) M(1)M −1(0) M(2) −
1
2M
−1
(0) M(2)M −1(0) M(1)
+ 13M
−1
(0) M(1)M −1(0) M(1)M −1(0) M(1)
)
. (4.462)
M(0), . . . ,M(3) are the contributions to
M = a4M(0) + a5M(1) + a6M(2) + a7M(3) + . . . , (4.463)
and M was defined in (4.450):
M(~q, ~q ′) = ∑
~q′′
A(~q, ~q ′′)B(~q ′′, ~q ′)
−14
∑
~q′′,~q′′′,~q(IV)
A(~q, ~q ′′)C(~q ′′, ~q ′′′)A−1(~q ′′′, ~q(IV))C(−~q ′,−~q(IV)) . (4.464)
161
4. Effective Field Theory for the t-J Model at Low Doping
Note that (4.464) features an unconventional matrix product in C(−~q ′,−~q(IV)). However,
when we define
C˜(~q, ~q ′) = C(−~q ′,−~q) = C(0)(~q − ~q ′) + C(1a)(~q − ~q ′)− (qα + q′α)C(1b)(~q − ~q ′) , (4.465)
we can write this as a standard matrix-product:
M(~q, ~q ′) =
(
AB − 14ACA
−1C˜
)
(~q, ~q ′)
=
∑
~q′′
A(~q, ~q ′′)B(~q ′′, ~q ′)− 14
∑
~q′′,~q′′′,~q(IV)
A(~q, ~q ′′)C(~q ′′, ~q ′′′)A−1(~q ′′′, ~q(IV))C˜(~q(IV), ~q ′) . (4.466)
The different contributions in (4.461) evaluate as follows:
Zeroth-order contribution
S
(0,log)
ζ =
1
2 Tr ln
(
a4M(0)
)
, (4.467)
where
M(0) = A(0)B(0) − 14A(0)C(0)A
−1
(0)C(0) = A(0)B(0) −
1
4C(0)C(0) , (4.468)
since C˜(0) = C(0). Thus,
S
(0,log)
ζ =
1
2 Tr ln
[
a4
(
A(0)B(0) − 14C(0)C(0)
)]
, (4.469)
and
A(0)B(0) − 14C(0)C(0) =
∑
~q′′
(
A(0)(~q, ~q ′′)B(0)(~q ′′, ~q ′)− C(0)(~q, ~q ′′)C(0)(~q ′′, ~q ′)
)
= 2
N
∑
j
ei(~q−~q′)~xj
(
∆2(0) − 4 Λ∗(0)Λ(0)
)
, (4.470)
where we plugged in (4.352)-(4.354). Now we pull the term out of the exponent
exp
(
− S(0,log)ζ
)
= exp
(
− 12 Tr ln
[
a4
2
N
∑
j
ei(~q−~q′)~xj
(
∆2(0) − 4 Λ∗(0)Λ(0)
)])
=
(
det
[
a4
2
N
∑
j
ei(~q−~q′)~xj
(
∆2(0) − 4 Λ∗(0)Λ(0)
)])− 12
162
4.5. Integrating out small bosonic fluctuations
= 1
a2
(
∆2(0) − 4 Λ∗(0)Λ(0)
)− 12
. (4.471)
Inserting (4.344), we can write it as
1
a2
(
∆2(0) − 4 Λ∗(0)Λ(0)
)− 12
= 18 a2 γ2
|z1|2
|z1|2 − |z2|2 . (4.472)
To summarize
Result 53 (Contribution to the measure from the logarithmic part)
exp
(
− S(0,log)ζ
)
= 18 a2 γ2
|z1|2
|z1|2 − |z2|2 . (4.473)
A comparison with the integral measure in (4.338) shows that the zeroth-order logarith-
mic contribution cancels part of the Jacobian and the path integral takes the expected
form of a CP 1-model
Z =
∫
D(z¯, z) (z1 − z∗1) δ
(
z¯z − 1
)
e−Seff . (4.474)
O(a1)-contribution:
Next we look for the 1st order contributions
S
(1,log)
ζ =
1
2 TrM
−1
(0) M(1) , (4.475)
where
M(1) =
(
AB − 14ACA
−1C˜
)
(1)
= A(1)B(0) +A(0)B(1) − 14
(
A(1)C(0)A−1(0)C˜(0) +A(0)C(1)A−1(0)C˜(0)
+A(0)C(0)A−1(1)C˜(0) +A(0)C(0)A−1(0)C˜(1)
)
. (4.476)
With the help of (4.370) we derive
A(1)C(0)A−1(0)C˜(0) +A(0)C(0)A−1(1)C˜(0) = A(1)C(0)A−1(0)C˜(0) − C(0)A(1)A−1(0)C˜(0) = 0 , (4.477)
163
4. Effective Field Theory for the t-J Model at Low Doping
where we used the fact that all occurring matrices are diagonal and commute with each
other. So we have
S
(1,log)
ζ =
1
2 Tr
(
M −1(0)
[
A(1)B(0) +A(0)B(1)
− 14
(
A(0)C(1)A−1(0)C˜(0) +A(0)C(0)A−1(0)C˜(1)
)])
. (4.478)
Here we use the cyclic property of the trace to obtain
S
(1,log)
ζ =
1
2 Tr
[
M −1(0)
(
A(1)B(0) +A(0)B(1) − 12 C(0)C(1a)
)]
. (4.479)
So let us take care of the first two summands in (4.479)
TrM −1(0)
(
A(1)B(0) +A(0)B(1)
)
(4.480)
=
∑
~q,~q′
δ~q,~q′
∑
~q′′,~q′′′
M −1(0) (~q, ~q ′′)
(
A(1)(~q ′′, ~q ′′′)B(0)(~q ′′′, ~q ′) +A(0)(~q ′′, ~q ′′′)B(1)(~q ′′′, ~q ′)
)
= 2
N
∑
q
∑
j
M −1(0) (~xj)
(
A(1)(~xj)B(0)(~xj) +A(0)(~xj)B(1)(~xj)
)
=
∑
j
M −1(0) (~xj)
(
A(1)(~xj)B(0)(~xj) +A(0)(~xj)B(1)(~xj)
)
. (4.481)
Note that the q-sum was just the volume of the Brillouin zone, given by N/2. So when
we plug in A and B from (4.352) and (4.353) we get
A(1)(~xj)B(0)(~xj) +A(0)(~xj)B(1)(~xj) = 2 ∆(0)∆(1) − 2 (Λ(0) + Λ∗(0))(Λ(1) − Λ∗(1)) . (4.482)
For the last summand in (4.479) we get
− 12 TrM
−1
(0) C(0)C(1a) = −
1
2
2
N
∑
q
∑
j
M −1(0) (~xj) C(0)(~xj)C(1a)(~xj)
= −12
∑
j
M −1(0) (~xj) C(0)(~xj)C(1a)(~xj) . (4.483)
When we plug in C from (4.354), we get for the first order contribution
S
(1,log)
ζ =
1
2
∑
j
M −1(0) (~xj)
(
A(1)(~xj)B(0)(~xj) +A(0)(~xj)B(1)(~xj)− 12C(0)(~xj)C(1a)(~xj)
)
164
4.5. Integrating out small bosonic fluctuations
=
∑
j
∆(0)∆(1,α) + 2 (Λ(0)Λ∗(1,α) − Λ∗(0)Λ(1,α))−∆(0)∆(1,τ))
∆2(0) − 4 Λ∗(0)Λ(0)
−∆(0)∆(1,τ) + 2 (Λ(0)Λ
∗
(1,τ) − Λ∗(0)Λ(1,τ))
∆2(0) − 4 Λ∗(0)Λ(0)
, (4.484)
where we used (4.433) in the last step. Plugging in (4.344) and (4.434), we get
Result 54 (First order contribution from the logarithmic part)
S
(1,log)
ζ = −
γ5
γ2
∑
j
(
z¯ ∂τz
)
− γ3
γ2
∑
j
(
|z1|4 − |z2|4
)(
z∗1∂αz1 − z1∂αz∗1
)
− 2 |z1|2|z2|2
(
z∗2∂αz2 − z2∂αz∗2
)
|z1|2
(
|z1|2 − |z2|2
) , (4.485)
where ∂α = ∂x − ∂y
O(a2)-contribution:
Next we look for the 2nd order contributions
S
(2,log)
ζ =
1
2
[
Tr
(
M −1(0) M(2)
)
− 12 Tr
(
M −1(0) M(1)M −1(0) M(1)
)]
, (4.486)
where
M(2) =
(
AB − 14ACA
−1C˜
)
(2)
= A(1)B(1) − 14
(
A(0)C(0)A−1(2)C˜(0) +A(1)C(1)A−1(0)C˜(0) +A(0)C(1)A−1(1)C˜(0)
+A(0)C(0)A−1(1)C˜(1) +A(1)C(0)A−1(1)C˜(0) +A(1)C(0)A−1(0)C˜(1) +A(0)C(1)A−1(0)C˜(1)
)
. (4.487)
We plug in (4.370), commute the diagonal matrices among each other and use the cyclic
property of the trace. Finally, we get
Tr
(
M −1(0) M(2)
)
= Tr
(
M −1(0) A(1)B(1)
)
− 14Tr
(
M −1(0) A(0)C(1)A−1(0)C˜(1)
)
. (4.488)
165
4. Effective Field Theory for the t-J Model at Low Doping
The second contribution in (4.486) reads
− 12TrM
−1
(0) M(1)M −1(0) M(1)
= −12Tr
(
M −1(0)
[
A(1)B(0) +A(0)B(1) − 14
(
A(0)C(1)A−1(0)C˜(0) +A(0)C(0)A−1(0)C˜(1)
)
· M −1(0)
[
A(1)B(0) +A(0)B(1) − 14
(
A(0)C(1)A−1(0)C˜(0) +A(0)C(0)A−1(0)C˜(1)
)])
. (4.489)
We commute the diagonal matrices, using the cyclic property of the trace
= −12Tr
[
M −1(0) M −1(0)
(
A(1)B(0) +A(0)B(1)
)2
−M −1(0) M −1(0)
(
A(1)B(0) +A(0)B(1)
)
C(0)C(1a)
+ 14M
−1
(0) M −1(0) C(0)C(0)C(1a)C(1a)
]
. (4.490)
So we get for (4.486)
S
(2,log)
ζ =
1
2
Tr[M −1(0) A(1)B(1)
]
− 14Tr
[
M −1(0) A(0)C(1)A−1(0)C˜(1)
]
− 12Tr
[
M −1(0) M −1(0)
(
A(1)B(0) +A(0)B(1)
)2]
+ 12Tr
[
M −1(0) M −1(0)
(
A(1)B(0) +A(0)B(1)
)
C(0)C(1a)
]
− 18Tr
[
M −1(0) M −1(0) C(0)C(0)C(1a)C(1a)
] . (4.491)
When we plug in (4.352)-(4.354), we get for the first summand
Tr
[
M −1(0) A(1)B(1)
]
=
(
∆(1) + Λ∗(1) − Λ(1)
)(
∆(1) − Λ∗(1) + Λ(1)
)
∆2(0) − 4 Λ∗(0)Λ(0)
. (4.492)
The second summand is the only contribution with a non-diagonal part, since
C(1)(~q, ~q ′) = C(1a)(~q − ~q ′) + [qα + q′α]C(1b)(~q − ~q ′)
166
4.5. Integrating out small bosonic fluctuations
C˜(1)(~q, ~q ′) = C(1a)(~q − ~q ′)− [qα + q′α]C(1b)(~q − ~q ′) . (4.493)
We write out the matrix product and obtain
−14Tr
[
M −1(0) A(0)C(1)A−1(0)C˜(1)
]
= −14
∑
~q,~q′,~q′′
M −1(0) (~q − ~q ′)C(1a)(~q ′ − ~q ′′)C(1a)(~q ′′ − ~q)
+14
∑
~q...~qIV
[
M −1(0) (~q − ~q ′)A(0)(~q ′ − ~q ′′)
[
q′′α + q′′′α
]
C(1b)(~q ′′ − ~q ′′′)
·A−1(0)(~q ′′′ − ~qIV )
[
qIVα + qα
]
C(1b)(~qIV − ~q)
]
. (4.494)
The first summand here simply gives
− 14Tr
[
M −1(0) A(0)C˜(1)A−1(0)C˜(1)
]
diag
= −14
C(1a)(~x)C(1a)(~x)
A(0)(~x)B(0)(~x)− 14C(0)(~x)C(0)(~x)
. (4.495)
The second summand is more complicated. With the help of integration by parts we
show that
1
4Tr
[
M −1(0) A(0)C˜(1)A−1(0)C˜(1)
]
non−diag
= 14N
∫
dx dq q2
(
2 C1b(x)
)2
, (4.496)
which is a divergent expression.
Discussion of the logarithmic contribution
We stopped the evaluation of the logarithmic contribution in 2nd order, since we obtain
rather obscure integrals with diverging contributions to the action and on top of that, a
rotational invariance breaking term in the 1st-order contribution that will consequently
dominate our effective field theory. This is unexpected, since we started from a rota-
tionally invariant action.
All this leads to the conclusion that there is a problem in the calculation, either a tech-
nical or a conceptual one. In order to find the source of this problem, we shall retrace
our steps that lead us to this result.
First of all, let us note that the terms in the logarithmic part have two different origins.
There is a part with only spatial derivatives which comes from expanding the trigono-
167
4. Effective Field Theory for the t-J Model at Low Doping
metric functions of ~q in the 2nd-order nearest neighbour term, see for example (4.271).
The other part originates from a product of three Green’s functions and is a combination
of the 1st-order fermionic Berry phase and a 2nd-order nearest neighbour kinetic term,
see for example (4.317). The term responsible for the breaking of rotational invariance
and the diverging integral in (4.496) comes from (4.271) only. An error in their deriva-
tions could not be found.
It would proof useful to know the exact step in which rotational invariance gets lost and
whether it is already broken before we integrate out the fluctuations. However, checking
for rotational invariance prior to the integration over ζ turns out to be difficult, since it is
a priori not clear how a rotation in real space affects the variables z and ζ. The attempt
to invert relation (4.31) failed, on account of the combined gauge fixing condition for z
and ζ.
Another possibility arises from the comparison of our effective action with the one ob-
tained for the same model but in a 2nd order gradient expansion only. There, no such
problematic terms arise and the effective action is fully rotationally invariant. So the
problem is a pure 3rd-order effect. At first glance, 1st- and 2nd-order contributions
arising from the inclusion of 3rd-order terms may sound contradictory but can be traced
back to the integral over fluctuations. To understand how these terms arise, one may
consider a simple Gaussian integral where the bilinear form splits into a 2nd- and a
3rd-order term
∞∫
−∞
dη e−η(a2A0+a3A1)η =
√
pi√
a2A0 + a3A1
=
√
pi√
a2A0
·
(
1− a2
A1
A0
+ 3a
2
8
A21
A20
− 5a
3
16
A31
A30
+O(a4)
)
=
√
pi√
a2A0
· exp
[
ln
(
1− a2
A1
A0
+ 3a
2
8
A21
A20
− 5a
3
16
A31
A30
+O(a4)
)]
. (4.497)
So terms of lower order can in principle appear. However, assuming that a gradient
expansion for our model can be performed consistently, there should be some physical
or symmetry related mechanism that leads to a cancellation of these terms.
We should also consider the possibility that there is a problem in the way we derive the
168
4.5. Integrating out small bosonic fluctuations
Gaussian integral over fluctuations. Note that an arbitrary scaling of the fields
∞∫
−∞
dη e−c2η(a2A0+a3A1)η =
√
pi
c
√
a2A0 + a3A1
(4.498)
changes the value of the integral only by a constant factor which can always be absorbed
into normalization. So the result does not take into account the fact that the exponent
is already of order a2 plus higher orders (or equivalently, that η is small). Introducing a
cutoff for the integration obviously prevents us from integrating over large values of η.
We have
L∫
−L
dη e−η(a2A0+a3A1)η =
√
pi√
a2A0 + a3A1
· erf
(
L
√
a2A0 + a3A1
)
=
√
pi√
a2A0
· erf
(
L
√
a2A0 ·
√
1 + aA1/A0
)
√
1 + aA1/A0
. (4.499)
If for the limit of integration we choose
L = a
2
2
√
a2A0
= a
2
√
2
· σ0 , (4.500)
where σ0 is the variance of the Gaussian e−η(a
2A0)η, we get the following result
a2√
2 ·σ0∫
− a2√2 ·σ0
dη e−η(A0+aA1)η =
√
pi√
a2A0
· erf
(
a2/2 ·
√
1 + aA1/A0
)
√
1 + aA1/A0
= a√
A0
·
(
1− a
4
12
(
1 + aA1/A0
)
+ a
8
160
(
1 + aA1/A0
)
+O(a12)
)
= a√
A0
· exp
[
ln
(
1 +O(a4)
)]
, (4.501)
such that we would obtain only a lowest order contribution. In our model this would
correspond to the part that affects the measure, (4.473). No additional contributions
from the third order terms would appear. We obtain the same expression if we set
A1 = 0 from the start, so the cutoff does not spoil the result from the second order
gradient expansion [10].
169
4. Effective Field Theory for the t-J Model at Low Doping
No matter what the correct formulation of the logarithmic contribution is, one thing is
for sure. If we look at (4.452) we see that all terms are at most first order in a except
A−1 which through (4.370) also turns out to be a combination of zeroth- and first-order
contributions. Further inspection reveals that all zeroth-order terms are purely real and
all first-order terms purely imaginary. Thus, the third order contribution will be purely
imaginary since it must be a product of three first order contributions. And since we
know from our analysis in Section 2.4.1 that the Hopf term is a real quantity, it cannot
emerge from the logarithmic contribution.
4.6. Summary and discussion of results
Finally, let us collect our results. From (4.321), we have the effective action that resulted
from integrating out the fermions. We extract the part without fluctuations, hence all
parts that do not include the ζ-fields
S
(2,3)

ζ
=
∫
dτ d2x
[
(Aψ)αβ1 (∂αz¯ ∂βz) + (Aψ)αβ2 (z¯ ∂αz)(z¯ ∂βz)
+ (Bψ)αβτ (z¯ ∂τz)
(
∂αz¯ ∂βz + (z¯ ∂αz)(z¯ ∂βz)
)
+ (Dψ)αβτ (z¯ ∂τz)(z¯ ∂α∂βz − z ∂α∂β z¯)
]
, (4.502)
where the first line constitutes theO(a2)-, the second and third line theO(a3)-contributions.
The coefficient matrices (Aψ)1, (Aψ)2, (Bψ)τ and (Dψ)τ can be read of (4.321).
Now let us include the terms that result from integrating out bosonic fluctuations. We
have an O(a2)-contribution, given in (4.398)
S
(2)
ζ =
∫
dτ d2x
[
(Aζ)αβ1 (∂αz¯ ∂βz) + (Aζ)αβ2 (z¯ ∂αz)(z¯ ∂βz)
]
, (4.503)
and an O(a3)-contribution from (4.449).
S
(3)
ζ =
∫
dτ d2x
[
(Aζ)αβτ ∂τ z¯ ∂α∂βz + (Aζ)αβx ∂xz¯ ∂α∂βz + (Aζ)αβy ∂yz¯ ∂α∂βz
+ (Bζ)αβτ (z¯ ∂τz)(∂αz¯ ∂βz) + (Bζ)αβx (z¯ ∂xz)(∂αz¯ ∂βz) + (Bζ)αβy (z¯ ∂yz)(∂αz¯ ∂βz)
170
4.6. Summary and discussion of results
+ (Cζ)αβτ (z¯ ∂τz)(z¯ ∂αz)(z¯ ∂βz) + (Dζ)αβτ (z¯ ∂τz)(z¯ ∂α∂βz − z ∂α∂β z¯)
]
. (4.504)
The logarithmic contribution discussed in Section 4.5.4 needs further considerations but
is not expected to give more contributions to the action.
In the end, we combine all contributions
S = S(2,3)

ζ
+ S(2)ζ + S
(3)
ζ , (4.505)
where the coefficients simplify even more. We obtain the final result:
4.6.1. Effective action up to O(a3) for the t-J model at low doping
Summary 3 (Effective action up to O(a3) for the t-J model at low doping)
S =
∫
dτ d2x
[
Aαβ (∂αz¯ ∂βz) +Bαβ (z¯ ∂αz)(z¯ ∂βz)
+ Cαβτ ∂τ z¯ ∂α∂βz + Cαβx ∂xz¯ ∂α∂βz + Cαβy ∂yz¯ ∂α∂βz
+Dαβτ (z¯ ∂τz)(∂αz¯ ∂βz) +Dαβx (z¯ ∂xz)(∂αz¯ ∂βz) +Dαβy (z¯ ∂yz)(∂αz¯ ∂βz)
+ Eαβτ (z¯ ∂τz)(z¯ ∂αz)(z¯ ∂βz) + Fαβτ (z¯ ∂τz)(z¯ ∂α∂βz − z ∂α∂β z¯)
]
, (4.506)
where the coefficient matrices are given by:
Aαβ =

a1 0 0
0 a2 0
0 0 a2
 , Bαβ =

b1 0 0
0 b2 0
0 0 b2
 , (4.507)
a1 =
(1 + 2ρ˜)2
2a2(J + 4 ρ˜ J + 16 t2ρ˜κ1)
,
a2 = J
(
1 + 4ρ˜
)
+ 16 t2ρ˜ (κ2 − κ1) + 8 t′ρ˜1 + 16 t′′ρ˜2 ,
b1 =
(1 + 2ρ˜)2
2a2(J + 4 ρ˜ J + 16 t2ρ˜κ1)
+ 2ρ˜κ
a2
,
b2 = J
(
1 + 4ρ˜
)
+ 16 t2ρ˜ (κ2 − κ1)− 16 t′2ρ˜κ3 − 64t′t′′ρ˜κ6 − 64t′′2ρ˜κ4 . (4.508)
171
4. Effective Field Theory for the t-J Model at Low Doping
Cαβτ =

0 12c1 −12c1
1
2c1 0 0
−12c1 0 0
 , Cαβx =

0 0 0
0 c2 0
0 0 c2
 , Cαβy =

0 0 0
0 −c2 0
0 0 −c2
 ,
Dαβτ =

d1 −c1 c1
−c1 d2 0
c1 0 d2
 , Dαβx =

−c1 0 0
0 −c2 0
0 0 −c2
 , Dαβy =

−c1 0 0
0 c2 0
0 0 c2
 ,
Eαβτ =

d1 0 0
0 d2 0
0 0 d2
 , Fαβτ =

0 0 0
0 f1 f2
0 f2 f1
 , (4.509)
c1 =
4
√
2 t2ρ˜κ1 (1 + 2ρ˜)2
a(J + 4Jρ˜+ 16 t2ρ˜κ1)2
,
c2 = 8
√
2 a t2 ρ˜ (κ2 − κ1)
d1 =
8 t2ρ˜λ4 (1 + 2ρ˜)2
a2(J + 4Jρ˜+ 16 t2ρ˜κ1)2
,
d2 = 16 t2 ρ˜ (λ4 − λ8) ,
f1 =
4 t2ρ˜ (κ2 − 2κ1) (1 + 2ρ˜)
J + 4Jρ˜+ 16 t2ρ˜κ1
+ 4 ρ˜ (t′κ7 + 2 t′′κ8) ,
f2 =
4 t2ρ˜κ1 (1 + 2ρ˜)
J + 4Jρ˜+ 16 t2ρ˜κ1
. (4.510)
The search for a Hopf term
The Hopf density is given by an antisymmetric combination of terms with three different
derivatives. It reads
H = αβγ(z¯ ∂αz)(∂β z¯ ∂γz)
= Dαβτ (z¯ ∂τz)(∂β z¯ ∂γz) +Dαβx (z¯ ∂xz)(∂β z¯ ∂γz) +Dαβy (z¯ ∂yz)(∂β z¯ ∂γz) . (4.511)
172
4.6. Summary and discussion of results
where
Dαβτ =

0 0 0
0 0 d
0 −d 0
 , Dαβx =

0 0 −d
0 0 0
d 0 0
 , Dαβy =

0 d 0
−d 0 0
0 0 0
 . (4.512)
Unfortunately, there are no antisymmetric contributions to D in the result we obtained,
see (4.506). The only term with three different derivatives we did obtain reads
Sτ,x,y =
∫
dτ d2x (z¯ ∂τz)(z¯ ∂x∂yz − z ∂x∂yz¯) , (4.513)
which by use of the constraint or integration by parts can only be transformed in sym-
metric combinations of ∂αz¯ ∂βz and ∂β z¯ ∂αz. To conclude, there is no Hopf term in
the effective long-wavelength action of the t-J model at low doping.
The limit of zero doping
In the zero doping limit, the density of holes ρ˜ and its modulated variants ρ˜1 and ρ˜2
vanish. Hence, the O(a3)-contribution vanishes completely and the remaining O(a2)-
contribution simplifies to
Summary 4 (Effective action in the limit of zero doping)
S =
∫
dτ d2x
[
Aαβ1 (∂αz¯ ∂βz) + Aαβ2 (z¯ ∂αz)(z¯ ∂βz)
]
, (4.514)
where
Aαβ1 =

1
2a2J 0 0
0 J 0
0 0 J
 , Aαβ2 =

1
2a2J 0 0
0 J 0
0 0 J
 . (4.515)
As expected, we obtain the O(3) non-linear σ-model in CP1-representation, correctly
describing the Heisenberg limit.
173

Conclusion
We carefully derived the long-wavelength effective action for the t-J model at low values
of doping through a gradient expansion up to third order in the lattice constant. In
second order, we recovered the massive CP 1-model, see Section 2.4.3. We are able to
give some corrections to the coupling constants that were obtained in [10].
Unfortunately, the third-order contribution did not yield the Hopf term, as we originally
hoped for. Terms of O(a3) that have no topological significance do not affect the be-
haviour of the model since they will always be dominated by the O(a2)-contributions.
The long and tedious calculation was performed by hand and thoroughly checked with
the help of Mathematica. In the derivation of the logarithmic part of the integral over
fluctuations, we encountered a possible inconsistency in the gradient expansion resulting
from the inclusion of third-order terms. After careful examination and reexamination of
our steps, we come to the conclusion that the integral over fluctuations, see Appendix
A.2, probably has to be performed in a different manner. For details on this point, we
refer to the discussion at the end of Section 4.5.4. The issue will require further consid-
eration.
Provided that the inconsistency in the gradient expansion results from the integral over
fluctuations, we are able to give a definite answer to the original scope of our work. As
a result of the discussion in Section 4.6 and the effective action obtained in (4.506), we
conclude that the long-wavelength action of the t-J model at low values of doping does
not give rise to a Hopf term and its low-energy behaviour is not governed by excitations
with fractional spin and statistics.
175

Appendix A
Supplementaries
A.1. Generalized non-linear σ-model
Any order parameter space T can be expressed in terms of a group G that acts transi-
tively on T and its isometry subgroup H [45]. Transitively means that, given any two
elements of T , say t1 and t2, there is an element of G that maps t1 into t2. The isometry
subgroup H of G is defined as the subgroup that maps a given element of T into itself.
The order parameter space T may then be identified with the coset space
T ' G/H . (A.1)
Such order parameter spaces are called homogeneous spaces.
It is instructive to work this out for the O(3) non-linear σ-model. The group that acts
transitively on vectors living on the S2 is given by SO(3). The SO(2)-subgroup which ro-
tates ~n around itself does not change ~n at all. Thus, we can identify S2 = SO(3)/SO(2).
Note that S2 ' CP1 ' SU(2)/U(1), and this is yet another example for a homogeneous
space. SU(2) acts transitively on elements of CP1 and U(1) is obviously the isometry
subgroup.
SU(2) representation of the CP1-model
We may express the CP1-model as a model in the coset space SU(2)/U(1) where the
order parameter takes values in SU(2) [97]. The action is expressed in terms of Maurer-
Cartan forms ωµ = g−1∂µg, such that
S = −1
g
∫
d3x Tr ωµ ωµ = −1
g
∫
ddx Tr (g−1∂µg)(g−1∂µg) . (A.2)
We also have to take into account that elements of the U(1)-subgroup are being identified.
177
A. Supplementaries
Note that every element of SU(2) may be expressed in terms of the hermitian generators
of its Lie algebra which are given by the Pauli matrices
σ1 =
 0 1
1 0
 , σ2 =
 0 −i
i 0
 , σ3 =
 1 0
0 −1
 , (A.3)
such that
g = exp
( 3∑
a=1
ga σa
)
, with ga ∈ R (A.4)
and the Maurer Cartan forms take values directly in this algebra ωµ =
∑3
a=1 ω
a
µσa. The
U(1)-subgroup of SU(2) is generated by σ3, so that the restriction to SU(2)/U(1) is
easily performed by writing ωµ
∣∣∣
r
= ∑2a=1 ωaµ σa. Thus, the non-linear σ-model for an
SU(2)/U(1) valued order parameter is given by
S = −1
g
∫
d3x Tr (g−1∂µg)
∣∣∣
r
(g−1∂µg)
∣∣∣
r
. (A.5)
And in fact, using the explicit parametrization for SU(2) in (2.99), we obtain
(g−1∂µg) =
 z¯ ∂µz −(z∗1∂µz∗2 − z∗2∂µz∗1)
(z1∂µz2 − z2∂µz1) −z¯ ∂µz
 =
 i Aµ −F ∗µ
Fµ −i Aµ
 (A.6)
and
(g−1∂µg)
∣∣∣
r
= g−1(∂µ − iAµσz) g =
 0 −F ∗µ
Fµ 0
 , (A.7)
such that
S = 2
g
∫
d3x F µ∗Fµ , (A.8)
and due to F µ∗Fµ = ∂µz¯ ∂µz + (z¯ ∂µz)2, we can see the equivalence of (A.8) and the
CP1-model (2.69).
The generalized winding number
The third homotopy group of SU(2) is characterized by the generalized winding number
[98]
H = 148pi2
∫
d3x µνλ(g−1∂µg)(g−1∂νg)(g−1∂λg) (A.9)
178
A.2. Gaussian functional integration
in the sense that it counts the number of times the mapping g covers SU(2) while S3
is traversed [99]. Note that for g ∈ SU(2), it evaluates exactly to the Hopf-invariant
in CP1-representation. This can be easily seen in terms of the parametrization (2.99),
where we obtain
H = i48pi2
∫
d3x µνλ
(
Aµ(F ∗νFλ − F ∗λFν) + Aν(F ∗λFµ − F ∗µFλ)
+Aλ(F ∗µFν − F ∗νFµ)
)
= 18pi2
∫
d3x µνλ(z¯ ∂µz)(∂ν z¯ ∂λz) . (A.10)
A.2. Gaussian functional integration
The functional integral over fluctuations
When integrating out the bosonic fluctuations we encounter an integral of the form
∫
DηDξ exp
(
−∑
~q,~q′
[
η∗(~q)A(~q − ~q′)η(~q′) + ξ∗(~q)B(~q − ~q′)ξ(~q′)
+ ξ∗(~q) C(~q, ~q′)η(~q′)
]
−∑
~q
[
d(−~q)η(~q) + e(−~q)ξ(~q)
])
, (A.11)
where η(~q) and ξ(~q) are complex-valued fields, defined as the Fourier representation of
real-valued fields
η(~x) =
√
2
N
∑
~q
e−i~q~x η(~q) and ξ(~x) =
√
2
N
∑
~q
e−i~q~x ξ(~q) , (A.12)
so they obey the property
η∗(~q) = η(−~q) , ξ∗(~q) = ξ(−~q) . (A.13)
Preparation: Let us focus for a moment on the integral measureDηDξ in (A.11) that
resulted from the change of variables from (ζ2)j := ζ∗2 (~xj) to real- and imaginary part,
ηj := η(~xj) and ξj := ξ(~xj), and the subsequent Fourier transformation to ηq := η(~q)
and ξq := ξ(~q). Decomposition into real- and imaginary part is a linear transformation
179
A. Supplementaries
with Jacobian 12 which gets absorbed by normalization. So we have
(ζ∗2 )j, (ζ2)j → ηj, ξj (A.14)
=⇒ D(ζ2, ζ∗2 ) :=
n
2∏
j=−n2
d(ζ2)j d(ζ∗2 )j → D(η, ξ) :=
n
2∏
j=−n2
dηj dξj . (A.15)
Note that (A.12) is a discrete Fourier transformation which is implemented as
ηj =
√
1
n+ 1
n
2∑
q=−n2
ei 2pi
jq
n+1 ηq (A.16)
and similarly for ξj. The cell-index j ∈ {−n/2, . . . , n/2}, so the size of the lattice is
n+ 1 cells. In our case n+ 1 = N2 , where N denotes the number of lattice-sites.
The Fourier transformation of the real functions ηj and ξj gives two complex functions
ηq and ξq that obey the symmetry conditions ηq = η∗−q and ξq = ξ∗−q. In terms of real-
and imaginary parts, we have
Re ηq = Re η−q , Im ηq = −Im η−q ,
Re ξq = Re ξ−q , Im ξq = −Im ξ−q . (A.17)
In other words, when we Fourier transform a real quantity, the degrees of freedom
seemingly get increased by a factor of two: ηj → (Re ηq, Im ηq), but the symmetry
conditions (A.17) compensate for that by reducing them again by a factor 12 . Also note
that the Fourier-transformation is linear with a constant Jacobian which we absorb again
in the normalization. We obtain
ηj, ξj → ηq, ξq ,
D(η, ξ) :=
n
2∏
j=−n2
dηjdξj → DηDξ :=
0∏
q=−n2
dRe ηq dRe ξq
n
2∏
q′=1
dIm ηq′ dIm ξq′ . (A.18)
To derive the result of the integral in (A.11) we proof the following
Theorem: Given a complex, positive definite, diagonalizable matrix A, two non-
zero complex vectors b and c, and a complex vector η with the property ηq = η∗−q, the
180
A.2. Gaussian functional integration
Gaussian integral over the set of non-redundant modes (see (A.18)) is
∫
Dη exp
(
− η†Aη + b†η + η†c
)
∼ detA− 12 exp
(
b†A−1c
)
= exp
(
− 12Tr lnA+ b
†A−1c
)
. (A.19)
Proof: For the identity to hold, A has to be positive definite which means that
Re(η†A η) > 0 for all non-zero complex vectors η. We will assume this condition to
be fulfilled. Additionally, A has to be diagonalizable. A general complex matrix A is
unitarily diagonalizable if, and only if, it is normal (spectral theorem):
A†A = AA† . (A.20)
This requirement is fulfilled in our case. So there exists a unitary matrix U such that
U †AU = D is diagonal with dq := Dqq, the complex eigenvalues of A.
∫
Dη exp
(
− η†A η + b†η + η†c
)
=
∫
Dη exp
(
− η†(UU †)A(UU †)η + b†(UU †)η + η†(UU †)c
)
. (A.21)
We do a change of variables, η˜ = U †η, b˜ = U †b and c˜ = U †c. Since U is unitary, the
Jacobian of this change of variables is 1. We get
∫
Dη˜ exp
(
− η˜†D η˜ + b˜†η˜ + η˜†c˜
)
=
∫
Dη˜ exp
(
−∑
q
dqη˜
∗
q η˜q + b˜∗q η˜q + η˜∗q c˜q
)
. (A.22)
Writing all variables in terms of real- and imaginary parts, η˜ = η˜1 + iη˜2, b˜ = b˜1 + ib˜2, c˜ =
c˜1 + ic˜2, we get
=
∫
Dη˜ exp
(
−∑
q
dq
[
(η˜1)2q + (η˜2)2q
]
+
[
(b˜1)q − i(b˜2)q
][
(η˜1)q + i(η˜2)q
]
+
[
(η˜1)q − i(η˜2)q
][
(c˜1)q + i(c˜2)q
])
. (A.23)
Now we complete the square η˜1 → η˜1 + 12d(b˜1 − ib˜2 + c˜1 + ic˜2) and η˜2 → η˜2 + 12d(ib˜1 +
b˜2 − ic˜1 + c˜2). This shift of integration variables absorbs the term linear in η˜q but does
181
A. Supplementaries
not change the value of the integral.
=
∫
Dη˜ exp
(
−∑
q
dq
[
(η˜1)2q + (η˜2)2q
]
+ (b˜1 − ib˜2)q(D−1)q(c˜1 + ic˜2)q
)
. (A.24)
Note that D−1 = (U †AU)−1 = U †A−1U . So we have
= exp
(
b†A−1 c
) ∫
Dη˜ exp
(
−∑
q
dq
[
(η˜1)2q + (η˜2)2q
])
. (A.25)
Finally, we apply a linear transformation that rearranges the elements of (η˜1)q and (η˜2)q,
utilizing the fact that η˜1 only depends on n2 + 1 degrees of freedom and η˜2 on
n
2 , so that
η˜1 → η¯1 =
[
(η¯1)−n2 , . . . , (η¯1)0, 0, . . . , 0
]
, η˜2 → η¯2 =
[
0, . . . , 0, (η¯2)1, . . . , (η¯2)n2
]
. (A.26)
Such a transformation definitely exists and its Jacobian is a constant. Applying this
transformation, the integral finally factorizes
= exp
(
b†A−1 c
) ∫ 0∏
p=−n2
d(η¯1)p
n
2∏
p′=1
d(η¯2)p′ exp
(
−
n
2∑
q=−n2
dq
[
(η¯1)2q + (η¯2)2q
])
= exp
(
b†A−1 c
) 0∏
q=−n2
∫
d(η¯1)q exp
(
− dq(η¯1)2q
) n2∏
q′=1
∫
d(η¯2)q′ exp
(
− dq′(η¯2)2q′
)
= exp
(
b†A−1 c
)
(
√
pi)n2 +1
n
2∏
q=−n2
1√
dq
∼ exp
(
b†A−1 c
) 1√
detA . (A.27)
To proof the second identity in (A.19), we note that
1√
detA =
∏
q
1√
dq
= exp
[
ln
(∏
q
1√
dq
)]
= exp
(
− 12 Tr lnA
)
, (A.28)
where we used the fact that when A has eigenvalues dq, the matrix logarithm ln(A) has
eigenvalues ln(dq). So we arrived at the desired result.
182
A.3. Matsubara frequency summation
A.3. Matsubara frequency summation
We evaluate sums over Matsubara frequencies by contour integration [100]. For the sake
of completeness, let us start with the well known case of a single Green’s function.
Single Green’s function
The sum we want to evaluate reads
S = 1
β
∑
n
G0(ν,~k) =
1
β
∑
n
1
iνn −
[
(~k) + J + µ
] , (A.29)
where νn = (2n+1)piβ are the fermionic Matsubara frequencies and n runs over all integers.
Consider the complex integral
I = lim
R→∞
∫ dz
2pi i s(z)n(z) , (A.30)
where the contour of integration is a circle of radius
R, as sketched in Fig.(A.1). s(z) is the analytic con-
tinuation of our summand
s(z) = 1
z −
[
(~k) + J + µ
] = 1
z − ζ , (A.31)
and n(z) is chosen to have poles at z = iνn = (2n+1)ipiβ
n(z) = 1eβz + 1 , (A.32)
where each of the poles has residue 1
β
.
Im z
Re z
C
z = ζ
R
z = (2n+1)ipiβ
Fig A.1: Integration contour
Thus, we have
I = lim
R→∞
1
2pi i
∫
dz
n(z)
z − ζ = limR→∞
1
2pi i
∫
dz f(z) , (A.33)
with f(z) having
• a simple pole at z = ζ with residue
Res
[
f(z), ζ
]
= lim
z→ζ
[
(z − ζ) n(z)
z − ζ
]
= n(ζ) = 1
eβ[(~k)+J+µ] + 1
= nF (~k) , (A.34)
183
A. Supplementaries
where nF (~k) is the Fermi-Dirac distribution function.
• and simple poles at z = iνn = (2n+1)ipiβ with residue
Res
[
f(z), iνn
]
=
Res
[
n(z), iνn
]
iνn − ζ = −
1
β(iνn − ζ) = −
1
β
s(iνn) . (A.35)
For R → ∞, we have I → 0 and according to Cauchy’s theorem we may write the
integral as a sum of residues
I = lim
R→∞
∫ dz
2pi i s(z)n(z) = nF (
~k)− 1
β
∑
n
s(iνn) = 0 . (A.36)
So we get for the frequency sum
=⇒ S = 1
β
∑
n
1
iνn −
[
(~k) + J + µ
] = 1
β
∑
n
s(iνn) = nF (~k) . (A.37)
Two Green’s functions
Here we have
S = 1
β
∑
n,n′
G0(νn, ~k)G0(νn′ , ~k′)
= 1
β
∑
n,n′
1
iνn −
[
(~k) + J + µ
] 1
iνn′ −
[
(~k′) + J + µ
] . (A.38)
We do a change of variables
ν ′n = νn + ωm − ωm∗ , ~k′ = ~k + ~q − ~q∗ , (A.39)
where ωm = 2npiβ are bosonic Matsubara frequencies. Then we get
S = 1
β
∑
n,m∗
1
iνn −
[
(~k) + J + µ
] 1
i(νn + ωm − ωm∗)−
[
(~k + ~q − ~q∗) + J + µ
] . (A.40)
Let us consider the complex integral
I = lim
R→∞
∫ dz
2pi i s(z)n(z) , (A.41)
184
A.3. Matsubara frequency summation
where n(z) is given by (A.32) and
s(z) = 1
z − ζ1
1
z − ζ2 , (A.42)
with
ζ1 = (~k) + J + µ , ζ2 = (~k + ~q − ~q∗) + J + µ− i(ωm − ωm∗) . (A.43)
Our integrand f(z) has
• a simple pole at z = ζ1 with residue
Res
[
f(z), ζ1
]
= n(ζ1)
ζ1 − ζ2 =
nF (~k)
i(ω − ω∗) +
[
(~k)− (~k + ~q − ~q∗)
] , (A.44)
• a simple pole at z = (~k + ~q + ~q∗) + J + µ− i(ωm − ωm∗) with residue
Res
[
f(z), ζ2
]
= − n(ζ2)
ζ1 − ζ2 = −
nF (~k + ~q − ~q∗)
i(ω − ω∗) +
[
(~k)− (~k + ~q − ~q∗)
] , (A.45)
• and simple poles at z = iνn = (2n+1)ipiβ with residue
Res
[
f(z), iνn
]
= − 1
β
s(iνn) . (A.46)
So for R→∞, we have I → 0 and we may write the integral as a sum of residues
lim
R→∞
∫ dz
2pi i f(z)n(z) = Res
[
f(z), ζ1
]
+ Res
[
f(z), ζ2
]
− 1
β
∞∑
n=−∞
s(iνn) = 0 . (A.47)
So we have for the frequency sum
=⇒ S = 1
β
∑
n,m∗
f(iνn) =
∑
m∗
nF (~k)− nF (~k + ~q − ~q∗)
i(ωm − ωm∗) +
[
(~k)− (~k + ~q − ~q∗)
] . (A.48)
Let us define ω˜m = ωm − ωm∗ and ~˜q = ~q − ~q∗. We consider the polarization which is
185
A. Supplementaries
given by the previous expression summed over ~k
χ0(ω˜m, ~˜q) =
2
N
∑
~k
nF (~k)− nF (~k + ~˜q)
iω˜m +
[
(~k)− (~k + ~˜q)
] . (A.49)
The quantities ωm, ωm∗ , ~q and ~q∗ can be considered small, since they appear in the
arguments of the z-fields and z is considered to be slowly varying. So it only has sizable
components for small energies and small wave vectors. We thus consider the static
polarization and expand in powers of ~˜q:
nF (~k + ~˜q) ' nF (~k) + ∂nF (
~k)
∂
~∇(~k) · ~˜q + . . . (A.50)
= nF (~k)− ∂nF (
~k)
∂µ
~∇(~k) · ~˜q + . . . (A.51)
(~k + ~˜q) ' (~k) + ~∇(~k) · ~˜q + . . . (A.52)
When we insert this into the sum, we get in lowest order
χ0(0, ~˜q) =
2
N
∑
~k
∂nF (~k)
∂µ
~∇(~k) · ~˜q
−~∇(~k) · ~˜q = −
2
N
∂
∂µ
∑
~k
nF (~k) = −2∂ρ˜
∂µ
= −2ρ˜κ , (A.53)
where κ denotes the electronic compressibility, defined by κ = 1
ρ˜
∂ρ˜
∂µ
.
Three Green’s functions
Here we have
S = 1
β
∑
n,n′,n′′
G0(νn, ~k)G0(νn′ , ~k′)G0(νn′′ , ~k′′)
= 1
β
∑
n,n′,n′′
1
iνn −
[
(~k) + J + µ
] 1
iνn′ −
[
(~k′) + J + µ
] 1
iνn′′ −
[
(~k′′) + J + µ
] . (A.54)
After the change of variables
νn′ = νn + ωm − ωm∗ , ~k′ = ~k + ~q − ~q∗ , (A.55)
νn′′ = νn + ωm − ωm∗∗ , ~k′′ = ~k + ~q − ~q∗∗ , (A.56)
186
A.3. Matsubara frequency summation
we get
S = 1
β
∑
n,m∗,m∗∗
1
iνn − [(~k) + J + µ]
1
i(νn + ωm − ωm∗)− [(~k + ~q − ~q∗) + J + µ]
· 1
i(νn + ωm − ωm∗∗)− [(~k + ~q − ~q∗∗) + J + µ]
. (A.57)
To evaluate the sum over νn we consider the contour integral
I = lim
R→∞
∫ dz
2pii s(z)n(z) , (A.58)
with the contour being a circle of radius R, n(z) given in (A.32) and
s(z) = 1
z − ζ1 ·
1
z − ζ2 ·
1
z − ζ3 , (A.59)
where
ζ1 = (~k) + J + µ ,
ζ2 =
[
(~k + ~q − ~q∗) + J + µ
]
− i(ωm − ωm∗) ,
ζ3 =
[
(~k + ~q − ~q∗∗) + J + µ
]
− i(ωm − ωm∗∗) . (A.60)
According to Cauchy’s theorem the integral may be written as a sum of residues
Res(ζ1) =
n(ζ1)
(ζ1 − ζ2)(ζ1 − ζ3) ,
Res(ζ2) =
n(ζ2)
(ζ2 − ζ1)(ζ2 − ζ3) ,
Res(ζ3) =
n(ζ3)
(ζ3 − ζ1)(ζ3 − ζ2) ,
Res((2n+ 1)ipi/β) = − 1
β
1
(iνn − ζ1)(iνn − ζ2)(iνn − ζ3) . (A.61)
For R→∞ we have I → 0 so that
1
β
S = n(ζ1)(ζ1 − ζ2)(ζ1 − ζ3) +
n(ζ2)
(ζ2 − ζ1)(ζ2 − ζ3) +
n(ζ3)
(ζ3 − ζ1)(ζ3 − ζ2) . (A.62)
187
A. Supplementaries
We plug in (A.60) and by a straightforward manipulation obtain
1
β
S =
n(~k)−n(~k+~q−~q∗∗)
i(ωm−ωm∗∗ )+(~k)−(~k+~q−~q∗∗)
− n(~k+~q−~q∗)−n(~k+~q−~q∗∗)
i(ωm∗−ωm∗∗ )+(~k+~q−~q∗)−(~k+~q−~q∗∗)
i(ωm − ωm∗) + (~k)− (~k + ~q − ~q∗)
. (A.63)
Let us define
ω˜m = ωm − ωm∗ , ~˜q = ~q − ~q∗ , (A.64)
˜˜ωm = ωm − ωm∗∗ , ˜˜~q = ~q − ~q∗∗ , (A.65)
so that
1
β
S =
n(~k)−n(~k+˜˜~q)
i ˜˜ωm+(~k)−(~k+˜˜~q)
− n(~k+~˜q)−n(~k+˜˜~q)
i(˜˜ωm−ω˜m)+(~k+~˜q)−(~k+˜˜~q)
iω˜m + (~k)− (~k + ~˜q)
. (A.66)
We examine the static limit, setting ω˜ and ˜˜ω to zero
1
β
S =
n(~k)−n(~k+˜˜~q)
(~k)−(~k+˜˜~q) −
n(~k+~˜q)−n(~k+˜˜~q)
(~k+~˜q)−(~k+˜˜~q)
(~k)− (~k + ~˜q) . (A.67)
We expand the Fermi distribution and the dispersion up to second order in ~q:
n(~k + ~˜q) = n(~k)− ∂n(
~k)
∂µ
∂α(~k)q˜α +
1
2
(
∂2n(~k)
∂µ2
∂β(~k)∂γ(~k)− ∂n(
~k)
∂µ
∂β∂γ(~k)
)
q˜β q˜γ ,
n(~k + ˜˜~q) = n(~k)− ∂n(
~k)
∂µ
∂α(~k)˜˜qα +
1
2
(
∂2n(~k)
∂µ2
∂β(~k)∂γ(~k)− ∂n(
~k)
∂µ
∂β∂γ(~k)
)
˜˜qβ ˜˜qγ ,
(~k + ~˜q) = (~k) + ∂α(~k)q˜α +
1
2∂β∂γ(
~k)q˜β q˜γ ,
(~k + ˜˜~q) = (~k) + ∂α(~k)˜˜qα +
1
2∂β∂γ(
~k)˜˜qβ ˜˜qγ . (A.68)
If we plug this into (A.67), we obtain in lowest order
1
β
S = 12
∂2
∂µ2
nF (~k) . (A.69)
Finally, we perform the sum over all wave vectors ~k
2
N
∑
~k
1
β
S = 2
N
1
2
∂2
∂µ2
∑
~k
nF (~k) =
∂2ρ˜
∂µ2
. (A.70)
188
A.4. Parametrization of z and ζ and gauge fixing
A.4. Parametrization of z and ζ and gauge fixing
The z-field can be parametrized as
z =
 cos θ2 e−i(ϕ/2−Λ)
sin θ2 e
+i(ϕ/2+Λ)
 , (A.71)
where θ and ϕ are two angular variables and Λ is a pure gauge field. Note that in this
parametrization z¯z = 1 is automatically fulfilled. For ζ we choose
ζ =
 ρ1 e−i(χ/2−Γ)
ρ2 e+i(χ/2+Γ)
 , (A.72)
where ρi = |ζi|, χ is an angular variable and Γ is again a pure gauge field. The constraint
z¯ ζ + ζ¯ z = 0 now translates into
z¯ ζ + ζ¯ z = 2ρ1 cos
θ
2 cos
(
χ
2 −
φ
2 + (Λ− Γ)
)
+ 2ρ2 sin
θ
2 cos
(
χ
2 −
φ
2 − (Λ− Γ)
)
= 0 . (A.73)
Gauge invariance of the effective action (4.321) implies that Γ − Λ = Φ, where Φ is a
global phase. It cannot be chosen to 0, since that would lead to a contradiction as ρi > 0
and 0 ≤ θ ≤ pi. However, we may choose Φ = pi/2, such that (A.73) leads to
ρ1 cos
θ
2 = ρ2 sin
θ
2 . (A.74)
The condition Γ = Λ + pi/2 also implies that
ζ1
|ζ1|
ζ2
|ζ2| = e
2iΓ = −e2iΛ = − z1|z1|
z2
|z2| , (A.75)
by which we obtained a gauge fixing condition for the ζ-field.
189

Appendix B
Definitions
B.1. Densities and compressibilities
The hole density and the compressibility are defined through
ρ˜ = 1
N
∑
~k
nF (~k) , κ =
1
ρ˜
∂ρ˜
∂µ
. (B.1)
We use the following conventions for the modulated quantities
ρ˜1 =
1
N
∑
~k
nF (~k) cos
√
2akx ,
ρ˜2 =
1
N
∑
~k
nF (~k) cos
√
2akx cos
√
2aky , (B.2)
κ1 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) cos2
√
2a
2 kx cos
2
√
2a
2 ky ,
κ2 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) cos2
√
2a
2 kx ,
κ3 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) sin2
√
2akx ,
κ4 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) sin2
√
2akx cos2
√
2aky ,
κ5 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) sin2
√
2a
2 kx cos
2
√
2a
2 ky ,
κ6 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) sin2
√
2akx cos
√
2aky ,
κ7 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) cos
√
2akx ,
191
B. Definitions
κ8 =
1
ρ˜
∂
∂µ
1
N
∑
~k
nF (~k) cos
√
2akx cos
√
2aky , (B.3)
λ4 =
1
ρ˜
∂2
∂µ2
1
N
∑
~k
nF (~k) cos2
√
2a
2 kx cos
2
√
2a
2 ky ,
λ8 =
1
ρ˜
∂2
∂µ2
1
N
∑
~k
nF (~k) cos2
√
2a
2 kx . (B.4)
B.2. F - and G-fields
We define
Fα = z1∂αz2 − z2∂αz1 , G = 2
(
z1ζ2 − z2ζ1
)
,
Fαβ = z1∂α∂βz2 − z2∂α∂βz1 , G1α = 2
(
[∂αz1]ζ2 − [∂αz2]ζ1
)
,
αFβ = ∂αz1 ∂βz2 − ∂αz2 ∂βz1 , G2α = 2
(
z1∂αζ2 − z2∂αζ1
)
, (B.5)
such that
∂αG = G1α +G2α ,
∂αFα = Fαα ,
∂αFβ = αFβ + Fαβ ,
∂βFα = βFα + Fαβ , (B.6)
and since αFβ = −βFα we have
∂αFβ + ∂βFα = 2Fαβ . (B.7)
From Fα, F ∗β to z¯, z: O(a2)-terms
We derive the simple identity:
F ∗αFβ = (z∗1∂αz∗2 − z∗2∂αz∗1)(z1∂βz2 − z2∂βz1)
= |z1|2∂αz∗2∂βz2 + |z2|2∂αz∗1∂βz1 − (z2∂αz∗2)(z∗1∂βz1)− (z1∂αz∗1)(z∗2∂βz2)
= ∂αz∗2∂βz2 + ∂αz∗1∂βz1 − (z2∂αz∗2)(z∗2∂βz2)− (z1∂αz∗1)(z∗1∂βz1)
192
B.2. F - and G-fields
−(z2∂αz∗2)(z∗1∂βz1)− (z1∂αz∗1)(z∗2∂βz2)
= ∂αz¯ ∂βz + (z¯∂αz)(z¯∂βz) , (B.8)
where we repeatedly used z¯z = 1.
Note that via integration by parts in the first summand, we can show that
F ∗αFβ = ∂αz¯ ∂βz + (z¯ ∂αz)(z¯ ∂βz) = ∂β z¯ ∂αz + (z¯ ∂βz)(z¯ ∂αz) = F ∗βFα . (B.9)
F -terms go nicely with z1 and z2:
z∗1z
∗
2 Fα = z∗1z∗2z1∂αz2 − z∗1z∗2z2∂αz1
= |z1|2(z∗2∂αz2)− |z2|2(z∗1∂αz1) ,
z1z2 F
∗
α = |z1|2(z2∂αz∗2)− |z2|2(z1∂αz∗1) . (B.10)
From Fα, F ∗β to z¯, z: O(a3)-terms
We have, similarly to the O(a2)-case,
F ∗αFβγ = (z∗1∂αz∗2 − z∗2∂αz∗1)(z1∂β∂γz2 − z2∂β∂γz1)
= ∂αz¯ ∂β∂γz + (z¯∂αz)(z¯ ∂β∂γz) ,
F ∗αβFγ = (z∗1∂α∂βz∗2 − z∗2∂α∂βz∗1)(z1∂γz2 − z2∂γz1)
= ∂α∂β z¯ ∂γz − (z¯∂γz)(z ∂α∂β z¯) , (B.11)
and
z∗1z
∗
2 Fαβ = |z1|2(z∗2∂α∂βz2)− |z2|2(z∗1∂α∂βz1) ,
z1z2 F
∗
αβ = |z1|2(z2∂α∂βz∗2)− |z2|2(z1∂α∂βz∗1) . (B.12)
193

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