08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access Hubbard and Kondo lattice models in two dimensions : a QMC study(2003) Feldbacher, Martin; Assaad, Fakher F. (Prof. Dr.)This thesis discusses mainly two Fermionic lattice systems, first a Kondo lattice with additional Hubbard interaction and second a Hubbard Hamiltonian augmented with additional spin and charge interactions. We first introduce the Quantum Monte Carlo technique, which is then employed to study the two respective systems. We present an innovation that allows to calculate time displaced Greens functions more efficiently. Compared with previously used numerically stable algorithms the new method gains an order of magnitude in speed, but is just as precise, and very simple to implement. In the second chapter we consider the Kondo lattice model in two dimensions at half filling. In addition to the Fermionic hopping integral t and the superexchange coupling J the role of a Coulomb repulsion U in the conduction band is investigated. We find the model to display a magnetic order-disorder transition in the U-J plane with a critical value of Jc which is decreasing as a function of U. The single-particle spectral function A(k,ω) is computed across this transition. We conclude that (i) the local screening of impurity spins determines the low-energy behavior of the spectral function and (ii) one cannot deform continuously the spectral function of the half-filled Hubbard model at J=0 to that of the Kondo insulator at J>Jc. In the third chapter we investigate the phase diagram of a new model that exhibits a first order transition between s-wave superconducting and antiferromagnetic phases. The model, a generalized Hubbard model augmented with competing spin-spin and pair-pair interactions, was investigated using the projector quantum Monte Carlo method. Upon varying the Hubbard U from attractive to repulsive, we find a first order phase transition between superconducting and antiferromagnetic states.Item Open Access Elektronische Kopplung und Transferprozesse in Donor-Brücke-Akzeptor-Systemen(2000) Roccasalvo, Giuseppe; Sigmund, Ernst (Prof. Dr.)Donor-Brücke-Akzeptor(DBA)-Molekülen kommt eine wichtige Bedeutung beim Design von funktionellen Einheiten in der Molekularen Elektronik zu, die u.a. das Ziel verfolgt, Ladung oder Energie unter kontrollierbaren Bedingungen zwischen unterschiedlichen Zuständen eines 'molekularelektronischen Bauelementes' zu transferieren. Im Rahmen dieser Arbeit wird mittels theoretischer DBA-Modellsystemen die Abhängigkeit von Transferprozessen im Hinblick auf unterschiedliche chemische Strukturelemente (Kopplungsparameter, Orbitalenergie-Konstellationen) sowie Reservoireinflüsse untersucht, wodurch gezielt Abschätzungen über die praktische Verwendbarkeit von Molekülsystemen angestellt werden können. Ein besonderer Augenmerk liegt dabei auf der Übernächsten-Nachbar(NNN)-Wechselwirkung in kettenartigen Brückensystemen mit zickzackartiger Struktur (z.B. Molekülklasse der Alkane). Methodisch wird die Transferrate näherungsweise durch die Reduktion auf ein effektives Zwei-Niveau-System und exakt durch explizite Reservoirankopplung berechnet. Letztere Methode wird mit Hilfe des Greenschen-Funktions(GF)-Formalismus auf Pole beliebiger Ordnung und beliebiger (energieabhängiger) Reservoirankopplung verallgemeinert. Die qualitativen Eigenschaften der behandelten Modellsysteme sind einerseits resonanzartige Verstärkungen und andererseits antiresonanzartige Minima in der elektronischen Kopplung bzw. Transferrate. Ersteres tritt im Tunnelregime in der Nähe der Brückeneigenwerte auf, wohingegen letzteres Verhalten durch quantenmechanische Interferenzeffekte bei mehr als einem möglichenTransferpfad (NNN-Wechselwirkung) zustande kommt. Betrachtet man die Abhängigkeit der Transferrate von der Anzahl der Monomereinheiten der Brücke, so können bereits kleine NNN-Wechselwirkungen im Zusammenspiel mit der NN-Wechselwirkung nichtmonotone Effekte hinsichtlich der monoton fallenden Abstandsgesetze bei reiner NN-Wechselwirkung bewirken.Item Open Access Nonequilibrium dynamics of strongly correlated quantum systems(2006) Manmana, Salvatore Rosario; Muramatsu, Alejandro (Prof. Dr.)In this thesis, strongly correlated quantum many-body systems in equilibrium and in out-of-equilibrium situations are investigated. This is done by applying and developing well established numerical methods. The focus of the thesis lies in the development and application of the density matrix renormalization group method (DMRG) to quantum many-body systems out of equilbrium. In this thesis, in addition to the DMRG, methods for the exact diagonalization of the Hamiltonian of the system, like the Lanczos- or the Jacobi-Davidson method, are treated. An extension of the Lanczos method makes it possible to treat the time evolution of strongly correlated quantum systems with an accuracy comparable to machine precision. This method is the basis for a possible extension of the DMRG for the treatment of systems out of equilibrium, the so-called adaptive time-dependent DMRG ("adaptive t-DMRG"). A second variant of the adaptive t-DMRG uses the Suzuki-Trotter decomposition of the time-evolution operator. An error analysis demonstrates that both methods, for a suitable choice of control parameters, have errors < 1% at the end of the time evolution in complicated quantities like, e.g., the momentum distribution function. We apply these numerical methods to investigate the quantum critical behavior of a variant of the Hubbard model and to treat two non-equilibrium situations of current interest. Extensive use of the DMRG makes it possible to clarify the quantum-critical behavior of the ionic Hubbard model; in particular, the numerical results demonstrate quite clearly that the correct scenario has two critical points; at the first critical point, only the charge degrees of freedom are critical, while at the second one only the spin degrees of freedom are critical. Quite surprisingly, the Mott-insulator phase in the strong-coupling regime shows a divergent electrical susceptibility. Next, the dynamics of a system of so-called soft-core bosons is treated. The particles initially are trapped in a deep box-potential and the dynamics is investigated after releasing them from this trapping potential. Similar to exact results obtained for so-called hard-core bosons, (quasi-)coherent matter waves emerge, demonstrating the possibility of realizing an atom laser in experiments on optical lattices. We find that the wave vector of the emerging matter wave can be tuned by changing the strength of the interaction between the particles. The second non-equilibrium situation is a so-called "quantum-quench" of a system of strongly correlated fermions, i.e., the evolution of the system after suddenly changing an intrinsic parameter like, e.g., the interaction strength between the particles, is investigated. In this case, we are mainly interested in the long-time behavior; in particular, the question arises whether, relying on Boltzmann's ergodic hypothesis, relaxation to a thermal state is obtained. We find that, in general, non-thermal final states are reached, which can be described by generalized Gibbs-Boltzmann ensembles.Item Open Access Single hole dynamics in the t-J model(2000) Brunner, Michael; Muramatsu, Alejandro (Prof. Dr.)In dieser Arbeit werden Ergebnisse für die Ein-Loch-Dynamik im t-J Modell vorgestellt. Diese Resultate konnten durch einen neuen Quanten Monte Carlo Algorithmus erzielt werden, der die Einteilchen-Greensfunktion in imaginärer Zeit bei halber Füllung berechnet. Die Greensfunktion wird benutzt um einerseits Quasiteilchendispersion und -gewicht direkt zu berechnen, andererseits kann mit Hilfe der Maximum Entropy-Methode die komplette Spektralfunktion berechnet werden. Es wurden Simulationen für das t-J Modell in einer und zwei Dimensionen, sowie in Doppel- und Dreifachleitern durchgeführt. In einer Dimension zeigt sich, dass die Ergebnisse mit einem einfachen Spin-Ladungstrennungs-Ansatz übereinstimmen, wobei das Minimum der Dispersion bei k=pi/2 liegt. Am supersymmetrischen Punkt J/t=2 beobachtet man ein verschwindendes Quasiteilchengewicht, was mit analytischen Rechnungen übereinstimmt. Die in zwei Dimensionen gefundene Dispersion stimmt mit Vorhersagen aus selbstkonsistenter Born-Näherung und Reihenentwicklung überein. Man beobachtet flache Bänder bei k=(pi,0) und ein Minimum bei k=(pi/2,pi/2). Das Quasiteilchengewicht im thermodynamischen Limes ist endlich. Die beiden Leitersysteme zeigen ein deutlich anderes Verhalten. Es ist bekannt, dass die Doppelleiter einen Spingap hat, während die Dreifachleiter gaplosee Spinanregungen besitzt. Die Dispersion der Doppelleiter bis hin zum isotropen Fall lässt sich analytisch gut verstehen, wenn man vom Limes starker Kopplung entlang der Sprossen ausgeht. Weiter beobachtet man ein großes Quasiteilchengewicht. Der Limes starker Kopplung in der Dreifachleiter führt zu einem effektiven Modell, das zur eindimensionalen t-J-Kette äquivalent ist. Falls die Kopplung zwischen den Ketten weit stärker als entlang der Leiter ist, so sind unsere Ergebnisse mit diesem effektiven Modell konsistent. Bei J/t=2 zeigen unsere Daten ein Verschwinden des Quasiteilchengewichts im thermodynamischen Limes.Item Open Access Dynamics of the doped one-dimensional t-J model from quantum Monte Carlo simulations(2003) Lavalle, Catia; Muramatsu, Alejandro (Prof. Dr.)In this work, we present results for the dynamics of the doped one-dimensional t-J model for very large systems and in a wide parameter range. These results are obtained by quantum Monte Carlo simulation based on a newly developed algorithm, the hybrid-loop Monte Carlo. Within this algorithm the loop-algorithm is used to update the spin degrees of freedom, assuring optimal efficiency, whereas the charge degrees of freedom are evolved exactly for a given spin background in the frame of the determinantal algorithm, so that both static and dynamical observables can be measured. The study of the one-particle spectral function in the whole Luttinger-liquid regime of the phase diagram shows a splitting of the spectral weight into two branches on the inverse photoemission side, in contradiction with what is expected for an ordinary metal. This feature can be understood if we assume that the elementary excitation content of the nearest-neighbor t-J model and their dispersion is the same as the one analytically obtained at J/t=2 from a t-J model with both hopping and exchange interaction decaying as 1/r². In this case the elementary excitations are spinons with charge Q=0 and spin S=1/2, holons with charge Q=-e and spin S=0, and antiholon with charge Q=2e and spin S=0. Since the antiholon has twice the mass of the holon and charge with opposite sign, holon and antiholon are not charge conjugate but two independent elementary excitations. The agreement between the analytic results derived from our assumptions and the Monte Carlo data for a wide range of parameters and for different values of doping strongly indicate, that antiholons are generic excitations in the nearest-neighbor t-J model. Since the quantum numbers of the antiholons are the same as those of for a Cooper pair, the question arises whether these new excitations are related to superconductivity, that is indeed one of the possible phases in the one-dimensional nearest-neighbor t-J model. A further study of the system in the phase-separated region shows that the spectral function is very discontinuous in the photoemission side. This feature reminds an interference scenario generated by superposition of a continuous dispersion with a regular pattern. The nature of a phase-separated regime in a simulation with periodic boundary condition can be the origin of the regular pattern.Item Open Access Quantum Monte Carlo studies of fermions with attractive interactions in optical traps(2008) Karim Pour, Farshid; Muramatsu, Alejandro (Prof. Dr.)About 50 years ago, Penrose and Onsager posed the question: Can a solid be superfluid? Supersolids are defined by the simultaneous presence of two types of long-range order: superfluidity' coexisting with a periodic modulation of the density', in other words, supersolids combine the main characteristics of solids and superfluids. Using quantum Monte Carlo simulations, we show here that density-density and pairing correlation functions of the one-dimensional attractive fermionic Hubbard model in a confinement potential are characterized by the anomalous dimension (Luttinger parameter) of a corresponding periodic system. This allows to determine conditions for a supersolid state inside a trap. We show explicitely, that under such conditions the structure form factors for both correlation functions scale with the same exponent upon increasing the system size, thus giving rise to a (quasi-)supersolid.Item Open Access Spin gap in doped ladder systems(2004) Campos Venuti, Lorenzo; Alejandro, Muramatsu (Prof. Dr.)A proper theoretical description of doped antiferromagnets is until now an unresolved issue. In this work we study chain and ladder systems, and in particularly we focus on the physically most interesting two leg ladder. Our starting point is the Spin Fermion model in which an antiferromagnetic background of localized spins interacts with mobile holes via a rotation invariant Kondo-like term. The interesting region of the phase diagram is that close to the Mott-insulator transition where the doping delta is zero. At zero doping in fact the system is an insulator and the spins organize themselves in a spin liquid, a rotational invariant state characterized by a finite correlation length and an energy gap (a spin gap) above the ground state. Such a state is also the ground state of a field theory, the non linear sigma model (NLsigmaM) which is recognized as the low energy effective theory for antiferromagnetic spin ladder. The first question we answer is how such a spin state evolves as one moves off from the Mott phase by increasing the doping. Integrating out the fermions in our model we obtain an effective theory for the spins which we are able to evaluate in the continuum limit. The effective theory is again a NLsigmaM with coupling constants which depend on the concentration of dopant holes. In contrast to existing mean field calculation our theory predicts a lowering of the spin gap with doping and a consequent increase in the correlation length. Indeed a lowering of the spin gap due to doping is also observed in numerical simulation and on NMR experiments on $Sr14-xCaxCu24O41 with which we obtain very good agreement. Secondly we concentrated on the behavior of the fermions. The general paradigm of interacting fermions in one dimension is that of the Luttinger Liquid characterized by bosonic excitations and spin charge separation. By generalizing our approach we are able to access the one particle fermion propagator as a quantity averaged over the NLsigmaM which controls the spin background. We see that in the limit of zero doping the quasiparticle weight Z is non-zero in a neighborhood of the Fermi energy. This in turn implies that the Luttinger liquid parameter Krho goes to one as the doping delta goes to zero as was first argued by Schulz. Our stronger result allows us to assert that in the very low doping regime the fermions constitute a Fermi liquid.Item Open Access Verallgemeinerungen und Anwendungen der Fulton-Gouterman-Transformation(2000) Rapp, Matthias; Wagner, Max (Prof. Dr. rer. nat.)In dieser Arbeit betrachten wir unitäre Transformationen in gekoppelten Elektron-Phonon-Systemen, die dadurch ausgezeichnet sind, dass sie die Symmetrie des Systems in optimaler Weise instrumentalisieren. Dadurch wird der Hamiltonoperator bezüglich des elektronischen Subsystems diagonalisiert und das Problem auf einen effektiven Phononen-Hamiltonoperator reduziert. Das Schwergewicht unserer Untersuchung liegt auf der Behandlung verallgemeinerter Formen der Fulton-Gouterman-Transformation (Geralised FGT, GFGT). Insbesondere diskutieren wir eine Erweiterung der GFGT auf mehr als ein elektronisches Band (Mehrband-Formulierung der GFGT, MFGT) und kontrastieren die MFGT mit der Lee-Low-Pines-Transformation (LLPT). Wir geben auch eine exponentielle Form der MFGT für zyklische Symmetrie an. Diese exponentielle Form kann als Ausgangspunkt für zukünftige Behandlungen von Vielteilchen-Systemen dienen. Um die erfolgreiche Wirkungsweise der MFGT in translatorischen Gittersystemen zeigen zu können, führen wir einen hinreichend allgemeinen Elektron-Phonon-Hamilton-operator ein, der die Diskretheit der periodischen Gitterstruktur und die phononische Anharmonizität inkorporiert. Dadurch werden Umklapp-Prozesse sowohl in der Elektron-Phonon-Kopplung wie auch in der phononischen Anharmonizität impliziert.Item Open Access Numerically exact studies of ultracold atoms on optical lattices(2004) Rigol Madrazo, Marcos; Muramatsu, Alejandro (Prof. Dr. rer. nat.)We study properties of ultracold quantum gases trapped in 1D and under the influence of an underlying optical lattice. We analyze both fermionic and bosonic systems following different numerically exact approaches, like quantum Monte Carlo for soft-core bosons and two-species fermions, and a recently developed exact approach for HCB. The exposition is organized as follows. In Chap. 2, we study single species degenerated fermions on 1D lattices. We show that a splitting of the system takes place in a region of the spectrum with eigenstates that have a non-vanishing weight only in a fraction of the system. We also show that if on top of the lattice an alternating potential is introduced, doubling the original periodicity, an additional slicing' of the system can be achieved. The width and number of such regions can be controlled in a given energy range by the amplitude of the new modulation. By filling these systems the ground state may exhibit insulating regions due to the presence of local gaps, which can be observed in the local density of states. As a complement to these 1D results we present in Appendix A results obtained in 2D lattices. In Chap. 3 we review in some detail the two quantum Monte Carlo (QMC) techniques that are relevant to our study of confined soft-core bosons and two-species fermions on optical lattices, the worldline QMC and the zero temperature projector QMC, respectively. The basis of the classical Monte Carlo method are also presented, included a proof of the Central Limit theorem (Appendix B) that establishes its foundations. Properties of soft-core bosons confined on optical lattices are presented in Chap. 4. QMC simulations show that due to the presence of the slowly varying confining potential the concept of commensurability, typical for periodic systems, loses its meaning. Mott domains appears for a continuous range of fillings and always coexist with superfluid phases, as it was mentioned before. The latter feature is reflected by the behavior of the global compressibility, which never vanishes. A local compressibility, proportional to the variance of the local density, is defined in order to characterize the local Mott-insulating phases. Finally, a phase diagram for trapped systems is also presented. In Chap. 5 we study the ground state properties of two-components fermions confined on optical lattices. Like for the bosonic case, Mott domains also appear for a continuous range of fillings and always coexist with compressible phases. We define a local order parameter (that we denominate local compressibility), which characterizes the Mott insulating phases in an unambiguous way. By means of this local compressibility, we study in detail the interphase between the metallic and insulating region finding that critical behavior sets in revealing a new critical exponent. Furthermore, the behavior of the local compressibility and the variance of the density are found to be universal in this case, independently of the confining potential and the strength of the interaction, as usual in critical phenomena. We present in Chap. 6 a recently developed exact numerical approach that allows to study ground state properties of HCB confined on 1-D lattices. This exact treatment is applied to study the off-diagonal behavior of the one-particle density matrix (OPDM) and related quantities in the equilibrium case. We find that the OPDM decays as a power-law x^-1/2 for large-x, irrespective of the confining potential chosen. The power-law above is shown to determine the scaling of the occupation of the lowest natural orbital in the thermodynamic limit. This scaling and its finite size corrections are also studied for arbitrary powers of the confining potential. In addition, we find a power-law decay of the NO occupations at low densities, and show that its exponent is also universal. The low density limit in the lattice, equivalent to continuous systems, is also analyzed in detail. The approach above is generalized in Chap. 7 to study the non-equilibrium dynamics of HCB in 1-D configurations with an underlying lattice. The presence of the lattice enables the creation of pure Fock states of HCB (a HCB per lattice site) where there is no coherence in the system. We show that quasi-long range correlations develop in the equal-time-one-particle density matrix when such states are allowed to evolve freely, and that they lead to the formation of quasi-condensates of HCB at finite momentum. In addition,we obtain an universal power-law describing the population of the quasi-condensate as a function of time. Finally, we discuss how such systems can be used to create atom lasers with a wave-length that can be controlled through the lattice parameter.Item Open Access Berechnung der zeitlichen Dynamik gekoppelter Exziton-Phonon-Systeme mit Hilfe unitärer Transformationen(2000) Herfort, Ulrich; Wagner, Max (Prof. Dr.)Das Phänomen der retardierten Lumineszenz in Alkalihalogeniden und Edelgaskristallen, aber auch generell die experimentelle Möglichkeit kurzzeitaufgelöster Beobachtung exzitonischer Evolutionen, lenkt die Aufmerksamkeit auf das Problem der zeitlichen Entwicklung gekoppelter Exziton-Phonon-Systeme. In früheren Arbeiten zu diesem Thema wurde meist eine semiklassische Näherung durchgeführt, das heißt, das Phononensystem wurde klassisch, das Exzitonsystem quantenmechanisch behandelt. Ein solches Verfahren führt jedoch in bestimmten Bereichen der Systemparameter zu Ergebnissen, die der exakten Lösung sogar qualitativ widersprechen. Einer exakten numerischen Lösung stehen jedoch, außer in den einfachsten Fällen, nahezu unüberwindliche Schwierigkeiten entgegen. In der vorliegenden Dissertation wird daher ein neues Verfahren zur Berechnung der zeitlichen Dynamik gekoppelter Exziton-Phonon-Systeme vorgestellt, das auf unitären Transformationen beruht. Das System wird dabei einer handhabbaren, parametrisiertenären Transformation unterworfen, wobei die Parameter so gewählt werden, daß der transformierte Hamiltonoperator möglichst diagonal bezüglich einer gegeben Zustandsbasis wird. Die Methode wird am Beispiel des Dimer-Oszillator-Modells demonstriert, des einfachsten Modells, das den für gekoppelte Exziton-Phonon-Systeme charakteristischen Antagonismus zwischen Selbsteinfang und Delokalisierung des Exzitons aufweist. Die Ergebnisse für die Energieeigenwerte und die zeitliche Entwicklung werden mit den durch numerische Diagonalisierung des Hamiltonoperators gewonnenen Lösungen verglichen. Insgesamt läßt sich sagen, daß sich durch unitäre Transformationen nicht nur die statischen Eigenschaften, sondern auch die zeitliche Entwicklung gekoppelter Exziton-Phonon-Systeme berechnen lassen. Der nächste Schritt wäre es, die Methode auf kompliziertere Systeme, wie z. B. ausgedehnte, translationsinvariante Systeme oder das 222-Modell zu verallgemeinern.