Browsing by Author "Alpin, Kirill"

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Open Access
From topological nodal planes and multifold crossings in 3D to strong correlations in 2D
(2024) Alpin, Kirill; Metzner, Walter (Prof. Dr.)
Topological semimetals are hosts of numerous exotic phenomena, like Fermi arc surface states, the anomalous Hall effect, the quantized circular photogalvanic effect, the chiral anomaly, the anomalous Nernst effect and the anomalous thermal Hall effect. These phenomena are directly dependent on the presence and the properties of topological band crossings in the band structures of these compounds. Therefore, this thesis deals with classifying topological crossings in semimetals, deriving constraints on their topological charges, studying the properties and finding material realizations of more exotic band degeneracies, like nodal planes (NPs) and multifold degeneracies. The main results of this work includes an algorithm for fully classifying topological band structures of materials by DFT. This algorithm is used throughout this thesis for many different semimetals. The most detailed topological classification is presented for the B20 compound CoSi, shown to host topologically enforced NPs when including spin-orbit coupling (SOC). On the other hand, in ferromagnetic MnSi NPs are enforced by the space-group symmetries to be topological regardless if SOC is present or not, which we confirmed by a direct computation of symmetry eigenvalues. Tuning the magnetic moment, it is further possible to turn the presence of NPs on and off, leading to divergent Berry curvatures. By performing a complete topological classification of all multifold crossings in all space groups using an algorithm able to derive analytical low-energy Hamiltonians automatically, a 4-fold crossing was found to host an unusually high topological charge of +/-5. A material candidate is presented to feature this new topological phase at the Gamma point of BaAsPt. In two hexagonal space groups, SG 182 and SG 173, the concept of representation-enforced topological NPs have been introduced and material candidates were presented and classified using the above mentioned algorithms. One of these materials, NaCu5S3, exhibits a strikingly simple material band structure and surface DOS, resembling the ones produced by a tight-binding toy model. In chapter 7, we switch to physics of strongly correlated matter and examine the repulsive Hubbard model in 2D at large U. A unitary transformation is presented to map the interaction part of the Hubbard model to a single particle term. Applying this transformation to the whole Hubbard model results in a Hamiltonian of unconstrained fermions, which can then be mapped to a system of fermions interacting with spins. Integrating out the fermions using variational perturbation theory and replacing spins with classical spins reveals a phase transition at non-zero chemical potential to a phase with a finite d-wave order parameter.
Open Access
Massively parallel computations of the Bose-Hubbard model with time-dependent potentials
(2016) Alpin, Kirill
The present thesis describes an efficient algorithm to determine the time evolution of the Bose-Hubbard model with time-dependent potentials. The Bose-Hubbard model was interpreted as a system of particles in a periodic structure at absolute zero. The system described here had M potential wells and N particles. The main part of the algorithm presented here was the calculation of a matrix-vector product, which was carried out matrix-free, i.e. without storing the matrix explicitly. Due to the 'on-the-fly' nature of the matrix-vector product, it was straightforward to incorporate a time dependence of the potentials into the system. The order of Fock states was an important part of the derivation of this algorithm. Using combinatorics a state indexing routine was found. For neighbouring site transitions of particles between two states the access time to the amplitudes of these states is of order O(1). Therefore the time complexity of the matrix-vector product is O(MD) with the Hilbert space dimension D. It is also possible to use the neighbouring site transitions multiple times to perform an arbitrary transition, so more complicated operators can be implemented. The time evolution itself was carried out using two different Runge-Kutta methods, namely the classical 4th order Runge-Kutta method and an embedded Runge-Kutta method, which can provide 4th and 5th order time steps simultaneously. A performance improvement was achieved by executing the presented algorithm in parallel utilizing a graphics card as a general purpose computational device. The GPU was chosen as a platform for these calculations, as it is capable to execute a large amount of tasks simultaneously. The implementation was done using CUDA, an API provided by Nvidia. The performance of this implementation was compared to cuSPARSE, a library for parallel sparse matrix computations also developed by Nvidia. The presented algorithm was found to outperform the well optimized cuSPARSE implementation by a factor of 165% on a consumer level graphics card. To prove the relevancy of the GPU implementation, a CPU one of the very same algorithm was made. The execution time of the CPU implementation surpassed the GPU one by a factor of 200% with consumer level hardware and 2300% on a professional graphics card. The presented algorithm was also compared to a different method of matrix-free matrix-vector multiplication involving hash functions and a Bisection search to retrieve the index of a given Fock state. Due to this required search algorithm, the time complexity of this method is O(MDlog D), which has a worse scaling behaviour than the presented algorithm. Last but not least, the functionality of the presented method was demonstrated by application to different systems and comparison to another work, where the Bogoliubov-Backreaction method was implemented.