Self-consistent study of Abelian and non-Abelian order in a two-dimensional topological superconductor

Abstract

In the past few years, there has been a burst of theoretical and experimental activity in the field of topological insulators and topological superconductors. These materials represent new states of matter which are classified by an integer invariant and exhibit topologically protected conducting edge states which appear when the material is physically terminated.One of the consequences of topological superconductivity that makes this field significant is the possible existence of Majorana fermions as an elementary excitation. Despite extensive research, however, these exotic particles remain elusive to this day.In spite of the field of topological superconductivity being a hot topic in condensed matter research, there has been a severe lack of microscopic mean-field studies. In this thesis, we adopt a two-dimensional s-wave topological superconductivity model and perform selfconsistent mean-field studies in the Bogoliubov-de Gennes (BdG) formalism. As the BdG equations for topological superconductivity have high numerical demand, we implement the method of Chebyshev polynomial expansion, allowing self-consistent determination of the mean fields with high parallel efficiency and without any diagonalization of the Hamiltonian.Furthermore, we apply the recently developed Sakurai-Sugiura method to efficiently obtain the eigenpairs of the converged mean-field Hamiltonian.We first demonstrate the differences between Abelian, non-Abelian and trivial topological order by computing the Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) number and investigating the bulk-boundary correspondence. We then shift our focus to self-consistent studies, illustrating how the electron-phonon coupling strength should be chosen for several different parameter sets. Adding boundaries in the system, we are then able to confirm the appearance of Majorana fermions in the non-Abelian topological phase and discuss under which circumstances they appear. We also examine the dependence of the critical temperature of the system on the TKNN number and compare results with a recent momentum-space study on impurity effects in an s-wave topological superconductor [1]. The effects of depositing a single non-magnetic impurity in the center of the sample are also investigated. In particular, we find that in the case of odd TKNN number, the order parameter is extremely sensitive to even weak non-magnetic impurities signifying unconventional p-wave-like behaviour.ii Finally, we investigate the possible interplay of charge density waves and topological superconductivity. We show that within our model, topological superconductivity and topological charge density waves can coexist in the Abelian topological phase. Studying separately the pure superconducting state, the pure density wave state and the mixed state, we find that these three states are degenerate with the same ground-state energy just as in the conventional s-wave case. Upon introducing open surfaces in the system, zero-energy bound states are also found within all three states. Fin...

First- and Second-Order Topological Superconductivity and Temperature-Driven Topological Phase Transitions in the Extended Hubbard Model with Spin-Orbit Coupling

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Ultracold Atoms in a Square Lattice with Spin-Orbit Coupling: Charge Order, Superfluidity, and Topological Signatures

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Yu-Shiba-Rusinov states of impurities in a triangular lattice ofNbSe2with spin-orbit coupling

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