Topological aspects of charge transport in quantum many-body systems
Motivated by the recent proposals and developments of topological insulators and topological superconductors for their potential applications in electronic devices and quantum computing, we have theoretically studied topological properties of quantum many-body systems. First, we calculate the gauge-invariant cumulants (and moments) associated with the Zak phase. The first cumulant corresponds to the Berry phase itself, the others turn out to be the associated spread, skew, kurtosis, etc. The cumulants are shown to be gauge invariant. We reconstruct the underlying probability distribution of the polarization by maximizing the information entropy and applying the moments as constraints in the Rice-Mele model and in the interacting, spinless Su-Schrieffer-Heeger model. When the Wannier functions are localized within one-unit cell, the probability distribution so obtained corresponds to that of the Wannier function. We follow the probability distribution of the polarization in cycles around the topologically nontrivial point of these models. Secondly, we have constructed a topological one-dimensional analog of the Haldane and Kane-Mele models in two dimensions, with hexagonal lattices. Our Haldane one-dimensional analog model belongs to the C and CI symmetry classes, depending on the parameters, but, due to re ection, it exhibits topological insulation. The model consists of two superimposed Creutz models with onsite potentials. The topological invariants of each Creutz model sum to give the mirror winding number, with winding numbers which are nonzero individually but equal and opposite in the topological phase, and both zero in the trivial phase. We also construct a topological one-dimensional ladder model following the steps which lead to the Kane-Mele model in two dimensions. We couple two Haldane-type ladder models, one for each spin channel, in such a way that time-reversal invariance is restored. We also add a Rashba spin-orbit coupling term. The model falls in the CII symmetry class. We demonstrate the presence of edge states and quantized Hall response in the topological region. Our model exhibits two distinct topological regions, distinguished by the different types of re ection symmetries. Thirdly, we consider the edge at the interface of a simple tight-binding model and a band insulator. We find that crossings in the band structure (one dimensional Dirac points) appear when an interface is present in the system. We calculate the hopping energy resolved along lines of bonds parallel to the interface as a function of distance from the interface. Similarly, we introduce a transport coe cient (Drude weight) for charge currents running parallel to the interface. We find that charge mobility (both the kinetic energy and the Drude weight) is significantly enhanced in the surface of the tight-binding part of the model near the interface. Finally, we study a variant of the generalized Aubry-Andre-Harper model with the effect of introducing next nearest-neighbor p-wave superconducting pairing with incommensurate and commensurate cosine modulations. We extend generalized Aubry-Andr e-Harper model with p-wave superconducting to topologically equivalent and nontrivial "anancestor" two-dimensional p-wave superconducting model. It is found that in incommensurate (commensurate) modulation, by varying next nearest-neighbor p-wave pairing order parameter, the system can switch between extended states and localized states (fully gapped phase and a gapless phase).