Berry phases and topological physics in one-dimensional systems
Abstract: Topological systems are one of the most active research areas in condensed matter physics. The topological characterization of a condensed matter system relies on mathematical constructs such as the Berry phase, the winding number, or the Chern number. In the first part of the talk, I will explain how the Berry phase can be understood as the first cumulant of a series. We calculate higher order cumulants and reconstruct the underlying distribution of the polarization for the Rice-Mele model. Our approach allows the visualization of a topological transition, how a system goes between phases with different quantization. In the second part of the talk, I will go through constructing one-dimensional analogs of the Haldane and Kane-Mele models. In the former, the overall winding number does not indicate topological behavior, but the model falls into two independent Creutz models with opposite windings, and the topological transition occurs within each one separately. The latter ladder also consists of two Creutz models, one for each spin-channel and falls in the CII symmetry class. In its analysis the thermodynamic derivation of the generalization of the Streda-Widom formula to the quantum spin Hall effect turned out to be useful.