Extensions of one-dimensional topological insulator models and their properties
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We mainly study the SSH model, a one dimensional topological insulator. As a start, we give a brief introduction about the model and theoretically showed that it should have at least 2 distinct states using Jackiw-Rebbi model. Instead of us-ing only the periodic boundary conditions, we also use open boundary conditions which revealed the zero energy edge states. Introducing the spectral symmetries, we show how a given system can be characterized using the periodic table of topo-logical insulators  and depending on the symmetries we discuss which invariant can be used to determine diﬀerent topological states. Using an enlarged system for a certain symmetry class Z, we show that polarization or Berry phase fails to distinguish diﬀerent topological states. Subsequently, we implement a similar idea that Haldane used , breaking the time-reversal symmetry via introducing the complex next nearest neighbor hopping and ﬁnd that the system is charac-terized by Z2 invariant. Moving away from the ”textbook” way of writing the Bloch states, we introduce the distance dependent SSH model where the distance between A and B sublattice is p/q with p and q are being co-primes. We ﬁnd that the polarization can be found using the inversion symmetry of the wannier centers, which characterize the topological index. Plotting the curve in the pa-rameter space, we come to conclusion that Brillouin zone must be extended q times in order for the system to conserve its periodicity, which brings the knot behaviour of the curves that can be used to distinguish the topological state. At last, we make the SSH model spinful by introducing the time-reversal symmetry protecting Rashba spin-orbit coupling. Due to the Kramers’ theorem, degenerate states occur and non-Abelian Berry connection must be constructed to analyze the system. We ﬁnd that Kato propagator is suitable and gauge invariant way of doing this and computed the time-reversal polarization of the system.
Topological phase tran-sitions