Symmetry and Topology

Abstract

A pedagogical introduction to the discrete symmetries and the linkage between topology and condensed matter physics is discussed. To emphasize the concepts concretely, we have considered systems in one dimension (1D) and two dimensions (2D) only. In 1D, we study two paradigmatic models, namely, the Su-Schrieffer-Heeger (SSH) model and a Kitaev chain with p-wave-superconducting correlations. The topological phases are ascertained by computing the edge modes and the topological invariants, such as the winding number. The systems are shown to undergo phase transitions from topological to trivial phases via gap closing scenarios in the parameter space that define the Hamiltonians. In 2D, we study graphene and do a thorough check therein on the topological properties should there be a spectral gap at the Dirac points by computing the Berry phase etc and the existence of edge modes. Following Haldane’s conjecture of a chiral second neighbor complex hopping, we study the Haldane model and ascertain the topological properties by computing the Chern number, edge modes and anomalous Hall conductivity with a plateau of magnitude e2h near the zero bias. Finally, in a bid to restore the lost time reversal symmetry in the Haldane model, Kane and Mele proposed linking the electron spin to the Haldane flux, which gave rise to a new type of topological insulator that is characterized by a different topological invariant, known as the Z2 index. The phase thus obtained in the presence of Rashba spin-orbit coupling should be beneficial to the realization of spintronic devices. In the appendix, we provide a periodic table that lists the topological invariants based on symmetries present in different spatial dimensions.