Topological invariants of edge states for periodic two-dimensional models

TL;DR

This paper develops a method to compute topological invariants of edge states in 2D periodic models, linking edge properties to bulk topological invariants and edge currents, advancing understanding of topological insulators.

Contribution

It introduces a numerical approach using Bott-Maslov indices to define and calculate topological invariants for edge states in 2D models, including time-reversal symmetric cases.

Findings

Edge state invariants relate to bulk Chern numbers.

A Z_2-invariant is defined for time-reversal symmetric systems.

Edge invariants connect to (spin) edge currents.

Abstract

Transfer matrix methods and intersection theory are used to calculate the bands of edge states for a wide class of periodic two-dimensional tight-binding models including a sublattice and spin degree of freedom. This allows to define topological invariants by considering the associated Bott-Maslov indices which can be easily calculated numerically. For time-reversal symmetric systems in the symplectic universality class this leads to a Z_2-invariant for the edge states. It is shown that the edge state invariants are related to Chern numbers of the bulk systems and also to (spin) edge currents, in the spirit of the theory of topological insulators.