Krein signatures of transfer operators for half-space topological insulators

TL;DR

This paper introduces a novel approach to topological invariants in half-space 2D tight-binding models using Krein signatures of transfer operators, providing a new perspective on surface modes and topological phases.

Contribution

It develops a framework based on Krein space theory to define topological invariants for half-space models, complementing existing methods.

Findings

Krein inertia characterizes eigenvalues on the unit circle in transfer operators.

Topological invariants derived from Krein signatures distinguish different phases.

Numerical results support the theoretical framework.

Abstract

We propose a complementary point of view on the topological invariants of two-dimensional tight-binding models restricted to half-spaces. The transfer operators for such systems are $J$-unitary on a infinite dimensional Krein space $(\mathcal{K},J)$ and, for energies in the bulk gap, only have discrete spectrum on the unit circle. These eigenvalues have Krein inertia which can be used to define topological invariants determining the nature of the surface modes and allowing to distinguish different topological phases. This is illustrated by numerical results.