Topological invariant for two-dimensional open systems

TL;DR

This paper introduces a topological invariant for two-dimensional open systems using Green's functions, extending topological classification to systems with environment coupling and exploring applications like time-reversal invariants and proximity effects.

Contribution

It proposes a new topological invariant based on Green's functions for open systems and demonstrates its equivalence to existing invariants in weak coupling regimes.

Findings

The invariant captures edge state differences via Green's function poles and zeros.

The invariant applies to general gapped systems and matches known invariants in weak coupling.

Application to time-reversal-invariant insulators and proximity effects illustrates its usefulness.

Abstract

We study the topology of two-dimensional open systems in terms of the Green's function. The Ishikawa-Matsuyama formula for the integer topological invariant is applied in open systems, which indicates the number difference of gapless edge bands arising from the poles and zeros of the Green's function. Meanwhile, we define another topological invariant via the single-particle density matrix, which works for general gapped systems and is equivalent to the former for the case of weak coupling to an environment. We also discuss two applications. For time-reversal-invariant insulators, the ${Z}_{2}$ index can be expressed by the invariant of each spin subsystem. As a second application, we consider the proximity effect when an ordinary insulator is coupled to a topological insulator.