Note on Green Function Formalism and Topological Invariants

TL;DR

This paper provides a geometric derivation of the topological order parameter from Green's function data, offering an alternative proof for identifying topological invariants in both non-interacting and interacting systems.

Contribution

It presents a geometric derivation of the topological invariant from Green's function data, clarifying the underlying mathematical structure.

Findings

Topological invariants can be derived geometrically from Green's functions.

The approach applies to both non-interacting and interacting systems.

Provides an alternative proof for the identification of topological order parameters.

Abstract

It has been discovered previously that the topological order parameter could be identified from the topological data of the Green's function, namely the (generalized) TKNN invariant in general dimensions, for both non-interacting and interacting systems. In this note, we show that this phenomenon has a clear geometric derivation. This proposal could be regarded as an alternative proof for the identification of the corresponding topological invariant and the topological order parameter.