Note on Green Function Formalism and Topological Invariants

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.

💡 Research Summary

In this paper the authors give a purely geometric proof that the “topological order parameter” defined from the full interacting Green’s function coincides with the generalized TK‑KNN invariant (the Chern number) for both non‑interacting and interacting systems. They start by considering a 2n‑dimensional crystal whose momentum space is a compact manifold M (typically a torus). The Matsubara Green’s function G(ω,k) is a map from M×ℝ to GL(N,ℂ). By decomposing G into a part proportional to the frequency and a Hermitian remainder, they write G(ω,k)=\bar{ω}A(ω,k)+B(ω,k) with A positive‑definite. This decomposition yields the asymptotic expansion G≈(1/ω)A₀(k)+O(1/ω²) for large |ω|, guaranteeing convergence of the integral that defines the topological order parameter

N₂ₙ = (1/2πi)·