Note on Green Function Formalism and Topological Invariants

TL;DR

This note provides a geometric proof that the Green's-function-derived topological order parameter ${\mathcal N}_{2n}$ equals the generalized TKNN invariant $C_n$ in even spatial dimensions, for both interacting and non-interacting systems. The authors formulate ${\mathcal N}_{2n}$ as a contour-integrated pullback of the Maurer–Cartan form and show convergence, then replace the imaginary-frequency integral with a compact contour on $M\times T$. A smooth homotopy to a non-interacting theory and a bundle-theoretic decomposition into $E_+$ and $E_-$ yield a twisted bundle $\widetilde{E}$ whose Chern character computations reproduce $C_n$, establishing ${\mathcal N}_{2n}=C_n$. A no-go theorem rules out certain naive generalizations, while a refined extension $N'_{j_1,\dots,j_l}$ connects higher Chern characters to the topology of $E_-$, suggesting practical invariants accessible via Green's function data. The approach offers a clean, purely geometric understanding of topological invariants in condensed matter and points to extensions toward interacting regimes and fractional quantum Hall phenomena.

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.

Figures (3)

• Figure 1: The integration contour for $\omega$ in the complex plane $\mathbb{C}$.
• Figure 2: Deformation for $E'_-$ sector.
• Figure 3: A commutative diagram.