Topology of Fermi Surfaces and anomaly inflows

TL;DR

This paper classifies topologically stable Fermi surfaces in non-interacting systems using $K$-theory, revealing their relation to anomaly inflow and extending results across various symmetry classes.

Contribution

It provides a rigorous $K$-theory classification of stable Fermi surfaces and connects these classifications to anomaly inflow interpretations across multiple symmetry classes.

Findings

Only topologically unstable Fermi surfaces exist on infinite crystals.

Stable Fermi surfaces are classified by $K^{-1}$ groups of the surface Brillouin zone.

The classification extends to symmetry classes AI, AII, C, and D.

Abstract

We derive a rigorous classification of topologically stable Fermi surfaces of non-interacting, discrete translation-invariant systems from electronic band theory, adiabatic evolution and their topological interpretations. For systems on an infinite crystal it is shown that there can only be topologically unstable Fermi surfaces. For systems on a half- space and with a gapped bulk, our derivation naturally yields a $\mathit{K}$-theory classification. Given the $d-1$-dimensional surface Brillouin zone $\mathrm{X}_{s}$ of a $d$-dimensional half-space, our result implies that different classes of globally stable Fermi surfaces belong in $\mathit{K^{-1}}\mathrm{(X_{s})}$ for systems with only discrete translation-invariance. This result has a chiral anomaly inflow interpretation, as it reduces to the spectral flow for $d = 2$. Through equivariant homotopy methods we extend these results for symmetry classes $AI,\,AII,\, C$ and $D$ and discuss their corresponding anomaly inflow interpretation.