Definition and Classification of Fermi Surface Anomalies

TL;DR

This paper introduces a universal classification scheme for Fermi surface anomalies using symmetry-protected topological phases in one dimension, linking gapless fermions to topological boundary modes of phase space Chern insulators.

Contribution

It establishes a novel classification framework for Fermi surface anomalies via cobordism, considering phase space as a non-commutative space and extending to codimension-p Fermi surfaces.

Findings

Classification reduces to a (0+1)-dimensional cobordism problem.

Provides concrete examples validating the classification scheme.

Connects Fermi surface anomalies to topological phases and symmetric mass generation.

Abstract

We propose that the Fermi surface anomaly of symmetry group $G$ in any dimension is universally classified by $G$-symmetric interacting fermionic symmetry-protected topological (SPT) phases in $(0+1)$-dimensional spacetime. The argument is based on the perspective that the gapless fermions on the Fermi surface can be viewed as the topological boundary modes of Chern insulators in the phase space (position-momentum space). Given the non-commutative nature of the phase space coordinates, we show that the momentum space dimensions should be counted as negative dimensions for SPT classification purposes. Therefore, the classification of phase-space Chern insulators (or, more generally fermionic SPT phases) always reduces to a $(0+1)$-dimensional problem, which can then be answered by the cobordism approach. In addition to the codimension-1 Fermi surface case, we also discuss the codimension-$p$ Fermi surface case briefly. We provide concrete examples to demonstrate the validity of our classification scheme, and make connections to the recent development of Fermi surface symmetric mass generation.