Stability of Fermi Surfaces and K-Theory

TL;DR

This paper demonstrates that the stability classes of Fermi surfaces in nonrelativistic Fermi liquids are classified by K-theory, revealing a deep mathematical structure and a surprising connection to string theory D-branes.

Contribution

It establishes a K-theoretic classification of stable Fermi surfaces and links low-energy excitations to relativistic invariance, highlighting a novel mathematical framework for Fermi liquids.

Findings

K-theory classifies stability of Fermi surfaces

Bott periodicity determines stability patterns

Low-energy modes near Fermi surfaces exhibit relativistic invariance

Abstract

Nonrelativistic Fermi liquids in d+1 dimensions exhibit generalized Fermi surfaces: (d-p)-dimensional submanifolds in the momentum-frequency space supporting gapless excitations. We show that the universality classes of stable Fermi surfaces are classified by K-theory, with the pattern of stability determined by Bott periodicity. The Atiyah-Bott-Shapiro construction implies that the low-energy modes near a Fermi surface exhibit relativistic invariance in the transverse p+1 dimensions. This suggests an intriguing parallel between norelativistic Fermi liquids and D-branes of string theory.