Revealing a topological connection between stabilities of Fermi surfaces and topological insulators/superconductors

TL;DR

This paper establishes a fundamental topological link between Fermi surface stabilities and topological insulators/superconductors, providing a complete classification table and a new index theorem relating boundary states to bulk topology.

Contribution

It introduces a rigorous one-to-one correspondence between Fermi surface types and topological insulators/superconductors, and proves a general index theorem connecting boundary Fermi surface charges to bulk topological invariants.

Findings

Derived a complete topological classification table for TIs/TSCs.

Proved a general index theorem relating boundary Fermi surfaces to bulk topology.

Established a topological connection between Fermi surface stability and topological phases.

Abstract

A topology-intrinsic connection between the stabilities of Fermi surfaces (FSs) and topological insulators/superconductors (TIs/TSCs) is revealed. In particular, a one-to-one relation between the topological types of FSs and TIs/TSCs is rigorously derived; combining it with a well-established topological theory of FSs, we produce a complete table illustrating precisely topological types of all TIs/TSCs, while a valid part of it was postulated before. Moreover, we propose and prove a general index theorem that relates the topological charge of FSs on the natural boundary of a strong TI/TSC to its bulk topological number. Implications of the general index theorem on the boundary quasi-particles are also addressed.