An index for invertible phases of two-dimensional quantum spin systems

TL;DR

This paper introduces an index for classifying invertible phases in 2D quantum spin systems, linking it to boundary conformal field theory invariants and demonstrating its effectiveness in identifying non-trivial phases in free fermionic systems.

Contribution

It defines a new index for 2D invertible quantum phases that correlates with boundary conformal invariants and proves its utility in classifying free fermionic phases.

Findings

The index relates to the boundary chiral central charge mod 24.

Free fermionic systems with Chern number mod 48 are non-trivial.

The index provides a microscopic invariant for invertible phases.

Abstract

We define an index for invertible phases of two-dimensional fermionic and bosonic quantum spin systems without any additional symmetry. Conjecturally, it provides a microscopic definition of an invariant related to the chiral central charge of the boundary modes $c_- \bmod 24$ when the effective conformal field theory description is valid. Using this index, we prove that free fermionic systems with Chern number $\nu \bmod 48 \neq 0$ are in a non-trivial invertible phase.