Spin TQFTs and fermionic phases of matter

TL;DR

This paper reveals that lattice models of fermionic phases of matter inherently depend on a discrete spin structure, which is essential for resolving ambiguities and is analogous to the spin-statistics relation.

Contribution

It uncovers the hidden dependence on spin structures in lattice constructions of fermionic phases, connecting fermionic topological orders to spin geometry.

Findings

Fermionic SPT order constructions depend on discrete spin structures.

Ambiguities in fermionic string-net models are resolved by spin structures.

Spin structure dependence is a general feature of lattice fermionic models.

Abstract

We study lattice constructions of gapped fermionic phases of matter. We show that the construction of fermionic Symmetry Protected Topological orders by Gu and Wen has a hidden dependence on a discrete spin structure on the Euclidean space-time. The spin structure is needed to resolve ambiguities which are otherwise present. An identical ambiguity is shown to arise in the fermionic analog of the string-net construction of 2D topological orders. We argue that the need for a spin structure is a general feature of lattice models with local fermionic degrees of freedom and is a lattice analog of the spin-statistics relation.