Fermionic Lieb-Schultz-Mattis Theorems and Weak Symmetry-Protected Phases

TL;DR

This paper extends Lieb-Schultz-Mattis theorems to two-dimensional fermionic systems, revealing new constraints on topological phases and the role of symmetries, with implications for non-Abelian anyons and surface states.

Contribution

It introduces fermionic LSM theorems in 2D, classifies weak SPT phases of fermions in 3D, and explores their impact on topological order and symmetry-enforced phenomena.

Findings

Fermionic LSM theorems for 2D systems established.

Constraints on gapped symmetry-enriched topological phases derived.

Lattice translations permute anyons, leading to non-Abelian dislocations.

Abstract

The Lieb-Schultz-Mattis (LSM) theorem and its higher-dimensional generalizations by Oshikawa and Hastings establish that a translation-invariant lattice model of spin-$1/2$'s can not have a non-degenerate ground state preserving both spin and translation symmetries. Recently it was shown that LSM theorems can be interpreted in terms of bulk-boundary correspondence of certain weak symmetry-protected topological (SPT) phases. In this work we discuss LSM-type theorems for two-dimensional fermionic systems, which have no bosonic analogs. They follow from a general classification of weak SPT phases of fermions in three dimensions. We further derive constraints on possible gapped symmetry-enriched topological phases in such systems. In particular, we show that lattice translations must permute anyons, thus leading to "symmetry-enforced" non-Abelian dislocations, or "genons". We also discuss surface states of other weak SPT phases of fermions.