Lieb-Schultz-Mattis theorems for symmetry protected topological phases

TL;DR

This paper extends Lieb-Schultz-Mattis theorems to show that certain symmetric ground states must be symmetry-protected topological phases with gapless edges, using magnetic translation symmetry.

Contribution

It introduces a new class of LSM-type theorems that link symmetry constraints to SPT phases via magnetic translation symmetry.

Findings

Any symmetry-allowed SRE ground state is an SPT phase with gapless edges.

Theorems apply to interacting bosons and fermions.

Provides new theoretical tools for analyzing SPT phases.

Abstract

The Lieb-Schultz-Mattis (LSM) theorem and its descendants represent a class of powerful no-go theorems that rule out any short-range-entangled (SRE) symmetric ground state irrespective of the specific Hamiltonian, based only on certain microscopic inputs such as symmetries and particle filling numbers. In this work, we introduce and prove a new class of LSM-type theorems, where any symmetry-allowed SRE ground state must be a symmetry-protected topological (SPT) phase with robust gapless edge states. The key ingredient is to replace the lattice translation symmetry in usual LSM theorems by magnetic translation symmetry. These theorems provide new insights into numerical models and experimental realizations of SPT phases in interacting bosons and fermions.