Dyonic Lieb-Shultz-Mattis Theorem and Symmetry Protected Topological Phases in Decorated Dimer Models

TL;DR

This paper proves a modified Lieb-Shultz-Mattis theorem for 2+1D lattice models with magnetic flux and fractional spin, revealing conditions for symmetry protected topological phases or topological order in gapped ground states.

Contribution

It introduces a generalized LSM theorem applicable to models with magnetic flux and fractional spin, and constructs models of SPT phases via decorated dimer models.

Findings

Gapped ground states can be SPT phases with protected edge states.

Gapped ground states can exhibit topological order with exotic excitations.

Explicit formula for allowed SPT phases based on fractional spin and magnetic flux.

Abstract

We consider 2+1D lattice models of interacting bosons or spins, with both magnetic flux and fractional spin in the unit cell. We propose and prove a modified Lieb-Shultz Mattis (LSM) theorem in this setting, which applies even when the spin in the enlarged magnetic unit cell is integral. There are two nontrivial outcomes for gapped ground states that preserve all symmetries. In the first case, one necessarily obtains a symmetry protected topological (SPT) phase with protected edge states. This allows us to readily construct models of SPT states by decorating dimer models of Mott insulators to yield SPT phases, which should be useful in their physical realization. In the second case, exotic bulk excitations, i.e. topological order, is necessarily present. While both scenarios require fractional spin in the lattice unit cell, the second requires that the symmetries protecting the fractional spin is related to that involved in the magnetic translations. Our discussion encompasses the general notion of fractional spin (projective symmetry representations) and magnetic flux (magnetic translations tied to a symmetry generator). The resulting SPTs display a dyonic character in that they associate charge with symmetry flux, allowing the flux in the unit cell to screen the projective representation on the sites. We provide an explicit formula that encapsulates this physics, which identifies a specific set of allowed SPT phases.