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

Lieb-Schultz-Mattis(LSM) theorem and its various generalizations provide a powerful guidance toward the search of novel phases of matter utilizing only information such as filling number and internal/spatial symmetries. Here we propose and prove a modified LSM theorem suitable for 2+1D lattice models of interacting bosons or spins, with both magnetic flux and fractional spin (projective symmetry representations) in the unit cell. 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 to yield SPT phases, which should be useful in their physical realization. In the second case, topological-ordered states are necessarily present. The resulting SPTs for the first case 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.