Bosonic SET to SPT Transition under Dyonic LSM Theorem

Employing large-scale quantum Monte Carlo simulatoins, we study the phase diagram of a quantum spin model which is subject to the recently developed dyonic Lieb-Shultz-Mattis (LSM) theorem. The theorem predicts there are symmetry enriched/protected topological (SET/SPT) phases in the phase diagram. Our numerical results reveal a first order quantum phase transition between SET and SPT phases, consistent with an anyon condensation mechanism that enforces SPT phase according to the theorem. Also there exists in the phase diagram a symmetry-breaking phase in the form of superfluid (SF). The transition between SET and SF is continuous and that between SPT and SF is first order. Interestingly, the SET, SPT and SF phases meet at a critical endpoint, whose presence can be universally explained via theory contains emergent gauge field coupled to vortex fields, and consequently reveals the exotic feature of our model even beyond the realm of dyonic LSM.