  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.
