Transitions to valence-bond solid order in a honeycomb lattice antiferromagnet

Abstract

We use quantum Monte Carlo methods to study the ground-state phase diagram of a $S=1/2$ honeycomb lattice magnet in which a nearest-neighbor antiferromagnetic exchange $J$ (favoring N\'eel order) competes with two different multispin interaction terms: a six-spin interaction ${Q}_{3}$ that favors columnar valence-bond solid (VBS) order, and a four-spin interaction ${Q}_{2}$ that favors staggered VBS order. For ${Q}_{3}\ensuremath{\sim}{Q}_{2}\ensuremath{\gg}J$, we establish that the competition between the two different VBS orders stabilizes N\'eel order in a large swath of the phase diagram even when $J$ is the smallest energy scale in the Hamiltonian. When ${Q}_{3}\ensuremath{\gg}({Q}_{2},J)$ $\mathrm{[}{Q}_{2}\ensuremath{\gg}({Q}_{3},J)\text{]}$, this model exhibits at zero temperature phase transition from the N\'eel state to a columnar (staggered) VBS state. We establish that the N\'eel-columnar VBS transition is continuous for all values of ${Q}_{2}$, and that critical properties along the entire phase boundary are well characterized by critical exponents and amplitudes of the noncompact ${\mathrm{CP}}^{1}$ $({\mathrm{NCCP}}^{1})$ theory of deconfined criticality, similar to what is observed on a square lattice. However, a surprising threefold anisotropy of the phase of the VBS order parameter at criticality, whose presence was recently noted at the ${Q}_{2}=0$ deconfined critical point, is seen to persist all along this phase boundary. We use a classical analogy to explore this by studying the critical point of a three-dimensional $XY$ model with a fourfold anisotropy field which is known to be weakly irrelevant at the three-dimensional $XY$ critical point. In this case, we again find that the critical anisotropy appears to saturate to a nonzero value over the range of sizes accessible to our simulations.