Emergence and spontaneous breaking of approximate O(4) symmetry at a weakly first-order deconfined phase transition

We investigate approximate emergent non-Abelian symmetry in a class of weakly first-order 'deconfined' phase transitions using Monte Carlo simulations and a renormalization group analysis. We study a transition in a 3D classical loop model that is analogous to a deconfined 2 + 1D quantum phase transition in a magnet with reduced lattice symmetry. The transition is between the Neel phase and a twofold degenerate valence bond solid (lattice-symmetry-breaking) phase. The combined order parameter at the transition is effectively a four-component superspin. It has been argued that in some weakly first-order 'pseudocritical' deconfined phase transitions, the renormalization group flow can take the system very close to the ordered fixed point of the symmetric O(N) sigma model, where N is the total number of 'soft' order parameter components, despite the fact that O(N) is not a microscopic symmetry. This yields a first-order transition with unconventional phenomenology. We argue that this occurs in the present model, with N = 4. This means that there is a regime of length scales in which the transition resembles a 'spin-flop' transition in the ordered O(4) sigma model. We give numerical evidence for (i) the first-order nature of the transition, (ii) the emergence of O(4) symmetry to an accurate approximation, and (iii) the existence of a regime in which the emergent O(4) is 'spontaneously broken,' with distinctive features in the order parameter probability distribution. These results may be relevant for other models studied in the literature, including 2 + 1D QED with two flavors, the 'easy-plane' deconfined critical point, and the Neel-VBS transition on the rectangular lattice.