Self-Dual Criticality in Three-Dimensional Gauge Theory with Matter

Abstract

The simplest topologically ordered phase in is the deconfined phase of gauge theory (realized in the toric code, for example). This phase permits a duality that exchanges electric and magnetic excitations (“” and “” particles). The phase transition where one of these particles condenses, while the other remains gapped, has 3D Ising exponents. But the transition out of the deconfined phase when self-duality symmetry is preserved is more mysterious. It has so far been unclear whether this transition is continuous, but if continuous, it may be the simplest critical point for which a useful continuum Lagrangian is still lacking. These questions are relevant to soft matter, too, since the gauge theory also describes classical membranes in 3D. Here, we study the self-dual transition with Monte Carlo simulations of the gauge-Higgs model on cubic lattices of linear size . Our results indicate a continuous transition, for example via a striking parameter-free scaling collapse. We use duality symmetry to distinguish the leading duality-odd and duality-even scaling operators and . We explain why standard techniques for locating the critical point are ineffective, and we develop an alternative using “renormalization group trajectories” of cumulants. We check that two- and three-point functions are scale invariant, with scaling dimensions and (autocorrelations in the Monte Carlo dynamics also yield a dynamical exponent ). Separately, we propose a general picture for emergent 1-form symmetries, in terms of “patching” of membranes or world surfaces. We relate this to the percolation of anyon worldlines in spacetime. The latter yields a fourth exponent for the self-dual transition. We propose variations of the model for further investigation.