Self-dual criticality in three-dimensional $\mathbb{Z}_2$ gauge theory with matter

The simplest topologically ordered phase in 2+1D is the deconfined phase of $Z_2$ gauge theory, realized for example in the toric code. This phase permits a duality that exchanges electric and magnetic excitations (``$e$'' and ``$m$'' particles). Condensing either particle while the other remains gapped yields a phase transition with 3D Ising exponents. More mysterious, however, is the transition out of the deconfined phase when self-duality symmetry is preserved. If this transition is continuous, which has so far been unclear, then it may be the simplest critical point for which we still lack any useful continuum Lagrangian description. This transition also has a soft matter interpretation, as a multicritical point for classical membranes in 3D. We study the self-dual transition with Monte Carlo simulations of the $Z_2$ gauge-Higgs model on cubic lattices of linear size $L\leq 96$. Our results indicate a continuous transition: for example, cumulants show a striking parameter-free scaling collapse. We estimate scaling dimensions by using duality symmetry to distinguish the leading duality-odd/duality-even scaling operators $A$ and $S$. All local operators have large scaling dimensions, making standard techniques for locating the critical point ineffective. We develop an alternative using ``renormalization group trajectories'' of cumulants. We check that two- and three-point functions, and temporal correlators in the Monte-Carlo dynamics, are scale-invariant, with scaling dimensions $x_A$ and $x_S$ and dynamical exponent $z$. We also give a picture for emergence of 1-form symmetries, in some parts of the phase diagram, in terms of ``patching'' of membranes/worldsurfaces. We relate this to the percolation of anyon worldlines in spacetime. Analyzing percolation yields a fourth exponent for the self-dual transition. We propose variations of the model for further investigation.