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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 permitsa duality that exchanges electric and magnetic excitations (``$e$'' and ``$m$''particles). Condensing either particle while the other remains gapped yields aphase transition with 3D Ising exponents. More mysterious, however, is thetransition 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 bethe simplest critical point for which we still lack any useful continuumLagrangian 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 resultsindicate a continuous transition: for example, cumulants show a strikingparameter-free scaling collapse. We estimate scaling dimensions by usingduality symmetry to distinguish the leading duality-odd/duality-even scalingoperators $A$ and $S$. All local operators have large scaling dimensions,making standard techniques for locating the critical point ineffective. Wedevelop an alternative using ``renormalization group trajectories'' ofcumulants. We check that two- and three-point functions, and temporalcorrelators in the Monte-Carlo dynamics, are scale-invariant, with scalingdimensions $x_A$ and $x_S$ and dynamical exponent $z$.  We also give a picture for emergence of 1-form symmetries, in some parts ofthe phase diagram, in terms of ``patching'' of membranes/worldsurfaces. Werelate this to the percolation of anyon worldlines in spacetime. Analyzingpercolation yields a fourth exponent for the self-dual transition. We proposevariations of the model for further investigation.

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