Worldsheet patching, 1-form symmetries, and "Landau-star" phase transitions

Abstract: The analysis of phase transitions of gauge theories has relied heavily on simplifications that arise at the boundaries of phase diagrams, where certain excitations are forbidden. Taking 2+1 dimensional $\mathbb{Z}_2$ gauge theory as an example, the simplification can be visualized geometrically: on the phase diagram boundaries the partition function is an ensemble of closed membranes. More generally, however, the membranes have "holes" in them, representing worldlines of virtual anyon excitations. If the holes are of a finite size, then typically they do not affect the universality class, but they destroy microscopic (higher-form) symmetries and microscopic (string) observables. We demonstrate how these symmetries and observables can be restored using a "membrane patching" procedure, which maps the ensemble of membranes back to an ensemble of closed membranes. (This is closely related to the idea of gauge fixing in the "minimal gauge", though not equivalent.) Membrane patching makes the emergence of higher symmetry concrete. Performing patching in a Monte Carlo simulation with an appropriate algorithm, we show that it gives access to numerically useful observables. For example, the confinement transition can be analyzed using a correlation function that is a power law at the critical point. We analyze the quasi-locality of the patching procedure and discuss what happens at a self-dual multicritical point in the gauge-Higgs model, where the lengthscale $\ell$ characterizing the holes diverges.