The simplest topologically ordered phase in 2+1D is the deconfined phase of Z2 lattice gauge theory. There are two reasonably well-understood ways to exit the deconfined phase: the Higgs transition, where electric charge (the "e" anyon) condenses, and the confinement transition, where magnetic charge (the "m" anyon) condenses. However, we can also exit the deconfined phase via the self-dual line in the phase diagram, where there is a symmetry between "e" and "m". What happens here is more mysterious. If this transition is continuous, it may be the simplest critical point with no useful continuum Lagrangian (as yet). After reviewing the formulation of the model as the statistical mechanics of membranes, I will describe clear Monte Carlo evidence for the continuity of the self-dual transition. I will sketch why it is not a conventional "Landau" critical point. Separately, I will use the membrane formulation to describe a very concrete and intuitive way of understanding the emergent higher-form symmetries which appear in part of the phase diagram (and which are the reason that the Higgs and confinement transitions can be understood using Landau theory, despite lacking local order parameters). Work with Andres Somoza and Pablo Serna (https://arxiv.org/abs/2012.15845 and https://arxiv.org/abs/2403.04025).
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Slava Rychkov
