Analytic framework for self-dual criticality in Zk gauge theory with matter

The deconfined phase of (2+1)-dimensional Zk gauge theory exhibits topological order, with e and m anyons that have a 2π/k braiding phase. Proliferating either e or m drives Higgs or confinement transitions, respectively. At the multicritical point where these transitions meet, the theory enjoys an additional duality symmetry that exchanges e and m anyons. This symmetry forces anyons with nontrivial braiding to close their gaps simultaneously, giving rise to a critical theory that mixes strong interactions with mutual statistics. We propose an effective U(1)×U(1) gauge theory with a mutual Chern-Simons term at level k to describe the vicinity of the multicritical point for k = 4. The emergence of a global U(1)top×U(1)top symmetry at the critical point imposes powerful constraints on universal properties of the phase transition. In particular, we show that (1) the lattice magnetic flux operator embeds as a conserved U(1) current with protected scaling dimension; (2) the f irst-order line emanating from the critical point for k = 2 disappears generically for sufficiently large k; (3) the correlation length exponent approaches that of the three-dimensional XY model with corrections of order 1/k2 in the large-k limit. These predictions can be tested in near-term numerical simulations and pave the way for a more general exploration of topological quantum criticality enriched with anyon-permuting symmetries.