The deconfined phase of ($2+1$)-dimensional ${\mathbb{Z}}_k$ gauge theory exhibits topological order, with $e$ and $m$ anyons that have a $2\pi /k$ 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 $\mathrm{U}\left(1\right)\times \mathrm{U}\left(1\right)$ gauge theory with a mutual Chern-Simons term at level $k$ to describe the vicinity of the multicritical point for $k\ge 4$. The emergence of a global $\mathrm{U}{\left(1\right)}^{\mathrm{top}}\times \mathrm{U}{\left(1\right)}^{\mathrm{top}}$ 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 first-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 $1/k^2$ 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.

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• Anyons
• Conformal field theory
• Lattice gauge theory
• Quantum criticality
• Topological phase transition
• Quantum spin models
• Strongly correlated systems
• Critical phenomena