Analytic framework for self-dual criticality in $\mathbb{Z}_k$ gauge theory with matter

The deconfined phase of 2+1D $\mathbb{Z}_k$ 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 ${\rm U(1)}\times {\rm U(1)}$ gauge theory with a mutual Chern-Simons term at level $k$ to describe the vicinity of the multicritical point for $k \geq 4$. The emergence of a global ${\rm U(1)}^{\rm{top}} \times {\rm U(1)}^{\rm{top}}$ 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 ${\rm U(1)}$ 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 3D 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.