The toric code, when deformed in a way that preserves the self-duality ${\mathbb{Z}}_2$ symmetry exchanging the electric and magnetic excitations, admits a transition to a topologically trivial state that spontaneously breaks the ${\mathbb{Z}}_2$ symmetry. Numerically, this transition was found to be continuous, which makes it particularly enigmatic given the longstanding absence of a continuum field-theoretic description. In this Letter we propose such a continuum field theory for the transition dubbed the $SO(4{)}_{2,-2}$ Chern-Simons-Higgs (CSH) theory. We show that our field theory provides a natural “mean-field” understanding of the phase diagram. Moreover, it can be generalized to an entire series of theories, namely the $SO(4{)}_{k,-k}$ CSH theories, labeled by an integer $k$. For each $k>2$, the theory describes an analogous transition involving different non-Abelian topological orders, such as the double Fibonacci order ($k=3$) and the $S_3$ quantum double ($k=4$). For $k=1$, we conjecture that the corresponding CSH transition is in fact infrared dual to the $3d$ Ising transition, in close analogy with the particle-vortex duality of a complex scalar.

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• Topological order
• Topological phase transition
• Strongly correlated systems