Self-Dual Higgs Transitions: Toric Code and Beyond

The toric code, when deformed in a way that preserves the self-duality Z_{2} symmetry exchanging the electric and magnetic excitations, admits a transition to a topologically trivial state that spontaneously breaks the 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.