Spontaneous symmetry breaking from anyon condensation

In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by a G-crossed braided extension $$ \mathcal{C}\subseteq {\mathcal{C}}_G^{\times } $$, we show that physical considerations require that a connected etale algebra A ∈ $$ \mathcal{C} $$ admit a G-equivariant algebra structure for symmetry to be preserved under condensation of A. Given any categorical action G → EqBr($$ \mathcal{C} $$) such that g(A) ≅ A for all g ∈ G, we show there is a short exact sequence whose splittings correspond to G-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation of A, while inequivalent splittings of the sequence correspond to different SETOs resulting from the anyon-condensation transition. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of $$ {\mathcal{C}}_A^{\mathrm{loc}} $$, and gauging this symmetry commutes with anyon condensation.