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 aG-crossed braided extension $$ \mathcal{C}\subseteq {\mathcal{C}}_G^{\times } $$C⊆CG× , we show that physical considerations require that a connected étale algebraA∈ $$ \mathcal{C} $$C admit aG-equivariant algebra structure for symmetry to be preserved under condensation ofA. Given any categorical actionG→EqBr( $$ \mathcal{C} $$C ) such thatg(A) ≅Afor allg∈G, we show there is a short exact sequence whose splittings correspond toG-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation ofA, 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 $$ {\mathcal{C}}_A^{\mathrm{loc}} $$CAloc , and gauging this symmetry commutes with anyon condensation.