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 𝒞⊆𝒞_G^× , we show that physical considerations require that a connected étale algebra A ∈ 𝒞 admit a G -equivariant algebra structure for symmetry to be preserved under condensation of A . Given any categorical action G → EqBr ( 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 𝒞_A^loc , and gauging this symmetry commutes with anyon condensation.