Abstract
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 étale 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.
Highlights
According to Landau theory, in a physical system whose Hamiltonian preserves a symmetry group G, the symmetry of its ground state can spontaneously break down to any subgroup H ⊂ G
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. This no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs)
Summary
For a (semisimple) tensor category C, Aut⊗(C) is the categorical group with End(∗) the set of tensor autoequivalences of C and 2-morphisms monoidal natural isomorphisms of tensor functors. For a (semisimple) braided tensor category C, Autb⊗r(C) is the categorical group with End(∗) the set of braided tensor autoequivalences of C and 2-morphisms monoidal natural isomorphisms of tensor functors. A G-crossed braided extension of C based on a categorical action (ρ, μ) : G → Autb⊗r(C) is a fusion category F equipped with the following structure:. For non-degenerately braided fusion categories, condensing O(G) and gauging a second level categorical G-symmetry (taking the equivariantization of a G-crossed braided extension) are mutually inverse; we refer the reader to [32, section 4] and [30] for more details. The recent article [35] provides an interesting step in this direction
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