A Gapless Phase with Haagerup Symmetry

Abstract

We construct a (1+1)d gapless theory which has Haagerup $\mathcal{H}_3$ symmetry. The construction relies on the recent exploration of the categorical Landau paradigm applied to fusion category symmetries. First, using the Symmetry Topological Field Theory, we construct all gapped phases with Haagerup symmetry. Extending this construction to gapless phases, we study the second order phase transition between gapped phases, and determine analytically a Haagerup-symmetric conformal field theory. This is given in terms of two copies of the three-state Potts model, on which we realise the full $\mathcal{H}_3$ symmetry action and determine the relevant deformations to the $\mathcal{H}_3$-symmetric gapped phases. This continuum analysis is corroborated by a lattice model construction of the gapped and gapless phases, using the anyon chain.