Holographic theory of symmetry protection of gaplessness and continuous phase transitions in 1+1D

It has been recently understood that phases of a bosonic lattice model with a finite symmetry are characterized by a topological order (TO) in one higher dimension. Lagrangian condensable algebras (LCAs) of a 2+1D TO correspond to its gapped boundaries, while non-Lagrangian ones (NLCAs) describe gapless ones. The above holographic point-of-view suggests that gapped and gapless states in 1+1D correspond to LCAs and NLCAs respectively. Specifically, a continuous phase transition between two gapped phases corresponds to an NLCA. This NLCA then constrains the possible CFTs that may describe the transition, if continuous. Moreover, automorphism symmetries of the 2+1D TO lead to dualities between different parts of the global phase diagram of the boundary theory. For the simplest non-Abelian symmetry group S₃, we find a duality that identifies the transition between the S₃ symmetric phase and the trivial phase with that between the Z₃ and Z₂ symmetric phases. So if one of them is continuous, the other must be too. We identify potential gapless CFTs that may describe this transition. Complementary to this, we numerically study the phase diagram of an S₃ symmetric tensor network model in 1+1D, realizing all the phases mentioned above. To investigate the implications of our holographic approach to a theory with a fusion category symmetry, we study the so-called “golden chain” using tools developed above.