Quantum phase transitions between a class of symmetry protected topological states

• We study the phase transitions between symmetry protected topological states.
• We provide a holographic interpretation.
• We engineer simple lattice models.

The subject of this paper is the phase transition between symmetry protected topological states (SPTs). We consider spatial dimension d and symmetry group G   so that the cohomology group, Hd+1(G,U(1))Hd+1(G,U(1)), contains at least one Z2nZ2n or Z   factor. We show that the phase transition between the trivial SPT and the root states that generate the Z2nZ2n or Z   groups can be induced on the boundary of a (d+1d+1)-dimensional G×Z2T-symmetric SPT by a Z2T symmetry breaking field. Moreover we show these boundary phase transitions can be “transplanted” to d dimensions and realized in lattice models as a function of a tuning parameter. The price one pays is for the critical value of the tuning parameter there is an extra non-local (duality-like) symmetry. In the case where the phase transition is continuous, our theory predicts the presence of unusual (sometimes fractionalized) excitations corresponding to delocalized boundary excitations of the non-trivial SPT on one side of the transition. This theory also predicts other phase transition scenarios including first order transition and transition via an intermediate symmetry breaking phase.