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Symmetry protected topological states (SPTs) have the same symmetry and thephase transition between them are beyond Landau's symmetry breaking formalism.In this paper we study (1) the critical theory of phase transition betweentrivial and non-trivial SPTs, and (2) the relation between such critical theoryand the gapless boundary theory of SPTs. Based on examples of SO(3) and SU(2)SPTs, we propose that under appropriate boundary condition the critical theorycontains the delocalized version of the boundary excitations. In addition, weprove that the boundary theory is the critical theory spatially confinedbetween two SPTs. We expect these conclusions to hold in general and, inparticular, for discrete symmetry groups as well.

Critical theories of phase transition between symmetry protected topological states and their relation to the gapless boundary theories