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

TL;DR

This paper explores the critical theories governing phase transitions between symmetry protected topological states and their connection to gapless boundary theories, revealing that boundary excitations are embedded within the critical bulk theory.

Contribution

It introduces a unified framework linking critical bulk theories of SPT phase transitions with their boundary theories, extending to discrete symmetry groups.

Findings

Critical theories contain delocalized boundary excitations.

Boundary theory is a confined critical theory between SPTs.

Results are applicable to both continuous and discrete symmetry groups.

Abstract

Symmetry protected topological states (SPTs) have the same symmetry and the phase transition between them are beyond Landau's symmetry breaking formalism. In this paper we study (1) the critical theory of phase transition between trivial and non-trivial SPTs, and (2) the relation between such critical theory and 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 theory contains the delocalized version of the boundary excitations. In addition, we prove that the boundary theory is the critical theory spatially confined between two SPTs. We expect these conclusions to hold in general and, in particular, for discrete symmetry groups as well.