2D symmetry protected topological orders and their protected gapless edge excitations

TL;DR

This paper constructs exactly solvable 2D models demonstrating nontrivial symmetry protected topological order with gapless boundary excitations, extending classification to interacting boson and fermion systems with discrete symmetries.

Contribution

It introduces a 2D interacting spin model with $Z_2$ symmetry protected topological order, generalizing gapless boundary behavior and linking to 3-cocycle group cohomology.

Findings

Boundary must be gapless if symmetry is preserved

Model has a unique gapped bulk state

Boundary excitations are gapless due to nontrivial 3-cocycle

Abstract

Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry protected topological orders exist. In this paper, we present a model in a 2D interacting spin system with nontrivial on-site $Z_2$ symmetry protected topological order. The order is nontrivial because we can prove that the 1D system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the $Z_2$ group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if $Z_2$ symmetry is not broken. We prove the latter by developing the tool of matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and revealing the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with on-site $Z_2$ symmetry protected topological order.