Symmetry Protected Topological States of Interacting Fermions and Bosons

TL;DR

This paper develops a unified framework for classifying interacting fermionic and bosonic symmetry protected topological states, revealing how interactions can alter classifications and extending the theory to include spatial symmetries.

Contribution

It introduces a general definition of SPT states, demonstrates how interactions modify free fermion classifications, and connects fermionic and bosonic SPT states, including those with spatial inversion symmetry.

Findings

Interactions reduce free fermion SPT classifications in all dimensions.

The formalism correctly classifies known SPT states with or without interactions.

Extension of the classification to SPT states with spatial inversion symmetry.

Abstract

We study the classification of interacting fermionic and bosonic symmetry protected topological (SPT) states. We define a SPT state as whether or not it is separated from the trivial state through a bulk phase transition, which is a general definition applicable to SPT states with or without spatial symmetries. We show that in all dimensions short range interactions can reduce the classification of free fermion SPT states, and we demonstrate these results by making connection between fermionic and bosonic SPT states. We first demonstrate that our formalism gives the correct classification for all the known SPT states, with or without interaction, then we will generalize our method to SPT states that involve the spatial inversion symmetry.