Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions

TL;DR

This paper investigates the physical properties and invariants of symmetry-protected topological (SPT) phases in interacting bosons and fermions, providing methods to detect and classify these states via fractionalization phenomena and group cohomology.

Contribution

It introduces physical SPT invariants related to fractionalization and defect states, linking them to group cohomology classifications for interacting systems.

Findings

Fractionalization of quantum numbers on defects characterizes SPT states.

SPT invariants can be measured to identify different SPT phases.

In 2+1D bosonic Z_n SPT states, monodromy defects carry quantized Z_n charges.

Abstract

Recently, it was realized that quantum states of matter can be classified as long-range entangled (LRE) states (i.e. with non-trivial topological order) and short-range entangled (SRE) states (\ie with trivial topological order). We can use group cohomology class ${\cal H}^d(SG,R/Z)$ to systematically describe the SRE states with a symmetry $SG$ [referred as symmetry-protected trivial (SPT) or symmetry-protected topological (SPT) states] in $d$-dimensional space-time. In this paper, we study the physical properties of those SPT states, such as the fractionalization of the quantum numbers of the global symmetry on some designed point defects, and the appearance of fractionalized SPT states on some designed defect lines/membranes. Those physical properties are SPT invariants of the SPT states which allow us to experimentally or numerically detect those SPT states, i.e. to measure the elements in ${\cal H}^d(G, R/Z)$ that label different SPT states. For example, 2+1D bosonic SPT states with $Z_n$ symmetry are classified by a $Z_n$ integer $m \in {\cal H}^3(Z_n, R/Z)=Z_n$. We find that $n$ identical monodromy defects, in a $Z_n$ SPT state labeled by $m$, carry a total $Z_n$-charge $2m$ (which is not a multiple of $n$ in general).