On topological phases of spin chains

TL;DR

This paper classifies symmetry-protected topological phases in one-dimensional spin chains with continuous symmetry groups, revealing a correspondence with the fundamental group of the symmetry group and extending Haldane's conjecture.

Contribution

It provides a systematic evaluation of cohomology groups for general spin chains and identifies the number of topological phases for various symmetry groups.

Findings

Topological phases correspond to elements of the fundamental group of G.

For PSU(N) symmetry, there are N distinct topological phases.

Up to four phases are found for orthogonal, symplectic, or exceptional groups.

Abstract

Symmetry protected topological phases of one-dimensional spin systems have been classified using group cohomology. In this paper, we revisit this problem for general spin chains which are invariant under a continuous on-site symmetry group G. We evaluate the relevant cohomology groups and find that the topological phases are in one-to-one correspondence with the elements of the fundamental group of G if G is compact, simple and connected and if no additional symmetries are imposed. For spin chains with symmetry PSU(N)=SU(N)/Z_N our analysis implies the existence of N distinct topological phases. For symmetry groups of orthogonal, symplectic or exceptional type we find up to four different phases. Our work suggests a natural generalization of Haldane's conjecture beyond SU(2).