Topological phases of spin chains

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 onsite 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₍, 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).