On topological phases of spin chains

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).

18 pages, 7 figures, 2 tables. Version v2 corresponds to the published version. It includes minor revisions, additional references and an application to cold atom systems

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• From symmetry-protected topological order to Landau order
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• Fragility of Symmetry Protected Topological Order on a Hubbard Ladder
• Chiral Haldane phases of SU(N) quantum spin chains in the adjoint representation
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• Unified Phase Diagram of Antiferromagnetic SU(N) Spin Ladders
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• The Haldane gap in the SU(3) [3 0 0] Heisenberg chain
• Symmetry-protected topological phases, conformal criticalities, and duality in exactly solvable SO($n$) spin chains
• Lowest-energy states in parity-transformation eigenspaces of SO(N) spin chain
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• Interrelations among frustration-free models via Witten's conjugation
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• Bosonic realization of SU(3) chiral Haldane phases