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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• Qudit quantum computation on matrix product states with global symmetry
• Fragility of Symmetry Protected Topological Order on a Hubbard Ladder
• A discriminating string order parameter for topological phases of gapped SU(N) spin chains
• Symmetry protected topological phases beyond groups: The q-deformed Affleck-Kennedy-Lieb-Tasaki model
• Novel families of $\mathrm{SU}(N)$ AKLT states with arbitrary self-conjugate edge states
• Exact diagonalization of $\mathrm{SU}(N)$ Heisenberg and AKLT chains using the full $\mathrm{SU}(N)$ symmetry
• Symmetry-protected topological phases in the SU(N) Heisenberg spin chain: a Majorana-fermion approach
• Interaction-induced exotic vortex states in an optical lattice clock with spin-orbit coupling
• Disentangling topological degeneracy in the entanglement spectrum of one-dimensional symmetry protected topological phases
• Identifying quantum phases from injectivity of symmetric matrix product states
• Symmetry protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain
• Charging a Dimerized Quantum XY Chain
• Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry