Distinct trivial phases protected by a point-group symmetry in quantum spin chains

Abstract

The ground state of the $S=1$ antiferromagnetic Heisenberg chain belongs to the Haldane phase -- a well known example of symmetry-protected topological phase. A staggered field applied to the $S=1$ antiferromagnetic chain breaks all the symmetries that protect the Haldane phase as a topological phase, reducing it to a trivial phase. That is, the Haldane phase is then connected adiabatically to an antiferromagnetic product state. Nevertheless, as long as the symmetry under site-centered inversion combined with a spin rotation is preserved, the phase is still distinct from another trivial phase. We demonstrate the existence of such distinct symmetry-protected trivial phases using a field-theoretical approach and numerical calculations. Furthermore, a general proof and a non-local order parameter are given in terms of an matrix-product state formulation.

5 pages, 1 figure, with 3 pages supplemental material

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• Boundary conditions and anomalies of conformal field theories in 1+1 dimensions
• The Ground State of the S=1 Antiferromagnetic Heisenberg Chain is Topologically Nontrivial if Gapped
• Anisotropy-induced spin parity effects
• Quantized edge magnetizations and their symmetry protection in one-dimensional quantum spin systems