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arXiv:0710.4198 · Physical Review B

Topological classification of gapped spin chains: Quantized Berry phase as a local order parameter

Takaaki Hirano, Hosho Katsura, Yasuhiro Hatsugai

Published Mar 26, 2008• 80 citations
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Abstract

We characterize several phases of gapped spin systems by local order parameters defined by quantized Berry phases [Y. Hatsugai, J. Phys. Soc. Jpn. 75, 123601 (2006)]. This characterization is topologically stable against any small perturbation as long as the energy gap remains finite. The models we pick up are $S=1,2$ dimerized Heisenberg chains and $S=2$ Heisenberg chains with uniaxial single-ion-type anisotropy. Analytically, we also evaluate the topological local order parameters for the generalized Affleck-Kennedy-Lieb-Tasaki model. The relation between the present Berry phases and the fractionalization in the integer spin chains are discussed as well.

DOI
10.1103/physrevb.77.094431

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Quantized Berry phases as local order parameters in t-J models are studied. A texture pattern of the local order parameters is topologically stable due to the quantization of non-Abelian Berry phases defined by low-energy states below a spin gap, which exists in the large J/t case with a few electrons. We have confirmed that itinerant singlets in the wide class of t-J models carry the nontrivial Berry phase pi. In the large J/t case for the one-dimensional t-J model, Berry phases are uniformly pi when the number of electrons is N =4n +2, (n=0,1,2,...n=0,1,2,...n=0,1,2,...).
We study S=1 quantum spin systems on the infinite chain with short ranged Hamiltonians that have a certain rotational and discrete symmetry. We define a Z_{2} index for any gapped unique ground state, and prove that it is invariant under a smooth deformation. By using the index, we provide the first rigorous proof of the existence of a "topological" phase transition, which cannot be characterized by any conventional order parameters, between the Affleck-Kennedy-Lieb-Tasaki (AKLT) model and trivial models. This rigorously establishes that the AKLT model is in a nontrivial symmetry protected topological phase.
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Spin-1/2 frustrated Heisenberg model show novel ground states due to the quantum fluctulation. To classify the ground states with toprogical invariant, we introduce fractionally quantized Berry phase. The Berry phase for the Heisenberg model is defined by local spin twists in the spin xy plane. As typical examples, we consider plateau states in frustrated magnets, e.g., a multimer ( N -mer) chains. Fractional quantization of the Berry phase comes from rotational symmetry of the twisted cluster. The Z N Berry phases are obtained from the numerical integration with using the ground states calculated by the Lanczos method. As a result, we obtained fractional quantization of Z 2 and Z 4 Berry phases in dimer ( N =2) and tetramer ( N =4) chains respectively.