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String order and degenerate entanglement spectrum of S = 1 / 2 and S = 1 Heisenberg bond-alternating chains

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Abstract

Generalized string orders and entanglement spectrum of S = 1/2 and S = 1 Heisenberg bond-alternating chains have been investigated by the infinite time-evolving block decimation (iTEBD) method. Generalized string order parameters with appropriate θ are capable of distinguishing all the topological phases. Central charges c ≃ 1 and critical exponents β ≃ 1/12 indicate all the topological QPTs belong to the Gaussian universality class. Interestingly, odd- and even-fold degeneracies of the entanglement spectrum are observed. Even-fold (doubly) degenerate entanglement spectra and the typical two-fold degenerate lowest-lying level are found to exist in both the spin-1/2 dimer and the S = 1 Haldane phases. However, odd-fold degenerate entanglement spectra with three-fold degenerate lowest-lying level are observed in both the S = 1 dimer and the S = 2 Haldane phase. The degeneracy of the lowest-lying entanglement spectrum level, which can be understood by entanglement spectra in the dimer limit (J 1 = 0), is adopted to estimate the lowest boundary of the bipartite entanglement. The entanglement spectrum and the generalized string orders are valuable for uncovering the underlying features of these symmetry-protect topological (SPT) states. Similar entanglement spectrum shows that the S = 1 (S = 2) Haldane phase is essentially the same as the S = 1/2 (S = 1) dimer phase.

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Authors and Affiliations

  1. Graduate School, Tianjin Polytechnic University, Tianjin, 300387, P.R. China

    Xiao-Yan Deng

  2. Department of Physics, Tianjin Polytechnic University, Tianjin, 300387, P.R. China

    Long-Juan Kong

  3. School of Physics, Peking University, Beijing, 100871, P.R. China

    Ling Qiang

Authors
  1. Xiao-Yan Deng
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  2. Long-Juan Kong
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  3. Ling Qiang
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Correspondence to Xiao-Yan Deng.

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Deng, XY., Kong, LJ. & Qiang, L. String order and degenerate entanglement spectrum of S = 1 / 2 and S = 1 Heisenberg bond-alternating chains. Eur. Phys. J. B 87, 247 (2014). https://doi.org/10.1140/epjb/e2014-50391-x

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