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

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 theta are capable of distinguishing all the topological phases. Central charges c similar or equal to 1 and critical exponents beta similar or equal to 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.