Quantum Phase Transitions,Entanglement Spectrum,and Schmidt Gap in Bond-Alternative S=1 Antiferromagnetic Chain

Bipartite entanglement, entanglement spectrum, and Schmidt gap in S=1 bond-alternative antiferromagnetic Heisenberg chain are investigated by the infinite time-evolving block decimation (iTEBD) method. The quantum phase transition (QPT) from the singlet-dimer phase to the Haldane phase can be detected by the singular behavior of bipartite entanglement, the sudden change of the entanglement spectrum, and the completely vanishing of the Schmidt gap. The critical point is determined to be around r_c≈ 0.587, and the second-order character of the QPT is verified. Doubly degenerate entanglement spectra of both even and odd bonds are observed in the Haldane phase, by which one can distinguish the Haldane phase from the singlet-dimer phase easily. Nearest-neighbor antiferromagnetic correlations and next-nearest-neighbor ferromagnetic correlations are found in the whole parameter region. At the critical massless point, although exponentially decaying antiferromagnetic correlation is observed, it approaches to a constant value finally. Therefore, long-range correlations exist and the correlation length becomes divergent at the critical point.