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The characterization of the quantum phase transition in a lowdimensional system has attracted a considerable amount of attention in quantum manybody systems. As one of the fundamental models in quantum magnetism, spin-1 models have richer phase diagrams and show more complex physical phenomena. In the spin-1 antiferromagnetic XXZ model, the Haldane phase and the Nel phase are the gapped topologic phases which cannot be characterized by the local order parameters. To characterize the nature in such phases, one has to calculate the non-local long range order parameters. Normally, the non-local order parameter in the topological phase is obtained from the extrapolation of finite-sized system in numerical study. However, it is difficult to extract the critical exponents with such an extrapolated non-local order parameter due to the numerical accuracy. In a recently developed tensor network representation, i.e., the infinite matrix product state (iMPS) algorithm, it was shown that the non-local order can be directly calculated from a very large lattice distance in an infinite-sized system rather than an extrapolated order parameter in a finite-sized system. Therefore, it is worthwhile using this convenient technique to study the non-local orders in the topological phases and characterize the quantum criticalities in the topological quantum phase transitions. In this paper, by utilizing the infinite matrix product state algorithm based on the tensor network representation and infinite time evolving block decimation method, the quantum entanglement, fidelity, and critical exponents of the topological phase transition are investigated in the one-dimensional infinite spin-1 bond-alternating XXZ Heisenberg model. It is found that there is always a local dimerization order existing in the whole parameter range when the bond-alternative strength parameter changes from 0 to 1. Also, due to the effect of the bond-alternating, there appears a quantum phase transition from the long-rang ordering topological Nel phase to the local ordering dimerization phase. The von Neumann entropy, fidelity per lattice site, and order parameters all give the same phase transition point at c = 0.638. To identify the type of quantum phase transition, the central charge c 0.5 is extracted from the ground state von Neumann entropy and the finite correlation length, which indicates that the phase transition belongs to the two-dimensional Ising universality class. Furthermore, it is found that the Nel order and the susceptibility of Nel order have power-law relations to |-c|. From the numerical fitting of the Nel order and its susceptibility, we obtain the characteristic critical exponents ' = 0.236 and ' = 0.838. It indicates that such critical exponents from our method characterize the nature of the quantum phase transition. Our critical exponents from the iMPS method can provide guidance for studying the properties of the phase transition in quantum spin systems.
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[1] Sachdev S 1999 Quantum Phase Transitions (Cambridge: Cambridge University Press)
[2] Kitazawa A, Nomura K, Okamoto K 1996 Phys. Rev. Lett. 76 4038
[3] Rizzi M, Rossini D, Chiara G D, Montangero S, Fazio R 2005 Phys. Rev. Lett. 95 240404
[4] Peters D, McCulloch I P, Selke W 2009 Phys. Rev. B 79 132406
[5] Katsura H, Tasaki H 2013 Phys. Rev. Lett. 110 130405
[6] Kennedy T, Tasaki H 1992 Phys. Rev. B 45 304
[7] Hatsugai Y 2007 J. Phys.: Condens. Matter 19 145209
[8] Pollmann F, Berg E, Turner A, Oshikawa M 2012 Phys. Rev. B 85 075125
[9] Ueda H, Nakano H, Kusakabe K 2008 Phys. Rev. B 78 224402
[10] Su Y H, Cho S Y, Li B, Wang H, Zhou H 2012 J. Phys. Soc. Jpn. 81 074003
[11] Vidal G 2007 Phys. Rev. Lett. 98 070201
[12] Su Y H, Hu B, Li S, Cho S Y 2013 Phys. Rev. E 88 032110
[13] Wang H, Li B, Cho S Y 2013 Phys. Rev. B 87 054402
[14] Wang H, Cho S Y 2015 J. Phys.: Condens. Matter 27 015603
[15] Kato Y, Tanaka A 1994 J. Phys. Soc. Jpn. 63 1277
[16] Osterloh A, Amico L, Falci G, Fazio R 2002 Nature 416 608
[17] Korepin V E 2004 Phys. Rev. Lett. 92 096402
[18] Amico L, Fazio R, Osterloh A, Vedral V 2008 Rev. Mod. Phys. 80 517
[19] Chung M, Landau D P 2011 Phys. Rev. B 83 113104
[20] Ma F, Liu S, Kong X 2011 Phys. Rev. A 83 062309
[21] Xu Y, Wang L, Kong X 2013 Phys. Rev. A 87 012312
[22] Zanardi P, Paunkovi N 2006 Phys. Rev. E 74 031123
[23] Rams M M, Damski B 2011 Phys. Rev. Lett. 106 055701
[24] Zhou H, Barjaktarevi J P 2008 J. Phys. A: Math. Theor. 41 412001
[25] Yu Y, Mller G, Viswanath V S 1996 Phys. Rev. B 54 9242
[26] Tagliacozzo L, de Oliveira T R, Iblisdir S, Latorre J I 2008 Phys. Rev. B 78 024410
[27] Pollmann F, Mukerjee S, Turner A, Moore J E 2009 Phys. Rev. Lett. 102 255701
[28] Su Y H, Chen A M, Xiang C, Wang H, Xia C, Wang J 2016 J. Stat. Mech. 2016 123102
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