Title

運用多尺度糾纏重整化方法研究保有對稱性的拓樸態

Translated Titles

Detection of Symmetry Protected Topological Phases with MERA

DOI

10.6342/NTU.2013.01318

Authors

張學文

Key Words

多尺度糾纏重整化方法 ; 保有對稱性的拓樸態 ; 鏡射對稱 ; MERA ; symmetry protected topological phase ; inversion symmetry

PublicationName

臺灣大學物理研究所學位論文

Volume or Term/Year and Month of Publication

2013年

Academic Degree Category

碩士

Advisor

高英哲

Content Language

繁體中文

English Abstract

Traditionally, we understand phases of matter by spontaneous symmetry breaking and therefore we have order parameters to characterize different phases. However, there are states of matter which fall beyond this type of characterization, we call these phases topological phases. The first experimental discovery of topological phase was at 1980s, however, we still do not have a clear understanding of topological phase nor do we know exactly what the topological orders are for characterizing these phases (in contrast with normal orders). Recently, using the ideas based on local unitary transformations, it is shown that an topological order is associated with the pattern of long-range entanglement in gapped quantum systems with finite topological entanglement entropy. On the other hand, several nonlocal string-order parameters are proposed for characterizing symmetry-protected topological states for different kinds of symmetries in 1D base on infinite-size matrix product state (iMPS) calculation . iMPS is a powerful tool to represent ground state of 1D gapped system. However, accuracy decrease near critical point. Multiscale entanglement renormalization ansatz (MERA) on the other hand, is more accurate near critical point. MERA is a numerical method that use the concept of entanglement renormalization, with the help of inserting disentanglers into the system so that short-range entanglement can be removed before renormalization. This prevents the accumulation of degrees of freedom during the renormalization so that this method can be addressed to large size of 1D or 2D system even at quantum critical points. Therefore, if one can use MERA to calculate the nonlocal string-order parameters proposed, not only will it be a confirmation, but also better results could be get. However, calculation of these parameters with MERA turned out to be highly nontrivial because the operator is nonlocal and it breaks the causal cone property of MERA, which makes the calculation exponentially hard. Nevertheless, we found that we can reduce the calculation by using symmetric MERA. We need Z_2 and inversion symmetry. The former is already proposed but there's no method to do inversion symmetry in any kind of tensor network until now. By inspecting the geometry of MERA, we successfully find the method to incorporate inversion symmetry in MERA with a brick-and-rope representation. These enable people to calculate the inversion string-order parameter with MERA. We calculate the proposed parameters and confirmed that the Haldane phase is protected by both inversion and time reversal symmetry. This thesis is structured as follows: In Chap. 1, we give a brief introduction to topological phase, including historical review and its relation with quantum entanglement and symmetry. Followed by the review of S = 1 Haldane chain. In Chap. 2, we review different kinds of tensor network methods and focus onMERA, with a detailed explanation and interpretation of the tensors in it. In Chap. 3, we discuss how to implement a Z2 symmetric MERA and introduce our algorithm on spatial inversion symmetric MERA based on a brick-and-rope representation. In Chap. 4, we present the results of stringorder parameters using both time-reversal and inversion symmetric MERA. We also observe the RG flow in the 2 phase goes to 2 different fixed point. We conclude our work in Chap. 5.

Topic Category 基礎與應用科學 > 物理
理學院 > 物理研究所
Reference
  1. [1] X. Chen, Z.-C. Gu, and X.-G.Wen, “Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order,” Phys.
    連結:
  2. [2] F. Pollmann and A.M. Turner, “Detection of symmetry-protected topological phases in one dimension,” Phys. Rev. B, vol. 86, p. 125441, Sep 2012.
    連結:
  3. [3] P. Corboz and G. Vidal, “Fermionic multiscale entanglement renormalization ansatz,” Phys. Rev. B, vol. 80, p. 165129, Oct 2009.
    連結:
  4. [4] G. Vidal, “Entanglement renormalization,” Phys. Rev. Lett., vol. 99, p. 220405, Nov 2007.
    連結:
  5. [5] X. G. Wen, “Topological orders in rigid states,” Int. J. Mod. Phys. B, vol. 4, p. 239, 1990.
    連結:
  6. [6] V. Kalmeyer and R. B. Laughlin, “Equivalence of the resonating-valence-bond and fractional quantum hall states,” Phys. Rev. Lett., vol. 59, pp. 2095–2098, Nov 1987.
    連結:
  7. [7] X. G.Wen, F.Wilczek, and A. Zee, “Chiral spin states and superconductivity,” Phys. Rev. B, vol. 39, pp. 11413–11423, Jun 1989.
    連結:
  8. [8] X. G. Wen, “Vacuum degeneracy of chiral spin states in compactified space,” Phys. Rev. B, vol. 40, pp. 7387–7390, Oct 1989.
    連結:
  9. [9] E. Witten, “Quantum field theory and the jones polynomial,” Comm. Math. Phys., vol. 121, p. 351, 1989.
    連結:
  10. [10] D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett., vol. 48, pp. 1559–1562, May 1982.
    連結:
  11. [11] R. B. Laughlin, “Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations,” Phys. Rev. Lett., vol. 50, pp. 1395–1398,May
    連結:
  12. [12] X. Chen, Z.-C. Gu, and X.-G. Wen, “Classification of gapped symmetric phases in one-dimensional spin systems,” Phys. Rev. B, vol. 83, p. 035107, Jan 2011.
    連結:
  13. [13] Z.-C. Gu and X.-G. Wen, “Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order,” Phys. Rev. B, vol. 80, p. 155131, Oct 2009.
    連結:
  14. [14] F. D.M. Haldane, “Nonlinear field theory of large-spin heisenberg antiferromagnets: Semiclassically quantized solitons of the one-dimensional easy-axis n’eel state,”
    連結:
  15. Phys. Rev. Lett., vol. 50, pp. 1153–1156, Apr 1983.
    連結:
  16. [15] F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, “Entanglement spectrum of a topological phase in one dimension,” Phys. Rev. B, vol. 81, p. 064439, Feb 2010.
    連結:
  17. [16] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Rigorous results on valence-bond ground states in antiferromagnets,” Phys. Rev. Lett., vol. 59, pp. 799–802, Aug 1987.
    連結:
  18. [17] M. den Nijs and K. Rommelse, “Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains,” Phys. Rev. B, vol. 40, pp. 4709–4734, Sep 1989.
    連結:
  19. [18] T. Kennedy and H. Tasaki, “Hidden z2×z2 symmetry breaking in haldane-gap antiferromagnets,” Phys. Rev. B, vol. 45, pp. 304–307, Jan 1992.
    連結:
  20. [19] M. den Nijs and K. Rommelse, “Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains,” Phys. Rev. B, vol. 40, pp. 4709–4734, Sep 1989.
    連結:
  21. [20] P. Corboz, R. Or’us, B. Bauer, and G. Vidal, “Simulation of strongly correlated fermions in two spatial dimensions with fermionic projected entangled-pair states,”
    連結:
  22. Phys. Rev. B, vol. 81, p. 165104, Apr 2010.
    連結:
  23. [21] S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett., vol. 69, pp. 2863–2866, Nov 1992.
    連結:
  24. [22] M. Troyer and U.-J.Wiese, “Computational complexity and fundamental limitations to fermionic quantum monte carlo simulations,” Phys. Rev. Lett., vol. 94, p. 170201, May 2005.
    連結:
  25. [23] S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett., vol. 69, pp. 2863–2866, Nov 1992.
    連結:
  26. [24] D. Gottesman and M. B. Hastings, “Entanglement versus gap for one-dimensional spin systems,” New Journal of Physics, vol. 12, no. 2, p. 025002, 2010.
    連結:
  27. [25] P. Zanardi and L. C. Venuti, “Entanglement susceptibility: Area laws and beyond.” arXiv:1205.2507, 2012.
    連結:
  28. [26] G. Evenbly and G. Vidal, “Algorithms for entanglement renormalization,” Phys. Rev. B, vol. 79, p. 144108, Apr 2009.
    連結:
  29. [27] S. Singh, R. N. C. Pfeifer, and G. Vidal, “Tensor network decompositions in the presence of a global symmetry,” Phys. Rev. A, vol. 82, p. 050301, Nov 2010.
    連結:
  30. [28] R. N. C. Pfeifer, G. Evenbly, and G. Vidal, “Entanglement renormalization, scale invariance, and quantum criticality,” Phys. Rev. A, vol. 79, p. 040301, Apr 2009.
    連結:
  31. [29] S. Singh and G. Vidal, “Symmetry protected entanglement renormalization.” Arxiv:1303.6716, 2013.
    連結:
  32. [30] C.-Y. Huang, X. Chen, and F.-L. Lin, “Symmetry protected quantum state renormalization.” arXiv:1303.4190, 2013.
    連結:
  33. Rev. B, vol. 82, p. 155138, Oct 2010.
  34. 1983.