Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
10 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Critical behavior of Fredenhagen-Marcu string order parameters at topological phase transitions with emergent higher-form symmetries (2402.00127v2)

Published 31 Jan 2024 in cond-mat.str-el

Abstract: A nonlocal string order parameter detecting topological order and deconfinement has been proposed by Fredenhagen and Marcu (FM). However, due to the lack of exact internal symmetries for lattice models and the nonlinear dependence of the FM string order parameter on ground states, it is a priori not guaranteed that it is a genuine order parameter for topological phase transitions. In this work, we find that the FM string order parameter exhibits universal scaling behavior near critical points of charge condensation transitions, by directly evaluating the FM string order parameter in the infinite string-length limit using infinite Projected Entangled Pair States (iPEPS) for the toric code in a magnetic field. Our results thus demonstrate that the FM string order parameter represents a quantitatively well-behaved order parameter. We find that only in the presence of an emergent 1-form symmetry the corresponding FM string order parameter can faithfully detect topological transitions.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (43)
  1. D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48, 1559 (1982).
  2. R. B. Laughlin, Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50, 1395 (1983).
  3. A. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2 (2003).
  4. A. Kitaev, Anyons in an exactly solved model and beyond, Annals of Physics 321, 2 (2006), january Special Issue.
  5. M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005).
  6. X.-G. Wen, A theory of 2+1D bosonic topological orders, National Science Review 3, 68 (2015), https://academic.oup.com/nsr/article-pdf/3/1/68/31565649/nwv077.pdf .
  7. X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89, 041004 (2017).
  8. K. Fredenhagen and M. Marcu, Charged states in z_2 gauge theories, Commun. Math. Phys 92 (1983).
  9. M. Marcu, (uses of) an order parameter for lattice gauge theories with matter fields, Lattice Gauge Theory: A Challenge in Large-Scale Computing , 267 (1986).
  10. R. Verresen, M. D. Lukin, and A. Vishwanath, Prediction of toric code topological order from rydberg blockade, Phys. Rev. X 11, 031005 (2021).
  11. P. Corboz, Variational optimization with infinite projected entangled-pair states, Phys. Rev. B 94, 035133 (2016).
  12. A. M. Somoza, P. Serna, and A. Nahum, Self-dual criticality in three-dimensional ℤ2subscriptℤ2\mathbb{Z}_{2}blackboard_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT gauge theory with matter, Phys. Rev. X 11, 041008 (2021).
  13. C. Bonati, A. Pelissetto, and E. Vicari, Multicritical point of the three-dimensional ℤ2subscriptℤ2\mathbb{Z}_{2}blackboard_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT gauge higgs model, Phys. Rev. B 105, 165138 (2022).
  14. F. Wu, Y. Deng, and N. Prokof’ev, Phase diagram of the toric code model in a parallel magnetic field, Phys. Rev. B 85, 195104 (2012).
  15. J. Vidal, S. Dusuel, and K. P. Schmidt, Low-energy effective theory of the toric code model in a parallel magnetic field, Phys. Rev. B 79, 033109 (2009).
  16. F. J. Wegner, Duality in generalized ising models and phase transitions without local order parameters, Journal of Mathematical Physics 12, 2259 (1971).
  17. H. W. J. Blöte and Y. Deng, Cluster monte carlo simulation of the transverse ising model, Phys. Rev. E 66, 066110 (2002).
  18. It is also called Ising* universality class because on a torus toric code model in a longitudinal field is mapped to the (2+1)21(2+1)( 2 + 1 ) transverse field Ising model in even sector, we ignore this difference and still say that the transition cross I⁢M𝐼𝑀IMitalic_I italic_M is (2+1)21(2+1)( 2 + 1 )D Ising universality class [65].
  19. M. B. Hastings and X.-G. Wen, Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance, Phys. Rev. B 72, 045141 (2005).
  20. S. D. Pace and X.-G. Wen, Exact emergent higher-form symmetries in bosonic lattice models, Phys. Rev. B 108, 195147 (2023).
  21. D. A. Huse and S. Leibler, Are sponge phases of membranes experimental gauge-higgs systems?, Phys. Rev. Lett. 66, 437 (1991).
  22. Z.-P. Cian, M. Hafezi, and M. Barkeshli, Extracting wilson loop operators and fractional statistics from a single bulk ground state (2022), arXiv:2209.14302 [cond-mat.str-el] .
  23. S. P. G. Crone and P. Corboz, Detecting a Z2subscript𝑍2{Z}_{2}italic_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT topologically ordered phase from unbiased infinite projected entangled-pair state simulations, Phys. Rev. B 101, 115143 (2020).
  24. M. Rader and A. M. Läuchli, Finite correlation length scaling in lorentz-invariant gapless ipeps wave functions, Phys. Rev. X 8, 031030 (2018).
  25. M. Iqbal, K. Duivenvoorden, and N. Schuch, Study of anyon condensation and topological phase transitions from a 𝕫4subscript𝕫4{\mathbb{z}}_{4}blackboard_z start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT topological phase using the projected entangled pair states approach, Phys. Rev. B 97, 195124 (2018).
  26. W.-T. Xu and N. Schuch, Characterization of topological phase transitions from a non-abelian topological state and its galois conjugate through condensation and confinement order parameters, Phys. Rev. B 104, 155119 (2021).
  27. W.-T. Xu, J. Garre-Rubio, and N. Schuch, Complete characterization of non-abelian topological phase transitions and detection of anyon splitting with projected entangled pair states, Phys. Rev. B 106, 205139 (2022).
  28. M. Iqbal and N. Schuch, Entanglement order parameters and critical behavior for topological phase transitions and beyond, Phys. Rev. X 11, 041014 (2021).
  29. N. Schuch, I. Cirac, and D. Pérez-García, Peps as ground states: Degeneracy and topology, Annals of Physics 325, 2153 (2010a).
  30. G.-Y. Zhu and G.-M. Zhang, Gapless coulomb state emerging from a self-dual topological tensor-network state, Phys. Rev. Lett. 122, 176401 (2019).
  31. F. Kos, D. Poland, and D. Simmons-Duffin, Bootstrapping the o (n) vector models, Journal of High Energy Physics 2014, 1 (2014).
  32. X.-G. Wen, Emergent anomalous higher symmetries from topological order and from dynamical electromagnetic field in condensed matter systems, Phys. Rev. B 99, 205139 (2019).
  33. J. McGreevy, Generalized symmetries in condensed matter, Annual Review of Condensed Matter Physics 14, 57 (2023), https://doi.org/10.1146/annurev-conmatphys-040721-021029 .
  34. The broken emergent 1-from symmetry defined on a non-contractible loop can be restored using minimally entangled states. Also, notice that the emergent Wilson loop symmetry defined on a contractible loop is unbroken for any ground state.
  35. W.-T. Xu, Q. Zhang, and G.-M. Zhang, Tensor network approach to phase transitions of a non-abelian topological phase, Phys. Rev. Lett. 124, 130603 (2020).
  36. R. Raussendorf, W. Yang, and A. Adhikary, Measurement-based quantum computation in finite one-dimensional systems: string order implies computational power, Quantum 7, 1215 (2023).
  37. R. Verresen, A. Vishwanath, and N. Schuch, in preparation; see also talk at the 2nd IQTN Plenary Meeting.
  38. W.-T. Xu, F. Pollmann, and M. Knap, Critical behavior of the Fredenhagen-Marcu order parameter for topological phase transitions, 10.5281/zenodo.10494400 (2024).
  39. A. Francuz, N. Schuch, and B. Vanhecke, Stable and efficient differentiation of tensor network algorithms (2023), arXiv:2311.11894 [quant-ph] .
  40. Z.-C. Gu, M. Levin, and X.-G. Wen, Tensor-entanglement renormalization group approach as a unified method for symmetry breaking and topological phase transitions, Phys. Rev. B 78, 205116 (2008).
  41. N. Schuch, I. Cirac, and D. Pérez-García, Peps as ground states: Degeneracy and topology, Annals of Physics 325, 2153 (2010b).
  42. W.-T. Xu, M. Knap, and F. Pollmann, Entanglement of gauge theories: from the toric code to the ℤ2subscriptℤ2\mathbb{Z}_{2}blackboard_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT lattice gauge higgs model (2023), arXiv:2311.16235 [cond-mat.str-el] .
  43. J. Huxford, D. X. Nguyen, and Y. B. Kim, Gaining insights on anyon condensation and 1-form symmetry breaking across a topological phase transition in a deformed toric code model (2023), arXiv:2305.07063 [cond-mat.str-el] .
Citations (8)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now