- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Open Access

Quantum phase transitions between symmetry-enriched fracton phases

Julian Boesl1,2, Yu-Jie Liu1,2,3, Wen-Tao Xu1,2, Frank Pollmann1,2, and Michael Knap1,2

PDF

Phys. Rev. B 112, 125152 – Published 23 September, 2025

DOI: https://doi.org/10.1103/fsr8-xd4n

Abstract

Topologically ordered phases exhibit further complexity in the presence of global symmetries: Their anyonic excitations may exhibit different transformation patterns under these symmetries, leading to a classification in terms of symmetry-enriched topological orders. We develop a generic scheme to study an analogous situation for three-dimensional fracton phases by means of isometric tensor network states (isoTNS) with finite bond dimension, which allow us to tune phase transitions between different symmetry fractionalization patterns. We focus on the X-Cube model, a paradigmatic fracton model hosting two types of excitations: lineons, which are mobile in a single direction only, and fractons that are immobile on their own. By deforming the local tensors of the fixed point ground state, we find a family of exact wavefunctions for which the symmetry fractionalization under an anti-unitary symmetry on both types of excitations is directly visible. These wavefunctions are nonstabilizer states and have nonvanishing correlation lengths. They even exhibit power-law correlations at criticality between two symmetry-enriched topological orders. Furthermore, the isoTNS description allows for the explicit construction of a linear-depth quantum circuit to sequentially realize these exotic 3D states on a quantum processor, including a holographic scheme using only a pair of two-dimensional qubit arrays alongside measurements. Our approach provides a construction to enrich phases with exotic topological or fracton order and to study 3D quantum phase transitions with exact wavefunctions, and offers a tractable route to implement and characterize fracton order on quantum devices.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (86)

  1. R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129, 393 (1990).
  2. X. G. Wen, Topological orders in rigid states, Int. J. Mod. Phys. B 04, 239 (1990).
  3. X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89, 041004 (2017).
  4. J. M. Leinaas and J. Myrheim, On the theory of identical particles, Nuovo Cim B 37, 1 (1977).
  5. F. Wilczek, Quantum mechanics of fractional-spin particles, Phys. Rev. Lett. 49, 957 (1982).
  6. A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  7. A. M. Essin and M. Hermele, Classifying fractionalization: Symmetry classification of gapped Z2 spin liquids in two dimensions, Phys. Rev. B 87, 104406 (2013).
  8. X. Chen, Symmetry fractionalization in two dimensional topological phases, Rev. Phys. 2, 3 (2017).
  9. F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Phys. Rev. B 81, 064439 (2010).
  10. D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48, 1559 (1982).
  11. M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, Symmetry fractionalization, defects, and gauging of topological phases, Phys. Rev. B 100, 115147 (2019).
  12. A. Mesaros and Y. Ran, Classification of symmetry enriched topological phases with exactly solvable models, Phys. Rev. B 87, 155115 (2013).
  13. B. Swingle, Interplay between short- and long-range entanglement in symmetry-protected phases, Phys. Rev. B 90, 035451 (2014).
  14. M. Levin and Z.-C. Gu, Braiding statistics approach to symmetry-protected topological phases, Phys. Rev. B 86, 115109 (2012).
  15. L. Haller, W.-T. Xu, Y.-J. Liu, and F. Pollmann, Quantum phase transition between symmetry enriched topological phases in tensor-network states, Phys. Rev. Res. 5, 043078 (2023).
  16. W. Ye and L. Zou, Classification of symmetry-enriched topological quantum spin liquids, Phys. Rev. X 14, 021053 (2024).
  17. C. Chamon, Quantum glassiness in strongly correlated clean systems: An example of topological overprotection, Phys. Rev. Lett. 94, 040402 (2005).
  18. J. Haah, Local stabilizer codes in three dimensions without string logical operators, Phys. Rev. A 83, 042330 (2011).
  19. S. Vijay, J. Haah, and L. Fu, A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations, Phys. Rev. B 92, 235136 (2015).
  20. S. Vijay, J. Haah, and L. Fu, Fracton topological order, generalized lattice gauge theory, and duality, Phys. Rev. B 94, 235157 (2016).
  21. H. Ma, A. T. Schmitz, S. A. Parameswaran, M. Hermele, and R. M. Nandkishore, Topological entanglement entropy of fracton stabilizer codes, Phys. Rev. B 97, 125101 (2018).
  22. S. Pai and M. Hermele, Fracton fusion and statistics, Phys. Rev. B 100, 195136 (2019).
  23. M. Pretko, X. Chen, and Y. You, Fracton phases of matter, Int. J. Mod. Phys. A 35, 2030003 (2020).
  24. G.-Y. Zhu, J.-Y. Chen, P. Ye, and S. Trebst, Topological fracton quantum phase transitions by tuning exact tensor network states, Phys. Rev. Lett. 130, 216704 (2023).
  25. H. Song, N. Tantivasadakarn, W. Shirley, and M. Hermele, Fracton self-statistics, Phys. Rev. Lett. 132, 016604 (2024).
  26. S. Bravyi and J. Haah, Quantum self-correction in the 3D cubic code model, Phys. Rev. Lett. 111, 200501 (2013).
  27. A. Prem, J. Haah, and R. Nandkishore, Glassy quantum dynamics in translation invariant fracton models, Phys. Rev. B 95, 155133 (2017).
  28. D. Aasen, D. Bulmash, A. Prem, K. Slagle, and D. J. Williamson, Topological defect networks for fractons of all types, Phys. Rev. Res. 2, 043165 (2020).
  29. Z. Song, A. Dua, W. Shirley, and D. J. Williamson, Topological defect network representations of fracton stabilizer codes, PRX Quantum 4, 010304 (2023).
  30. H. Ma, E. Lake, X. Chen, and M. Hermele, Fracton topological order via coupled layers, Phys. Rev. B 95, 245126 (2017).
  31. A. Prem, S.-J. Huang, H. Song, and M. Hermele, Cage-net fracton models, Phys. Rev. X 9, 021010 (2019).
  32. W. Shirley, K. Slagle, Z. Wang, and X. Chen, Fracton models on general three-dimensional manifolds, Phys. Rev. X 8, 031051 (2018).
  33. W. Shirley, K. Slagle, and X. Chen, Foliated fracton order in the checkerboard model, Phys. Rev. B 99, 115123 (2019).
  34. K. Slagle, D. Aasen, and D. Williamson, Foliated field theory and string-membrane-net condensation picture of fracton order, SciPost Phys. 6, 043 (2019).
  35. W. Shirley, K. Slagle, and X. Chen, Twisted foliated fracton phases, Phys. Rev. B 102, 115103 (2020).
  36. Y. You, T. Devakul, F. J. Burnell, and S. L. Sondhi, Symmetric fracton matter: Twisted and enriched, Ann. Phys. 416, 168140 (2020).
  37. N. Tantivasadakarn and S. Vijay, Searching for fracton orders via symmetry defect condensation, Phys. Rev. B 101, 165143 (2020).
  38. B. C. Rayhaun and D. J. Williamson, Higher-form subsystem symmetry breaking: Subdimensional criticality and fracton phase transitions, SciPost Phys. 15, 017 (2023).
  39. D. J. Williamson, N. Bultinck, M. Mariën, M. B. Şahinoğlu, J. Haegeman, and F. Verstraete, Matrix product operators for symmetry-protected topological phases: Gauging and edge theories, Phys. Rev. B 94, 205150 (2016).
  40. S. K. Shukla, M. B. Şahinoğlu, F. Pollmann, and X. Chen, Boson condensation and instability in the tensor network representation of string-net states, Phys. Rev. B 98, 125112 (2018).
  41. Y.-J. Liu, K. Shtengel, and F. Pollmann, Simulating two-dimensional topological quantum phase transitions on a digital quantum computer, Phys. Rev. Res. 6, 043256 (2024).
  42. X.-H. Yu, J. I. Cirac, P. Kos, and G. Styliaris, Dual-isometric projected entangled pair states, Phys. Rev. Lett. 133, 190401 (2024).
  43. F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Criticality, the area law, and the computational power of projected entangled pair states, Phys. Rev. Lett. 96, 220601 (2006).
  44. J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  45. M. P. Zaletel and F. Pollmann, Isometric tensor network states in two dimensions, Phys. Rev. Lett. 124, 037201 (2020).
  46. R. Haghshenas, M. J. O'Rourke, and G. K.-L. Chan, Conversion of projected entangled pair states into a canonical form, Phys. Rev. B 100, 054404 (2019).
  47. W. Kadow, F. Pollmann, and M. Knap, Isometric tensor network representations of two-dimensional thermal states, Phys. Rev. B 107, 205106 (2023).
  48. D. Malz and R. Trivedi, Computational complexity of isometric tensor-network states, PRX Quantum 6, 020310 (2025).
  49. T. Soejima, K. Siva, N. Bultinck, S. Chatterjee, F. Pollmann, and M. P. Zaletel, Isometric tensor network representation of string-net liquids, Phys. Rev. B 101, 085117 (2020).
  50. C. Schön, E. Solano, F. Verstraete, J. I. Cirac, and M. M. Wolf, Sequential generation of entangled multiqubit states, Phys. Rev. Lett. 95, 110503 (2005).
  51. M. M. Wolf, G. Ortiz, F. Verstraete, and J. I. Cirac, Quantum phase transitions in matrix product systems, Phys. Rev. Lett. 97, 110403 (2006).
  52. N. G. Jones, J. Bibo, B. Jobst, F. Pollmann, A. Smith, and R. Verresen, Skeleton of matrix-product-state-solvable models connecting topological phases of matter, Phys. Rev. Res. 3, 033265 (2021).
  53. A. Smith, B. Jobst, A. G. Green, and F. Pollmann, Crossing a topological phase transition with a quantum computer, Phys. Rev. Res. 4, L022020 (2022).
  54. Z.-Y. Wei, D. Malz, and J. I. Cirac, Sequential generation of projected entangled-pair states, Phys. Rev. Lett. 128, 010607 (2022).
  55. X. Chen, A. Dua, M. Hermele, D. T. Stephen, N. Tantivasadakarn, R. Vanhove, and J.-Y. Zhao, Sequential quantum circuits as maps between gapped phases, Phys. Rev. B 109, 075116 (2024).
  56. K. J. Satzinger, Y.-J. Liu, A. Smith, C. Knapp, M. Newman, C. Jones, Z. Chen, C. Quintana, X. Mi, A. Dunsworth et al., Realizing topologically ordered states on a quantum processor, Science 374, 1237 (2021).
  57. Y.-J. Liu, K. Shtengel, A. Smith, and F. Pollmann, Methods for simulating string-net states and anyons on a digital quantum computer, PRX Quantum 3, 040315 (2022).
  58. T. A. Cochran, B. Jobst, E. Rosenberg, Y. D. Lensky, G. Gyawali, N. Eassa, M. Will, A. Szasz, D. Abanin, R. Acharya et al., Visualizing dynamics of charges and strings in (2+1)D lattice gauge theories, Nature (London) 642, 315 (2025).
  59. G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar et al., Probing topological spin liquids on a programmable quantum simulator, Science 374, 1242 (2021).
  60. R. Verresen, M. D. Lukin, and A. Vishwanath, Prediction of toric code topological order from Rydberg blockade, Phys. Rev. X 11, 031005 (2021).
  61. M. Iqbal, N. Tantivasadakarn, R. Verresen, S. L. Campbell, J. M. Dreiling, C. Figgatt, J. P. Gaebler, J. Johansen, M. Mills, S. A. Moses et al., Non-Abelian topological order and anyons on a trapped-ion processor, Nature (London) 626, 505 (2024).
  62. N. Tantivasadakarn, R. Verresen, and A. Vishwanath, Shortest route to non-Abelian topological order on a quantum processor, Phys. Rev. Lett. 131, 060405 (2023).
  63. A. H. Nevidomskyy, H. Bernien, and A. Canright, Realizing fracton order from long-range quantum entanglement in programmable Rydberg atom arrays, arXiv:2407.05885.
  64. P. Chen, B. Yan, and S. X. Cui, Quantum circuits for toric code and X-cube fracton model, Quantum 8, 1276 (2024).
  65. K. Slagle and Y. B. Kim, X-cube model on generic lattices: Fracton phases and geometric order, Phys. Rev. B 97, 165106 (2018).
  66. M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005).
  67. C.-H. Lin, M. Levin, and F. J. Burnell, Generalized string-net models: A thorough exposition, Phys. Rev. B 103, 195155 (2021).
  68. H. He, Y. Zheng, B. A. Bernevig, and N. Regnault, Entanglement entropy from tensor network states for stabilizer codes, Phys. Rev. B 97, 125102 (2018).
  69. W. Shirley, K. Slagle, and X. Chen, Fractional excitations in foliated fracton phases, Ann. Phys. 410, 167922 (2019).
  70. F. Pollmann and A. M. Turner, Detection of symmetry-protected topological phases in one dimension, Phys. Rev. B 86, 125441 (2012).
  71. M. P. Zaletel, Detecting two-dimensional symmetry-protected topological order in a ground-state wave function, Phys. Rev. B 90, 235113 (2014).
  72. C.-Y. Huang, X. Chen, and F. Pollmann, Detection of symmetry-enriched topological phases, Phys. Rev. B 90, 045142 (2014).
  73. Y. Zhang, T. Grover, A. Turner, M. Oshikawa, and A. Vishwanath, Quasiparticle statistics and braiding from ground-state entanglement, Phys. Rev. B 85, 235151 (2012).
  74. K. Slagle and Y. B. Kim, Quantum field theory of x-cube fracton topological order and robust degeneracy from geometry, Phys. Rev. B 96, 195139 (2017).
  75. R. J. Baxter, Exactly solved models in statistical mechanics, Integrable Systems in Statistical Mechanics, Series on Advances in Statistical Mechanics Vol. 1 (World Scientific, Singapore, 1985), pp. 5–63
  76. A. J. Friedman, A. Chan, A. De Luca, and J. Chalker, Spectral statistics and many-body quantum chaos with conserved charge, Phys. Rev. Lett. 123, 210603 (2019).
  77. P. Kos, B. Bertini, and T. Prosen, Chaos and ergodicity in extended quantum systems with noisy driving, Phys. Rev. Lett. 126, 190601 (2021).
  78. N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Computational complexity of projected entangled pair states, Phys. Rev. Lett. 98, 140506 (2007).
  79. R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett. 86, 5188 (2001).
  80. R. Raussendorf, D. E. Browne, and H. J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A 68, 022312 (2003).
  81. M. Foss-Feig, D. Hayes, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, J. M. Pino, and A. C. Potter, Holographic quantum algorithms for simulating correlated spin systems, Phys. Rev. Res. 3, 033002 (2021).
  82. E. Chertkov, J. Bohnet, D. Francois, J. Gaebler, D. Gresh, A. Hankin, K. Lee, D. Hayes, B. Neyenhuis, R. Stutz, A. C. Potter, and M. Foss-Feig, Holographic dynamics simulations with a trapped-ion quantum computer, Nat. Phys. 18, 1074 (2022).
  83. X. Chen and M. Hermele, Symmetry fractionalization and anomaly detection in three-dimensional topological phases, Phys. Rev. B 94, 195120 (2016).
  84. C. Lee, Y. Hu, G. Y. Cho, and H. Watanabe, Zn generalizations of three-dimensional stabilizer codes, arXiv:2504.09847.
  85. D. Haag, F. Baccari, and G. Styliaris, Typical correlation length of sequentially generated tensor network states, PRX Quantum 4, 030330 (2023).
  86. Simulation codes are available at 10.5281/zenodo.14719699.

Outline

Information

Phys. Rev. B 112, 125152– Published 23 September, 2025
  • Received 30 January 2025
  • Revised 4 August 2025
  • Accepted 4 September 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation