- 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
  • Open Access

Quantum doubles in symmetric blockade structures

Hans Peter Büchler*,†, Tobias F. Maier†, Simon Fell†, and Nicolai Lang†

  • *Contact author: hans-peter.buechler@itp3.uni-stuttgart.de
  • †These authors contributed equally to this work.
PDF

Phys. Rev. B 114, 065113 – Published 9 July, 2026

DOI: https://doi.org/10.1103/99sy-kggw

Abstract

Exactly solvable models of topologically ordered phases with non-Abelian anyons typically require complicated many-body interactions that do not arise naturally. This motivates the “inverse problem” of quantum many-body physics: How to design a microscopic Hamiltonian that realizes a desired topological phase with a given, restricted type of two-body interactions? Here, we address this problem for systems where elementary two-level degrees of freedom couple via simple blockade interactions, a platform motivated by Rydberg atoms. Within this framework, we construct Hamiltonians that realize topological orders described by non-Abelian quantum double models. We analytically prove the existence of topological order in the ground states and present efficient schemes to prepare these states. We also introduce protocols for the controlled adiabatic braiding of anyonic excitations that probe their non-Abelian statistics. Our construction is generic and applies to quantum doubles D(G) for arbitrary finite groups G. We illustrate braiding for the simplest non-Abelian quantum double D(S3).

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (85)

  1. A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006).
  2. M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96, 110405 (2006).
  3. X.-G. Wen, Topological Order: From long-range entangled quantum matter to a unified origin of light and electrons, Int. Scholarly Res. Not. 2013, 198710 (2013).
  4. X.-G. Wen, Colloquium : Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89, 041004 (2017).
  5. A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  6. H. Bombin and M. A. Martin-Delgado, Topological quantum distillation, Phys. Rev. Lett. 97, 180501 (2006).
  7. B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015).
  8. M. H. Freedman, A. Kitaev, M. J. Larsen, and Z. Wang, Topological quantum computation, Bull. Am. Math. Soc. 40, 31 (2003).
  9. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
  10. Z. Wang, Topological Quantum Computation, Regional Conference Series in Mathematics/Conference Board of the Mathematical Sciences (American Mathematical Society, Providence, Rhode Island, 2010).
  11. H. Bartolomei, M. Kumar, R. Bisognin, A. Marguerite, J.-M. Berroir, E. Bocquillon, B. Plaçais, A. Cavanna, Q. Dong, U. Gennser, Y. Jin, and G. Fève, Fractional statistics in anyon collisions, Science 368, 173 (2020).
  12. J. Nakamura, S. Liang, G. C. Gardner, and M. J. Manfra, Direct observation of anyonic braiding statistics, Nat. Phys. 16, 931 (2020).
  13. K. Pakrouski, M. R. Peterson, T. Jolicoeur, V. W. Scarola, C. Nayak, and M. Troyer, Phase diagram of the ν=5/2 fractional quantum Hall effect: Effects of Landau-level mixing and nonzero width, Phys. Rev. X 5, 021004 (2015).
  14. A. Stern, Non-Abelian states of matter, Nature (London) 464, 187 (2010).
  15. S. C. Zhang, T. H. Hansson, and S. Kivelson, Effective-field-theory model for the fractional quantum Hall effect, Phys. Rev. Lett. 62, 82 (1989).
  16. A. Lopez and E. Fradkin, Fractional quantum Hall effect and Chern-Simons gauge theories, Phys. Rev. B 44, 5246 (1991).
  17. B. I. Halperin, P. A. Lee, and N. Read, Theory of the half-filled Landau level, Phys. Rev. B 47, 7312 (1993).
  18. M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005).
  19. L. Fidkowski, M. Freedman, C. Nayak, K. Walker, and Z. Wang, From string nets to nonabelions, Commun. Math. Phys. 287, 805 (2009).
  20. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2, 2 (2006).
  21. N. Lang and H. P. Büchler, Topological states in a microscopic model of interacting fermions, Phys. Rev. B 92, 041118(R) (2015).
  22. R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures, Phys. Rev. Lett. 105, 077001 (2010).
  23. L. Fidkowski, R. M. Lutchyn, C. Nayak, and Matthew P. A. Fisher, Majorana zero modes in one-dimensional quantum wires without long-ranged superconducting order, Phys. Rev. B 84, 195436 (2011).
  24. N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B 61, 10267 (2000).
  25. D. A. Ivanov, Non-Abelian statistics of half-quantum vortices in p-Wave superconductors, Phys. Rev. Lett. 86, 268 (2001).
  26. N. Regnault and B. A. Bernevig, Fractional Chern insulator, Phys. Rev. X 1, 021014 (2011).
  27. K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Nearly flatbands with nontrivial topology, Phys. Rev. Lett. 106, 236803 (2011).
  28. E. Tang, J.-W. Mei, and X.-G. Wen, High-temperature fractional quantum Hall states, Phys. Rev. Lett. 106, 236802 (2011).
  29. Y. Jia, J. Yu, J. Liu, J. Herzog-Arbeitman, Z. Qi, H. Pi, N. Regnault, H. Weng, B. A. Bernevig, and Q. Wu, Moiré fractional Chern insulators. I. First-principles calculations and continuum models of twisted bilayer MoTe2, Phys. Rev. B 109, 205121 (2024).
  30. L.-M. Duan, E. Demler, and M. D. Lukin, Controlling spin exchange interactions of ultracold atoms in optical lattices, Phys. Rev. Lett. 91, 090402 (2003).
  31. A. Micheli, G. K. Brennen, and P. Zoller, A toolbox for lattice-spin models with polar molecules, Nat. Phys. 2, 341 (2006).
  32. N. C. and, Rapidly rotating atomic gases, Adv. Phys. 57, 539 (2008).
  33. C. V. Kraus, M. Dalmonte, M. A. Baranov, A. M. Läuchli, and P. Zoller, Majorana edge states in atomic wires coupled by pair hopping, Phys. Rev. Lett. 111, 173004 (2013).
  34. A. Bühler, N. Lang, C. V. Kraus, G. Möller, S. D. Huber, and H. P. Büchler, Majorana modes and p-wave superfluids for fermionic atoms in optical lattices, Nat. Commun. 5, 4504 (2014).
  35. N. Y. Yao, A. V. Gorshkov, C. R. Laumann, A. M. Läuchli, J. Ye, and M. D. Lukin, Realizing fractional Chern insulators in dipolar spin systems, Phys. Rev. Lett. 110, 185302 (2013).
  36. S. Weber, R. Bai, N. Makki, J. Mögerle, T. Lahaye, A. Browaeys, M. Daghofer, N. Lang, and H. P. Büchler, Experimentally accessible scheme for a fractional Chern insulator in Rydberg atoms, PRX Quantum 3, 030302 (2022).
  37. C. Chamon, D. Green, and Z.-C. Yang, Constructing quantum spin liquids using combinatorial gauge symmetry, Phys. Rev. Lett. 125, 067203 (2020).
  38. Z.-C. Yang, D. Green, H. Yu, and C. Chamon, Z3 Quantum Double in a Superconducting Wire Array, PRX Quantum 2, 030327 (2021).
  39. D. Green and C. Chamon, Constructing non-Abelian quantum spin liquids using combinatorial gauge symmetry, SciPost Phys. 15, 067 (2023).
  40. H. Yu, D. Green, A. E. Ruckenstein, and C. Chamon, Abelian combinatorial gauge symmetry, SciPost Phys. Core 7, 014 (2024).
  41. H. Yu, D. Green, and C. Chamon, Non-Abelian combinatorial gauge theory, Phys. Rev. B 111, 235115 (2025).
  42. S. de Léséleuc, V. Lienhard, P. Scholl, D. Barredo, S. Weber, N. Lang, H. P. Büchler, T. Lahaye, and A. Browaeys, Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms, Science 365, 775 (2019).
  43. R. Verresen, M. D. Lukin, and A. Vishwanath, Prediction of toric code topological order from Rydberg blockade, Phys. Rev. X 11, 031005 (2021).
  44. G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, et al., Probing topological spin liquids on a programmable quantum simulator, Science 374, 1242 (2021).
  45. R. Sahay, A. Vishwanath, and R. Verresen, Quantum spin puddles and lakes: NISQ-era spin liquids from non-equilibrium dynamics, arXiv:2211.01381.
  46. G. Giudici, M. D. Lukin, and H. Pichler, Dynamical preparation of quantum spin liquids in Rydberg atom arrays, Phys. Rev. Lett. 129, 090401 (2022).
  47. F. Nogrette, H. Labuhn, S. Ravets, D. Barredo, L. Béguin, A. Vernier, T. Lahaye, and A. Browaeys, Single-atom trapping in holographic 2D arrays of microtraps with arbitrary geometries, Phys. Rev. X 4, 021034 (2014).
  48. D. Barredo, S. de Léséleuc, V. Lienhard, T. Lahaye, and A. Browaeys, An atom-by-atom assembler of defect-free arbitrary two-dimensional atomic arrays, Science 354, 1021 (2016).
  49. H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature (London) 551, 579 (2017).
  50. D. Barredo, V. Lienhard, S. de Léséleuc, T. Lahaye, and A. Browaeys, Synthetic three-dimensional atomic structures assembled atom by atom, Nature (London) 561, 79 (2018).
  51. D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Fast quantum gates for neutral atoms, Phys. Rev. Lett. 85, 2208 (2000).
  52. M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010).
  53. D. Barredo, H. Labuhn, S. Ravets, T. Lahaye, A. Browaeys, and C. S. Adams, Coherent excitation transfer in a spin chain of three Rydberg atoms, Phys. Rev. Lett. 114, 113002 (2015).
  54. H. Pichler, S.-T. Wang, L. Zhou, S. Choi, and M. D. Lukin, Computational complexity of the Rydberg blockade in two dimensions, arXiv:1809.04954.
  55. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Weak ergodicity breaking from quantum many-body scars, Nat. Phys. 14, 745 (2018).
  56. C.-J. Lin and O. I. Motrunich, Exact quantum many-body scar states in the Rydberg-blockaded atom chain, Phys. Rev. Lett. 122, 173401 (2019).
  57. D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Semeghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho, S. Choi, M. Serbyn, et al., Controlling quantum many-body dynamics in driven Rydberg atom arrays, Science 371, 1355 (2021).
  58. S. Ebadi, A. Keesling, M. Cain, T. T. Wang, H. Levine, D. Bluvstein, G. Semeghini, A. Omran, J.-G. Liu, R. Samajdar, X.-Z. Luo, B. Nash, et al., Quantum optimization of maximum independent set using Rydberg atom arrays, Science 376, 1209 (2022).
  59. M.-T. Nguyen, J.-G. Liu, J. Wurtz, M. D. Lukin, S.-T. Wang, and H. Pichler, Quantum optimization with arbitrary connectivity using Rydberg atom arrays, PRX Quantum 4, 010316 (2023).
  60. S. Stastny, H. P. Büchler, and N. Lang, Functional completeness of planar Rydberg blockade structures, Phys. Rev. B 108, 085138 (2023).
  61. T. F. Maier, H. P. Büchler, and N. Lang, Topological order in symmetric blockade structures, PRX Quantum 6, 030340 (2025).
  62. S. Fell, T. F. Maier, H. P. Büchler, and N. Lang (unpublished).
  63. D. K. Ruttley, T. R. Hepworth, A. Guttridge, and S. L. Cornish, Long-lived entanglement of molecules in magic-wavelength optical tweezers, Nature (London) 637, 827 (2025).
  64. A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  65. R. M. Karp, Reducibility Among Combinatorial Problems (Springer, Berlin, 2009), pp. 219–241.
  66. B. N. Clark, C. J. Colbourn, and D. S. Johnson, Unit disk graphs, Discrete Math. 86, 165 (1990).
  67. S. H. Simon, Topological Quantum (Oxford University Press, Oxford, 2023).
  68. H. Bombin and M. A. Martin-Delgado, Family of non-Abelian Kitaev models on a lattice: Topological condensation and confinement, Phys. Rev. B 78, 115421 (2008).
  69. S. X. Cui, S.-M. Hong, and Z. Wang, Universal quantum computation with weakly integral anyons, Quant. Info. Proc. 14, 2687 (2015).
  70. A. Kómár and O. Landon-Cardinal, Anyons are not energy eigenspaces of quantum double Hamiltonians, Phys. Rev. B 96, 195150 (2017).
  71. S. X. Cui, D. Ding, X. Han, G. Penington, D. Ranard, B. C. Rayhaun, and Z. Shangnan, Kitaev's quantum double model as an error correcting code, Quantum 4, 331 (2020).
  72. R. Dijkgraaf, V. Pasquier, and P. Roche, Quasi Hopf algebras, group cohomology and orbifold models, Nucl. Phys. B Proc. Suppl. 18, 60 (1991).
  73. S. Majid, Quantum double for quasi-Hopf algebras, Lett. Math. Phys. 45, 1 (1998).
  74. See Supplemental Material at http://link.aps.org/supplemental/10.1103/99sy-kggw for details on the construction of the blockade graphs, their automorphism group, and on the proof of ground state topological order.
  75. On a torus, not all ground state configurations |g〉∈HG0 (satisfying the no-flux constraint on every site) can be generated by applying plaquette (or loop) automorphisms on the state |e〉 with all gl=e. This partitions HG0 into topological sectors, and the amplitudes in the (unique) ground state between these sectors are not necessarily fixed by symmetries.
  76. S. Bravyi, M. B. Hastings, and S. Michalakis, Topological quantum order: Stability under local perturbations, J. Math. Phys. 51, 093512 (2010).
  77. S. Michalakis and J. P. Zwolak, Stability of frustration-free Hamiltonians, Commun. Math. Phys. 322, 277 (2013).
  78. S. Bravyi and M. B. Hastings, A short proof of stability of topological order under local perturbations, Commun. Math. Phys. 307, 609 (2011).
  79. X. Chen, Z.-C. Gu, and X.-G. Wen, Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order, Phys. Rev. B 82, 155138 (2010).
  80. S. Bravyi, D. P. DiVincenzo, and D. Loss, Schrieffer-Wolff transformation for quantum many-body systems, Ann. Phys. 326, 2793 (2011).
  81. V. G. Drinfel'd, Quantum groups, J. Sov. Math. 41, 898 (1988).
  82. K. G. Wilson, Confinement of quarks, Phys. Rev. D 10, 2445 (1974).
  83. K. Fredenhagen and M. Marcu, Charged states in Z2 gauge theories, Commun. Math. Phys. 92, 81 (1983).
  84. K. Gregor, D. A. Huse, R. Moessner, and S. L. Sondhi, Diagnosing deconfinement and topological order, New J. Phys. 13, 025009 (2011).
  85. The Sage Developers, sagemath, the Sage mathematics software system, Zenodo, 2026, https://doi.org/10.5281/zenodo.14600115.

Outline

Information

Phys. Rev. B 114, 065113– Published 9 July, 2026
  • Received 6 November 2025
  • Revised 11 June 2026
  • Accepted 12 June 2026
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