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Compact localized fermions and Ising anyons in a chiral spin liquid

Tim Bauer1,2,3 and Johannes Reuther1,2

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Phys. Rev. B 113, 094442 – Published 20 March, 2026

DOI: https://doi.org/10.1103/dgyy-rbcn

Abstract

Quasiparticle hybridization remains a major challenge in realizing and controlling exotic states of matter in existing quantum simulation platforms. We report the absence of hybridization for compact localized states (CLSs) emerging in the chiral spin liquid described by the Yao-Kivelson model. The CLS forms due to destructive quantum interference at fine-tuned coupling constants and populates perfectly flat quasiparticle bands on an effective kagome lattice. Using a formalism for general Majorana-hopping Hamiltonians, we derive exact expressions for CLSs for various flux configurations and both for the topological and trivial phases of the model. In addition to finite-energy matter fermions with characteristic spin-spin correlations, we construct compact localized Majorana zero modes attached to π-flux excitations, which enable non-Abelian braiding of Ising anyons with minimal separation. Our results inform the quantum simulation of topologically ordered states of matter and open avenues for exploring flat-band physics in quantum spin liquids.

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References (64)

  1. 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).
  2. K. 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).
  3. M. Kalinowski, N. Maskara, and M. D. Lukin, Non-Abelian Floquet spin liquids in a digital Rydberg simulator, Phys. Rev. X 13, 031008 (2023).
  4. M. Will, T. A. Cochran, E. Rosenberg, B. Jobst, N. M. Eassa, P. Roushan, M. Knap, A. Gammon-Smith, and F. Pollmann, Probing non-equilibrium topological order on a quantum processor, Nature (London) 645, 348 (2025).
  5. S. J. Evered, M. Kalinowski, A. A. Geim, T. Manovitz, D. Bluvstein, S. H. Li, N. Maskara, H. Zhou, S. Ebadi, M. Xu, et al., Probing the Kitaev honeycomb model on a neutral-atom quantum computer, Nature (London) 645, 341 (2025).
  6. Google Quantum AI and Collaborators, Non-Abelian braiding of graph vertices in a superconducting processor, Nature (London) 618, 264 (2023).
  7. S. Xu, Z.-Z. Sun, K. Wang, L. Xiang, Z. Bao, Z. Zhu, F. Shen, Z. Song, P. Zhang, W. Ren, et al., Digital simulation of projective non-Abelian anyons with 68 superconducting qubits, Chin. Phys. Lett. 40, 060301 (2023).
  8. S. Xu, Z.-Z. Sun, K. Wang, H. Li, Z. Zhu, H. Dong, J. Deng, X. Zhang, J. Chen, Y. Wu, et al., Non-Abelian braiding of Fibonacci anyons with a superconducting processor, Nat. Phys. 20, 1469 (2024).
  9. 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).
  10. M. Iqbal, A. Lyons, C. F. B. Lo, N. Tantivasadakarn, J. Dreiling, C. Foltz, T. M. Gatterman, D. Gresh, N. Hewitt, C. A. Holliman, et al., Qutrit toric code and parafermions in trapped ions, Nat. Commun. 16, 6301 (2025).
  11. W. Song, X. Liu, J. Sun, O. You, S. Wu, C. Chen, S. Zhu, T. Li, and S. Zhang, Shortcuts to adiabatic non-Abelian braiding on silicon photonic chips, Sci. Adv. 11, eadt7224 (2025).
  12. L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2017).
  13. C. Broholm, R. J. Cava, S. Kivelson, D. Nocera, M. Norman, and T. Senthil, Quantum spin liquids, Science 367, eaay0668 (2020).
  14. S. Kirkpatrick and T. P. Eggarter, Localized states of a binary alloy, Phys. Rev. B 6, 3598 (1972).
  15. B. Sutherland, Localization of electronic wave functions due to local topology, Phys. Rev. B 34, 5208 (1986).
  16. S. M. Zhang and L. Jin, Compact localized states and localization dynamics in the dice lattice, Phys. Rev. B 102, 054301 (2020).
  17. D. L. Bergman, C. Wu, and L. Balents, Band touching from real-space topology in frustrated hopping models, Phys. Rev. B 78, 125104 (2008).
  18. J. Vidal, R. Mosseri, and B. Douçot, Aharonov-Bohm cages in two-dimensional structures, Phys. Rev. Lett. 81, 5888 (1998).
  19. S. Mukherjee, M. Di Liberto, P. Öhberg, R. R. Thomson, and N. Goldman, Experimental observation of Aharonov-Bohm cages in photonic lattices, Phys. Rev. Lett. 121, 075502 (2018).
  20. H. Li, Z. Dong, S. Longhi, Q. Liang, D. Xie, and B. Yan, Aharonov-Bohm caging and inverse Anderson transition in ultracold atoms, Phys. Rev. Lett. 129, 220403 (2022).
  21. T. Chen, C. Huang, I. Velkovsky, T. Ozawa, H. Price, J. P. Covey, and B. Gadway, Interaction-driven breakdown of Aharonov–Bohm caging in flat-band Rydberg lattices, Nat. Phys. 21, 221 (2025).
  22. J.-X. Yin, B. Lian, and M. Z. Hasan, Topological kagome magnets and superconductors, Nature (London) 612, 647 (2022).
  23. Y. Wang, H. Wu, G. T. McCandless, J. Y. Chan, and M. N. Ali, Quantum states and intertwining phases in kagome materials, Nat. Rev. Phys. 5, 635 (2023).
  24. Q. Wang, H. Lei, Y. Qi, and C. Felser, Topological quantum materials with kagome lattice, Acc. Mater. Res. 5, 786 (2024).
  25. Y. Xie, A. T. Pierce, J. M. Park, D. E. Parker, E. Khalaf, P. Ledwith, Y. Cao, S. H. Lee, S. Chen, P. R. Forrester, et al., Fractional Chern insulators in magic-angle twisted bilayer graphene, Nature (London) 600, 439 (2021).
  26. S. Bhowmik, A. Ghosh, and U. Chandni, Emergent phases in graphene flat bands, Rep. Prog. Phys. 87, 096401 (2024).
  27. J. Schulenburg, A. Honecker, J. Schnack, J. Richter, and H.-J. Schmidt, Macroscopic magnetization jumps due to independent magnons in frustrated quantum spin lattices, Phys. Rev. Lett. 88, 167207 (2002).
  28. H.-J. Schmidt, J. Richter, and R. Moessner, Linear independence of localized magnon states, J. Phys. A: Math. Gen. 39, 10673 (2006).
  29. M. Goda, S. Nishino, and H. Matsuda, Inverse Anderson transition caused by flatbands, Phys. Rev. Lett. 96, 126401 (2006).
  30. J. T. Chalker, T. S. Pickles, and P. Shukla, Anderson localization in tight-binding models with flat bands, Phys. Rev. B 82, 104209 (2010).
  31. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. (NY) 321, 2 (2006).
  32. H. Yao and S. A. Kivelson, Exact chiral spin liquid with non-Abelian anyons, Phys. Rev. Lett. 99, 247203 (2007).
  33. G. Khaliullin, Orbital order and fluctuations in Mott insulators, Prog. Theor. Phys. Suppl. 160, 155 (2005).
  34. G. Jackeli and G. Khaliullin, Mott insulators in the strong spin-orbit coupling limit: From Heisenberg to a quantum compass and Kitaev models, Phys. Rev. Lett. 102, 017205 (2009).
  35. S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. Gegenwart, and R. Valentí, Models and materials for generalized Kitaev magnetism, J. Phys.: Condens. Matter 29, 493002 (2017).
  36. M. Hermanns, I. Kimchi, and J. Knolle, Physics of the Kitaev model: Fractionalization, dynamic correlations, and material connections, Annu. Rev. Condens. Matter Phys. 9, 17 (2018).
  37. S. Trebst and C. Hickey, Kitaev materials, Phys. Rep. 950, 1 (2022).
  38. Z. Zou and Q. Luo, High Chern numbers and topological flat bands in high-field polarized Kitaev magnets on the star lattice, Phys. Rev. B 112, 054409 (2025).
  39. K. Fukui and Y. Motome, Topological Majorana flat bands in the Kitaev model on a Bishamon-kikko lattice, Phys. Rev. B 112, 144411 (2025).
  40. K. B. Yogendra, G. Baskaran, and T. Das, Fractional Wannier orbitals and tight-binding gauge fields in Kitaev honeycomb superlattices with flat Majorana bands, Phys. Rev. X 15, 031020 (2025).
  41. F. L. Pedrocchi, S. Chesi, and D. Loss, Physical solutions of the Kitaev honeycomb model, Phys. Rev. B 84, 165414 (2011).
  42. F. Zschocke and M. Vojta, Physical states and finite-size effects in Kitaev's honeycomb model: Bond disorder, spin excitations, and NMR line shape, Phys. Rev. B 92, 014403 (2015).
  43. E. H. Lieb, Flux phase of the half-filled band, Phys. Rev. Lett. 73, 2158 (1994).
  44. S. Dusuel, K. P. Schmidt, J. Vidal, and R. L. Zaffino, Perturbative study of the Kitaev model with spontaneous time-reversal symmetry breaking, Phys. Rev. B 78, 125102 (2008).
  45. K. S. Tikhonov and M. V. Feigel'man, Quantum spin metal state on a decorated honeycomb lattice, Phys. Rev. Lett. 105, 067207 (2010).
  46. J.-N. Fuchs, S. Patil, and J. Vidal, Parity of Chern numbers in the Kitaev honeycomb model and the sixteenfold way, Phys. Rev. B 102, 115130 (2020).
  47. R. Chari, R. Moessner, and J. G. Rau, Magnetoelectric generation of a Majorana-Fermi surface in Kitaev's honeycomb model, Phys. Rev. B 103, 134444 (2021).
  48. K. Nakazawa, Y. Kato, and Y. Motome, Asymmetric modulation of Majorana excitation spectra and nonreciprocal thermal transport in the Kitaev spin liquid under a staggered magnetic field, Phys. Rev. B 105, 165152 (2022).
  49. G. Cassella, P. d'Ornellas, T. Hodson, W. M. Natori, and J. Knolle, An exact chiral amorphous spin liquid, Nat. Commun. 14, 6663 (2023).
  50. X.-F. Shi, Y. Chen, and J. Q. You, Exotic phase diagram of a topological quantum system, Phys. Rev. B 82, 174412 (2010).
  51. G. Baskaran, S. Mandal, and R. Shankar, Exact results for spin dynamics and fractionalization in the Kitaev model, Phys. Rev. Lett. 98, 247201 (2007).
  52. J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner, Dynamics of a two-dimensional quantum spin liquid: Signatures of emergent Majorana fermions and fluxes, Phys. Rev. Lett. 112, 207203 (2014).
  53. S.-S. Zhang, Z. Wang, G. B. Halász, and C. D. Batista, Vison crystals in an extended Kitaev model on the honeycomb lattice, Phys. Rev. Lett. 123, 057201 (2019).
  54. S. Chulliparambil, L. Janssen, M. Vojta, H.-H. Tu, and U. F. P. Seifert, Flux crystals, Majorana metals, and flat bands in exactly solvable spin-orbital liquids, Phys. Rev. B 103, 075144 (2021).
  55. D. J. Alspaugh, J.-N. Fuchs, A. Ritz-Zwilling, and J. Vidal, Effective models for dense vortex lattices in the Kitaev honeycomb model, Phys. Rev. B 109, 115107 (2024).
  56. 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).
  57. A. T. Bolukbasi and J. Vala, Rigorous calculations of non-Abelian statistics in the Kitaev honeycomb model, New J. Phys. 14, 045007 (2012).
  58. C. Harada, A. Ono, and J. Nasu, Real-time control of non-Abelian anyons in Kitaev spin liquid under energy dissipation, Phys. Rev. B 110, 214426 (2024).
  59. D. Aasen, R. S. K. Mong, B. M. Hunt, D. Mandrus, and J. Alicea, Electrical probes of the non-Abelian spin liquid in Kitaev materials, Phys. Rev. X 10, 031014 (2020).
  60. R. G. Pereira and R. Egger, Electrical access to Ising anyons in Kitaev spin liquids, Phys. Rev. Lett. 125, 227202 (2020).
  61. T. Bauer, L. R. D. Freitas, R. G. Pereira, and R. Egger, Scanning tunneling spectroscopy of Majorana zero modes in a Kitaev spin liquid, Phys. Rev. B 107, 054432 (2023).
  62. K. Klocke, Y. Liu, G. B. Halász, and J. Alicea, Spin-liquid-based topological qubits, arXiv:2411.08093.
  63. J. Reuther and T. Bauer, Data for “Compact localized fermions and Ising anyons in a chiral spin liquid,” Zenodo (2026), doi: 10.5281/zenodo.18731932.
  64. J.-P. Blaizot and G. Ripka, Quantum Theory of Finite Systems (MIT Press, Cambridge, 1986).

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Phys. Rev. B 113, 094442– Published 20 March, 2026
  • Received 18 November 2025
  • Revised 2 February 2026
  • Accepted 4 March 2026
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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.

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