- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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

Topological states and flat bands in exactly solvable decorated Cayley trees

Wanda P. Duss*, Askar Iliasov†, and Tomáš Bzdušek‡

  • *Contact author: wanda.duss@physik.uzh.ch
  • †Contact author: askar.iliasov@uzh.ch
  • ‡Contact author: tomas.bzdusek@uzh.ch
PDF

Phys. Rev. Research 8, 033036 – Published 10 July, 2026

DOI: https://doi.org/10.1103/t8gg-xqls

Abstract

We derive the full spectrum of decorated Cayley trees that constitute tree analogs of selected two-dimensional Euclidean lattices, namely, the Lieb, the double-Lieb, the kagome, and the star lattice. The common feature of these Euclidean lattices is that their nearest-neighbor models give rise to flat energy bands interpretable through compact localized states. We find that the tree analogs exhibit similar flat or nearly flat energy bands at the corresponding energies. Interestingly, such flat bands in the decorated Cayley trees acquire an interpretation that is absent in their Euclidean counterparts: as edge states localized to the inner or the outer boundary of the tree branches. In particular, we establish an exact correspondence between the Lieb-Cayley tree and an ensemble of one-dimensional Su-Schrieffer-Heeger chains, which maps topological edge states on one side of the chains to flat-band states localized in the bulk of the tree, furnishing the flat energy band with a topological stability. Similar mapping to topological edge states or to states bound to edge defects in one-dimensional chains is shown for flat-band states in all the considered tree decorations. We finally show that the persistence of exact flat bands on infinite decorated trees (i.e., Bethe lattices) arises naturally from a covering interpretation of tree graphs. Our findings reveal a rich landscape of flat-band and topological phenomena in non-Euclidean systems, where geometry alone can generate and stabilize unconventional quantum states.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (104)

  1. A. Georges and G. Kotliar, Hubbard model in infinite dimensions, Phys. Rev. B 45, 6479 (1992).
  2. M. Jarrell, Hubbard model in infinite dimensions: A quantum Monte Carlo study, Phys. Rev. Lett. 69, 168 (1992).
  3. M. Eckstein, M. Kollar, K. Byczuk, and D. Vollhardt, Hopping on the Bethe lattice: Exact results for densities of states and dynamical mean-field theory, Phys. Rev. B 71, 235119 (2005).
  4. M. Kollar, M. Eckstein, K. Byczuk, N. Blümer, P. van Dongen, M. Radke de Cuba, W. Metzner, D. Tanasković, V. Dobrosavljević, G. Kotliar, and D. Vollhardt, Green functions for nearest- and next-nearest-neighbor hopping on the Bethe lattice, Ann. Phys. 517, 642 (2005).
  5. R. Peters and T. Pruschke, Half-filled Hubbard model on a Bethe lattice with next-nearest-neighbor hopping, Phys. Rev. B 79, 045108 (2009).
  6. G. Semerjian, M. Tarzia, and F. Zamponi, Exact solution of the Bose-Hubbard model on the Bethe lattice, Phys. Rev. B 80, 014524 (2009).
  7. P. Lunts, A. Georges, E. M. Stoudenmire, and M. Fishman, Hubbard model on the Bethe lattice via variational uniform tree states: Metal-insulator transition and a Fermi liquid, Phys. Rev. Res. 3, 023054 (2021).
  8. M.-B. Lepetit, M. Cousy, and G. M. Pastor, Density-matrix renormalization study of the Hubbard model on a Bethe lattice, Eur. Phys. J. B 13, 421 (2000).
  9. J.-L. Chen, Z. Fan, B. Zhan, J. Hu, T. Liu, J. Ji, K. Wang, H.-J. Liao, and T. Xiang, Thermodynamics of the Hubbard model on the Bethe lattice, Phys. Rev. B 112, 125130 (2025).
  10. V. Bashmakov, A. Iliasov, T. Bzdušek, and A. A. Bagrov, Superconductivity in hyperbolic spaces: Regular hyperbolic lattices and Ginzburg-Landau theory, arXiv:2509.09330.
  11. M. Pavliuk, T. Bzdušek, and A. Iliasov, Superconductivity in hyperbolic spaces: Cayley trees, hyperbolic continuum, and BCS theory, arXiv:2510.26528.
  12. J. Vidal and R. Mosseri, Kitaev model in regular hyperbolic tilings, Phys. Rev. B 112, 195106 (2025).
  13. D. J. Thouless, Spin-glass on a Bethe lattice, Phys. Rev. Lett. 56, 1082 (1986).
  14. C. Laumann, A. Scardicchio, and S. L. Sondhi, Cavity method for quantum spin glasses on the Bethe lattice, Phys. Rev. B 78, 134424 (2008).
  15. M. Mézard and G. Parisi, The Bethe lattice spin glass revisited, Eur. Phys. J. B 20, 217 (2001).
  16. S. Savitz, C. Peng, and G. Refael, Anderson localization on the Bethe lattice using cages and the Wegner flow, Phys. Rev. B 100, 094201 (2019).
  17. R. Abou-Chacra, D. J. Thouless, and P. W. Anderson, A selfconsistent theory of localization, J. Phys. C 6, 1734 (1973).
  18. M. Aizenman and S. Warzel, The canopy graph and level statistics for random operators on trees, J. Math. Phys. Anal. Geom. 9, 291 (2006).
  19. T. Rizzo and M. Tarzia, Localized phase of the Anderson model on the Bethe lattice, Phys. Rev. B 110, 184210 (2024).
  20. T. P. Eggarter, Cayley trees, the Ising problem, and the thermodynamic limit, Phys. Rev. B 9, 2989 (1974).
  21. H. Bing and C. Shu, Correlation function of Potts model on Bethe lattice, Chin. Phys. Lett. 17, 549 (2000).
  22. T. K. Kopeć and K. D. Usadel, Short-range ±J interaction Ising spin glass in a transverse field on a Bethe lattice: A quantum-spherical approach, Phys. Status Solidi (b) 243, 502 (2006).
  23. C.-K. Hu and N. S. Izmailian, Exact correlation functions of Bethe lattice spin models in external magnetic fields, Phys. Rev. E 58, 1644 (1998).
  24. K. S. Tikhonov and A. D. Mirlin, Fractality of wave functions on a Cayley tree: Difference between tree and locally treelike graph without boundary, Phys. Rev. B 94, 184203 (2016).
  25. M. Sonner, K. S. Tikhonov, and A. D. Mirlin, Multifractality of wave functions on a Cayley tree: From root to leaves, Phys. Rev. B 96, 214204 (2017).
  26. J. Sun, C.-A. Li, Q. Guo, W. Zhang, S. Feng, X. Zhang, H. Guo, and B. Trauzettel, Non-Hermitian quantum fractals, Phys. Rev. B 110, L201103 (2024).
  27. S. Hamanaka, A. A. Iliasov, T. Neupert, T. Bzdušek, and T. Yoshida, Multifractal statistics of non-Hermitian skin effect on the Cayley tree, Phys. Rev. B 111, 075162 (2025).
  28. J. Sun, C.-A. Li, P. Li, S. Feng, and H. Guo, Inner non-Hermitian skin effect on the Bethe lattice, Phys. Rev. B 111, 075120 (2025).
  29. N. Hatano, H. Katsura, and K. Kawabata, Quantum transport on Bethe lattices with non-Hermitian sources and a drain, J. Phys. A: Math. Theor. 59, 175002 (2026).
  30. S. S. Gubser, J. Knaute, S. Parikh, A. Samberg, and P. Witaszczyk, p-Adic AdS/CFT, Commun. Math. Phys. 352, 1019 (2017).
  31. N. Manoj and V. B. Shenoy, Arboreal topological and fracton phases, Phys. Rev. B 107, 165136 (2023).
  32. C. R. Laumann, S. A. Parameswaran, and S. L. Sondhi, Absence of Goldstone bosons on the Bethe lattice, Phys. Rev. B 80, 144415 (2009).
  33. O. Breach, B. Placke, P. W. Claeys, and S. Parameswaran, Solvable quantum circuits in tree+1 dimensions, PRX Quantum 6, 040316 (2025).
  34. B. Placke, G. M. Sommers, N. P. Breuckmann, T. Rakovszky, and V. Khemani, Expansion creates spin-glass order in finite-connectivity models: A rigorous and intuitive approach from the theory of LDPC codes, arXiv:2507.13342.
  35. P. Basteiro, R. N. Das, G. D. Giulio, and J. Erdmenger, Aperiodic spin chains at the boundary of hyperbolic tilings, SciPost Phys. 15, 218 (2023).
  36. G. D. Mahan, Energy bands of the Bethe lattice, Phys. Rev. B 63, 155110 (2001).
  37. D. Aryal and S. Kettemann, Complete solution of the tight binding model on a Cayley tree: Strongly localised versus extended states, J. Phys. Commun. 4, 105010 (2020).
  38. M. Ostilli, C. G. Bezerra, and G. M. Viswanathan, Spectrum of the tight-binding model on Cayley trees and comparison with Bethe lattices, Phys. Rev. E 105, 034123 (2022).
  39. H. Yorikawa, Density of states of the Cayley tree, J. Phys. Commun. 2, 125009 (2018).
  40. X. Wang, Z. Nussinov, and G. Ortiz, Emergence of a boundary-sensitive phase in hyperbolic Ising models, Phys. Rev. B 112, 214102 (2025).
  41. F. R. Lux and E. Prodan, Converging periodic boundary conditions and detection of topological gaps on regular hyperbolic tessellations, Phys. Rev. Lett. 131, 176603 (2023).
  42. M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Observation of topological phase transitions in photonic quasicrystals, Phys. Rev. Lett. 110, 076403 (2013).
  43. J. Fan and H. Huang, Topological states in quasicrystals, Front. Phys. 17, 13203 (2022).
  44. C. W. Duncan, S. Manna, and A. E. B. Nielsen, Topological models in rotationally symmetric quasicrystals, Phys. Rev. B 101, 115413 (2020).
  45. R. Chen, C.-Z. Chen, J.-H. Gao, B. Zhou, and D.-H. Xu, Higher-order topological insulators in quasicrystals, Phys. Rev. Lett. 124, 036803 (2020).
  46. Z.-G. Chen, C. Lou, K. Hu, and L.-K. Lim, Fractal surface states in three-dimensional topological quasicrystals, Phys. Rev. Res. 6, 043054 (2024).
  47. D. Varjas, A. Lau, K. Pöyhönen, A. R. Akhmerov, D. I. Pikulin, and I. C. Fulga, Topological phases without crystalline counterparts, Phys. Rev. Lett. 123, 196401 (2019).
  48. C. Ouyang, Q. He, D.-H. Xu, and F. Liu, Higher-order topology in Fibonacci quasicrystals, Phys. Rev. B 110, 075425 (2024).
  49. D. Johnstone, M. J. Colbrook, A. E. B. Nielsen, P. Öhberg, and C. W. Duncan, Bulk localized transport states in infinite and finite quasicrystals via magnetic aperiodicity, Phys. Rev. B 106, 045149 (2022).
  50. F. Balling-Ansø, J. L. Krogh, E. E. Lassen, and A. E. B. Nielsen, Identification and properties of topological states in the bulk of quasicrystals, Phys. Rev. B 113, 075134 (2026).
  51. S. Pai and A. Prem, Topological states on fractal lattices, Phys. Rev. B 100, 155135 (2019).
  52. R. Canyellas, C. Liu, R. Arouca, L. Eek, G. Wang, Y. Yin, D. Guan, Y. Li, S. Wang, H. Zheng, C. Liu, J. Jia, and C. Morais Smith, Topological edge and corner states in bismuth fractal nanostructures, Nat. Phys. 20, 1421 (2024).
  53. M. Brzezińska, A. M. Cook, and T. Neupert, Topology in the Sierpiński-Hofstadter problem, Phys. Rev. B 98, 205116 (2018).
  54. M. N. Ivaki, I. Sahlberg, K. Pöyhönen, and T. Ojanen, Topological random fractals, Commun. Phys. 5, 327 (2022).
  55. Z. F. Osseweijer, L. Eek, A. Moustaj, M. Fremling, and C. Morais Smith, Haldane model on the Sierpiński gasket, Phys. Rev. B 110, 245405 (2024).
  56. B. Pal and K. Saha, Flat bands in fractal-like geometry, Phys. Rev. B 97, 195101 (2018).
  57. S. Manna, S. Nandy, and B. Roy, Higher-order topological phases on fractal lattices, Phys. Rev. B 105, L201301 (2022).
  58. S. Manna, S. K. Das, and B. Roy, Noncrystalline topological superconductors, Phys. Rev. B 109, 174512 (2024).
  59. D. J. Salib, A. J. Mains, and B. Roy, Topological insulators on fractal lattices: A general principle of construction, Phys. Rev. B 110, L241302 (2024).
  60. L. Eek, Z. F. Osseweijer, and C. Morais Smith, Fractality-induced topology, Phys. Rev. Lett. 134, 246601 (2025).
  61. S. Yu, X. Piao, and N. Park, Topological hyperbolic lattices, Phys. Rev. Lett. 125, 053901 (2020).
  62. D. M. Urwyler, P. M. Lenggenhager, I. Boettcher, R. Thomale, T. Neupert, and T. Bzdušek, Hyperbolic topological band insulators, Phys. Rev. Lett. 129, 246402 (2022).
  63. Z.-R. Liu, C.-B. Hua, T. Peng, and B. Zhou, Chern insulator in a hyperbolic lattice, Phys. Rev. B 105, 245301 (2022).
  64. Y.-L. Tao and Y. Xu, Higher-order topological hyperbolic lattices, Phys. Rev. B 107, 184201 (2023).
  65. A. Chen, Y. Guan, P. M. Lenggenhager, J. Maciejko, I. Boettcher, and T. Bzdušek, Symmetry and topology of hyperbolic Haldane models, Phys. Rev. B 108, 085114 (2023).
  66. T. Tummuru, A. Chen, P. M. Lenggenhager, T. Neupert, J. Maciejko, and T. Bzdušek, Hyperbolic non-Abelian semimetal, Phys. Rev. Lett. 132, 206601 (2024).
  67. T. Bzdušek and J. Maciejko, Flat bands and band-touching from real-space topology in hyperbolic lattices, Phys. Rev. B 106, 155146 (2022).
  68. A. Westström, W. Duan, K. Yao, X. Wang, J. Liu, and J. Li, Topological phases on quantum trees, arXiv:2302.03166.
  69. G. Singh, S. Bera, and V. B. Shenoy, Arboreal obstructed atomic insulating and metallic phases of fermions, Phys. Rev. B 110, 205143 (2024).
  70. O. Petrova and R. Moessner, Coulomb potential V(r)=1/r problem on the Bethe lattice, Phys. Rev. E 93, 012115 (2016).
  71. D. R. DeFord and D. N. Rockmore, On the spectrum of finite, rooted homogeneous trees, Linear Algebra Appl. 598, 165 (2020).
  72. D. L. Bergman, C. Wu, and L. Balents, Band touching from real-space topology in frustrated hopping models, Phys. Rev. B 78, 125104 (2008).
  73. E. H. Lieb, Two theorems on the Hubbard model, Phys. Rev. Lett. 62, 1201 (1989).
  74. M. Niţă, B. Ostahie, and A. Aldea, Spectral and transport properties of the two-dimensional Lieb lattice, Phys. Rev. B 87, 125428 (2013).
  75. S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, Observation of a localized flat-band state in a photonic Lieb lattice, Phys. Rev. Lett. 114, 245504 (2015).
  76. B. Sutherland, Localization of electronic wave functions due to local topology, Phys. Rev. B 34, 5208 (1986).
  77. W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42, 1698 (1979).
  78. D. Zhang, Y. Zhang, H. Zhong, C. Li, Z. Zhang, Y. Zhang, and M. R. Belić, New edge-centered photonic square lattices with flat bands, Ann. Phys. 382, 160 (2017).
  79. A. Bhattacharya and B. Pal, Flat bands and nontrivial topological properties in an extended Lieb lattice, Phys. Rev. B 100, 235145 (2019).
  80. G. Centała and J. W. Kłos, Compact localized states in magnonic Lieb lattices, Sci. Rep. 13, 12676 (2023).
  81. H. Hanafi, P. Menz, A. McWilliam, J. Imbrock, and C. Denz, Localized dynamics arising from multiple flat bands in a decorated photonic Lieb lattice, APL Photon. 7, 111301 (2022).
  82. V. M. Martinez Alvarez and M. D. Coutinho-Filho, Edge states in trimer lattices, Phys. Rev. A 99, 013833 (2019).
  83. A. Anastasiadis, G. Styliaris, R. Chaunsali, G. Theocharis, and F. K. Diakonos, Bulk-edge correspondence in the trimer Su-Schrieffer-Heeger model, Phys. Rev. B 106, 085109 (2022).
  84. M. Ghuneim and R. Weda Bomantara, Topological phases of tight-binding trimer lattice in the BDI symmetry class, J. Phys.: Condens. Matter 36, 495402 (2024).
  85. F. Harary and G. E. Uhlenbeck, On the number of Husimi trees, Proc. Natl. Acad. Sci. USA 39, 315 (1953).
  86. H.-M. Guo and M. Franz, Topological insulator on the Kagome lattice, Phys. Rev. B 80, 113102 (2009).
  87. E. Tang, J.-W. Mei, and X.-G. Wen, High-temperature fractional quantum Hall states, Phys. Rev. Lett. 106, 236802 (2011).
  88. K. Ohgushi, S. Murakami, and N. Nagaosa, Spin anisotropy and quantum Hall effect in the kagomé lattice: Chiral spin state based on a ferromagnet, Phys. Rev. B 62, R6065 (2000).
  89. A. J. Kollár, M. Fitzpatrick, P. Sarnak, and A. A. Houck, Line-graph lattices: Euclidean and non-Euclidean flat bands, and implementations in circuit quantum electrodynamics, Commun. Math. Phys. 376, 1909 (2020).
  90. C. Godsil and G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics (Springer, New York, NY, 2001), Vol. 207.
  91. W. P. Duss, Exploration of topological bands on Cayley trees, Master’s thesis, University of Zurich, Zurich, Switzerland, 2025.
  92. H. Yao and S. A. Kivelson, Exact chiral spin liquid with non-Abelian anyons, Phys. Rev. Lett. 99, 247203 (2007).
  93. W.-C. Chen, R. Liu, Y.-F. Wang, and C.-D. Gong, Topological quantum phase transitions and topological flat bands on the star lattice, Phys. Rev. B 86, 085311 (2012).
  94. R. Fan, L. Sun, X. Shao, Y. Li, and M. Zhao, Two-dimensional Dirac materials: Tight-binding lattice models and material candidates, ChemPhysMater 2, 30 (2023).
  95. O. Rojas, Geometrically frustrated Ising-Heisenberg spin model on expanded Kagomé lattice, J. Magn. Magn. Mater. 473, 442 (2019).
  96. A. J. Kollár, M. Fitzpatrick, and A. A. Houck, Hyperbolic lattices in circuit quantum electrodynamics, Nature (London) 571, 45 (2019).
  97. P. M. Lenggenhager, A. Stegmaier, L. K. Upreti, T. Hofmann, T. Helbig, A. Vollhardt, M. Greiter, C. H. Lee, S. Imhof, H. Brand, T. Kießling, I. Boettcher, T. Neupert, R. Thomale, and T. Bzdušek, Simulating hyperbolic space on a circuit board, Nat. Commun. 13, 4373 (2022).
  98. I. Boettcher, P. Bienias, R. Belyansky, A. J. Kollár, and A. V. Gorshkov, Quantum simulation of hyperbolic space with circuit quantum electrodynamics: From graphs to geometry, Phys. Rev. A 102, 032208 (2020).
  99. S. Dey, A. Chen, P. Basteiro, A. Fritzsche, M. Greiter, M. Kaminski, P. M. Lenggenhager, R. Meyer, R. Sorbello, A. Stegmaier, R. Thomale, J. Erdmenger, and I. Boettcher, Simulating holographic conformal field theories on hyperbolic lattices, Phys. Rev. Lett. 133, 061603 (2024).
  100. S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kießling, F. Schindler, C. H. Lee, M. Greiter, T. Neupert, and R. Thomale, Topolectrical-circuit realization of topological corner modes, Nat. Phys. 14, 925 (2018).
  101. T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kießling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits, Nat. Phys. 16, 747 (2020).
  102. L. Boyle, M. Dickens, and F. Flicker, Conformal quasicrystals and holography, Phys. Rev. X 10, 011009 (2020).
  103. L. Boyle and J. Kulp, Holographic foliations: Self-similar quasicrystals from hyperbolic honeycombs, Phys. Rev. D 111, 046001 (2025).
  104. W. Duss, cayleytree, github.com/wandafulworld/cayleytree: Initial release, Zenodo, 2026, doi: https://doi.org/10.5281/zenodo.20086335

Outline

Information

Phys. Rev. Research 8, 033036– Published 10 July, 2026
  • Received 11 December 2025
  • Accepted 21 May 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