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Evolution of flat bands in a two-dimensional fused pentagon network

Tomonari Mizoguchi1,*, Mina Maruyama1,2, Yasuhiro Hatsugai1, and Susumu Okada1,2

  • *Contact author: mizoguchi@rhodia.ph.tsukuba.ac.jp
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Phys. Rev. B 112, 085421 – Published 25 August, 2025

DOI: https://doi.org/10.1103/2wy3-hgqn

Abstract

The theoretical quest for flat-band tight-binding models usually relies on lattice structures on which electrons reside. Typical examples of candidate lattice structures include the Lieb-type lattices and the line graphs. Meanwhile, there can be accidental flat-band systems that belong to neither of such typical classes and deriving flat-band energies and wave functions for such systems is not straightforward. In this work, we investigate the characteristic band structure for the tight-binding model on a network composed of pentagonal rings, which is inspired by the theoretically predicted carbon-based material. Although the lattice does not belong to conventional classes of flat-band models, the exact flat bands appear only for fine-tuned parameters. We analytically derive the exact eigenenergies and eigenstates of the flat bands. By using the analytic form of the Bloch wave function, we construct the corresponding Wannier function and reveal its characteristic real-space profile. We also find that, even away from the exact-flat-band limits, the nearly flat band exists near the Fermi level for the half-filled systems, which indicates that the present system will be a suitable platform for the quest for flat-band-induced correlated electron physics if it is realized in a real material. This is supported by the density functional theory calculations for the carbon-based materials where we find the ferromagnetic state.

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References (103)

  1. A. Mielke, Ferromagnetism in the Hubbard model on line graphs and further considerations, J. Phys. A: Math. Gen. 24, 3311 (1991).
  2. A. Mielke, Exact ground states for the Hubbard model on the kagome lattice, J. Phys. A: Math. Gen. 25, 4335 (1992).
  3. H. Tasaki, Ferromagnetism in the Hubbard models with degenerate single-electron ground states, Phys. Rev. Lett. 69, 1608 (1992).
  4. Z. Gulácsi, A. Kampf, and D. Vollhardt, Route to ferromagnetism in organic polymers, Phys. Rev. Lett. 105, 266403 (2010).
  5. V. A. Khodel and V. R. Shaginyan, Superfluidity in system with fermion condensate, JETP Lett. 51, 553 (1990).
  6. M. Imada and M. Kohno, Superconductivity from flat dispersion designed in doped Mott insulators, Phys. Rev. Lett. 84, 143 (2000).
  7. P. Esquinazi, T. T. Heikkilä, Y. V. Lysogorskiy, D. A. Tayurskii, and G. E. Volovik, On the superconductivity of graphite interfaces, JETP Lett. 100, 336 (2014).
  8. S. Peotta and P. Törmä, Superfluidity in topologically nontrivial flat bands, Nat. Commun. 6, 8944 (2015).
  9. T. T. Heikkilä and G. E. Volovik, Flat bands as a route to high-temperature superconductivity in graphite, in Basic Physics of Functionalized Graphite, edited by P. D. Esquinazi (Springer International Publishing, Cham, 2016), pp. 123–143.
  10. G. E. Volovik, Graphite, graphene, and the flat band superconductivity, JETP Lett. 107, 516 (2018).
  11. H. Aoki, Theoretical possibilities for flat band superconductivity, J. Supercond. Novel Magn. 33, 2341 (2020).
  12. K.-E. Huhtinen, J. Herzog-Arbeitman, A. Chew, B. A. Bernevig, and P. Törmä, Revisiting flat band superconductivity: Dependence on minimal quantum metric and band touchings, Phys. Rev. B 106, 014518 (2022).
  13. S. D. Huber and E. Altman, Bose condensation in flat bands, Phys. Rev. B 82, 184502 (2010).
  14. H. Katsura, N. Kawashima, S. Morita, A. Tanaka, and H. Tasaki, Mott-insulator-like Bose-Einstein condensation in a tight-binding system of interacting Bosons with a flat band, Phys. Rev. Res. 3, 033190 (2021).
  15. E. Tang, J.-W. Mei, and X.-G. Wen, High-temperature fractional quantum Hall states, Phys. Rev. Lett. 106, 236802 (2011).
  16. K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Nearly flatbands with nontrivial topology, Phys. Rev. Lett. 106, 236803 (2011).
  17. T. Neupert, L. Santos, C. Chamon, and C. Mudry, Fractional quantum Hall states at zero magnetic field, Phys. Rev. Lett. 106, 236804 (2011).
  18. Z. Gulácsi, A. Kampf, and D. Vollhardt, Exact many-electron ground states on diamond and triangle Hubbard chains, Prog. Theor. Phys. Suppl. 176, 1 (2008).
  19. Y. Kuno, T. Mizoguchi, and Y. Hatsugai, Flat band quantum scar, Phys. Rev. B 102, 241115(R) (2020).
  20. T. Mizoguchi, Y. Kuno, and Y. Hatsugai, Construction of interacting flat-band models by molecular-orbital representation: Correlation functions, energy gap, and entanglement, Prog. Theor. Exp. Phys. 2022, 023I02 (2022).
  21. M. Goda, S. Nishino, and H. Matsuda, Inverse Anderson transition caused by flatbands, Phys. Rev. Lett. 96, 126401 (2006).
  22. S. Nishino, H. Matsuda, and M. Goda, Flat-band localization in weakly disordered system, J. Phys. Soc. Jpn. 76, 024709 (2007).
  23. J. T. Chalker, T. S. Pickles, and P. Shukla, Anderson localization in tight-binding models with flat bands, Phys. Rev. B 82, 104209 (2010).
  24. D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyatnikov, Flat band states: Disorder and nonlinearity, Phys. Rev. B 88, 224203 (2013).
  25. P. Shukla, Disorder perturbed flat bands: Level density and inverse participation ratio, Phys. Rev. B 98, 054206 (2018).
  26. P. Shukla, Disorder perturbed flat bands. II. Search for criticality, Phys. Rev. B 98, 184202 (2018).
  27. T. Bilitewski and R. Moessner, Disordered flat bands on the kagome lattice, Phys. Rev. B 98, 235109 (2018).
  28. X. Mao, J. Liu, J. Zhong, and R. A. Römer, Disorder effects in the two-dimensional Lieb lattice and its extensions, Phys. E (Amsterdam, Neth.) 124, 114340 (2020).
  29. Y. Hatsugai, Revisiting flat bands and localization, Ann. Phys. 435, 168453 (2021).
  30. T. Kuroda, T. Mizoguchi, H. Araki, and Y. Hatsugai, Machine learning study on the flat-band states constructed by molecular-orbital representation with randomness, J. Phys. Soc. Jpn. 91, 044703 (2022).
  31. A. Nandy, Tunable caging of excitation in decorated Lieb-ladder geometry with long-range connectivity, Chin. Phys. B 32, 127201 (2023).
  32. T. Mizoguchi and Y. Hatsugai, Molecular-orbital representation with random U(1) variables, Phys. Rev. B 107, 094201 (2023).
  33. Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature (London) 556, 43 (2018).
  34. Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature (London) 556, 80 (2018).
  35. M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning superconductivity in twisted bilayer graphene, Science 363, 1059 (2019).
  36. J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene, Nature (London) 590, 249 (2021).
  37. B. R. Ortiz, L. C. Gomes, J. R. Morey, M. Winiarski, M. Bordelon, J. S. Mangum, I. W. H. Oswald, J. A. Rodriguez-Rivera, J. R. Neilson, S. D. Wilson, E. Ertekin, T. M. McQueen, and E. S. Toberer, New kagome prototype materials: Discovery of KV3Sb5,RbV3Sb5, and CsV3Sb5, Phys. Rev. Mater. 3, 094407 (2019).
  38. J.-X. Yin, S. S. Zhang, G. Chang, Q. Wang, S. S. Tsirkin, Z. Guguchia, B. Lian, H. Zhou, K. Jiang, I. Belopolski, N. Shumiya, D. Multer, M. Litskevich, T. A. Cochran, H. Lin, Z. Wang, T. Neupert, S. Jia, H. Lei, and M. Z. Hasan, Negative flat band magnetism in a spin-orbit-coupled correlated kagome magnet, Nat. Phys. 15, 443 (2019).
  39. M. Kang, S. Fang, L. Ye, H. C. Po, J. Denlinger, C. Jozwiak, A. Bostwick, E. Rotenberg, E. Kaxiras, J. G. Checkelsky, and R. Comin, Topological flat bands in frustrated kagome lattice CoSn, Nat. Commun. 11, 4004 (2020).
  40. Z. Sun, H. Zhou, C. Wang, S. Kumar, D. Geng, S. Yue, X. Han, Y. Haraguchi, K. Shimada, P. Cheng, L. Chen, Y. Shi, K. Wu, S. Meng, and B. Feng, Observation of topological flat bands in the kagome semiconductor Nb3Cl8, Nano Lett. 22, 4596 (2022).
  41. L. Ye, S. Fang, M. Kang, J. Kaufmann, Y. Lee, C. John, P. M. Neves, S. Y. F. Zhao, J. Denlinger, C. Jozwiak, A. Bostwick, E. Rotenberg, E. Kaxiras, D. C. Bell, O. Janson, R. Comin, and J. G. Checkelsky, Hopping frustration-induced flat band and strange metallicity in a kagome metal, Nat. Phys. 20, 610 (2024).
  42. A. Kumar, K. Banerjee, A. S. Foster, and P. Liljeroth, Two-dimensional band structure in honeycomb metal–organic frameworks, Nano Lett. 18, 5596 (2018).
  43. Y. Shuku, R. Suizu, S. Nakano, M. Tsuchiizu, and K. Awaga, Engineering Dirac cones and topological flat bands with organic molecules, Phys. Rev. B 107, 155123 (2023).
  44. R. Nemoto, R. Arafune, S. Nakano, M. Tsuchiizu, N. Takagi, R. Suizu, T. Uchihashi, and K. Awaga, Chiral honeycomb lattices of nonplanar π-conjugated supramolecules with protected Dirac and flat bands, ACS Nano 18, 19663 (2024).
  45. S. Taie, H. Ozawa, T. Ichinose, T. Nishio, S. Nakajima, and Y. Takahashi, Coherent driving and freezing of bosonic matter wave in an optical Lieb lattice, Sci. Adv. 1, e1500854 (2015).
  46. H. Ozawa, S. Taie, T. Ichinose, and Y. Takahashi, Interaction-driven shift and distortion of a flat band in an optical Lieb lattice, Phys. Rev. Lett. 118, 175301 (2017).
  47. J. H. Kang, J. H. Han, and Y. Shin, Realization of a cross-linked chiral ladder with neutral fermions in a 1D optical lattice by orbital-momentum coupling, Phys. Rev. Lett. 121, 150403 (2018).
  48. Y. He, R. Mao, H. Cai, J.-X. Zhang, Y. Li, L. Yuan, S.-Y. Zhu, and D.-W. Wang, Flat-band localization in Creutz superradiance lattices, Phys. Rev. Lett. 126, 103601 (2021).
  49. S. Xia, A. Ramachandran, S. Xia, D. Li, X. Liu, L. Tang, Y. Hu, D. Song, J. Xu, D. Leykam, S. Flach, and Z. Chen, Unconventional flatband line states in photonic Lieb lattices, Phys. Rev. Lett. 121, 263902 (2018).
  50. D. Leykam and S. Flach, Perspective: Photonic flatbands, APL Photonics 3, 070901 (2018).
  51. L. Tang, D. Song, S. Xia, S. Xia, J. Ma, W. Yan, Y. Hu, J. Xu, D. Leykam, and Z. Chen, Photonic flat-band lattices and unconventional light localization, Nanophotonics 9, 1161 (2020).
  52. J. Yang, Y. Li, Y. Yang, X. Xie, Z. Zhang, J. Yuan, H. Cai, D.-W. Wang, and F. Gao, Realization of all-band-flat photonic lattices, Nat. Commun. 15, 1484 (2024).
  53. C. Danieli, A. Andreanov, D. Leykam, and S. Flach, Flat band fine-tuning and its photonic applications, Nanophotonics 13, 3925 (2024).
  54. E. D. Benedetto, A. Gonzalez-Tudela, and F. Ciccarello, Dipole-dipole interactions mediated by a photonic flat band, Quantum 9, 1671 (2025).
  55. B. Sutherland, Localization of electronic wave functions due to local topology, Phys. Rev. B 34, 5208 (1986).
  56. E. H. Lieb, Two theorems on the Hubbard model, Phys. Rev. Lett. 62, 1201 (1989).
  57. X. Ni, J. Yan, and F. Liu, Electronic structures of a diagonally striped lattice: Multiple (N−1)-fold degenerate flat bands, Phys. Rev. B 102, 235117 (2020).
  58. S. Miyahara, K. Kubo, H. Ono, Y. Shimomura, and N. Furukawa, Flat-bands on partial line graphs: Systematic method for generating flat-band lattice structures, J. Phys. Soc. Jpn. 74, 1918 (2005).
  59. J. M. Lee, C. Geng, J. W. Park, M. Oshikawa, S.-S. Lee, H. W. Yeom, and G. Y. Cho, Stable flatbands, topology, and superconductivity of magic honeycomb networks, Phys. Rev. Lett. 124, 137002 (2020).
  60. H. Liu, G. Sethi, S. Meng, and F. Liu, Orbital design of flat bands in non-line-graph lattices via line-graph wave functions, Phys. Rev. B 105, 085128 (2022).
  61. T. Mizoguchi, Y. Gao, M. Maruyama, Y. Hatsugai, and S. Okada, Unconventional gapless semiconductor in an extended Martini lattice in covalent honeycomb materials, Phys. Rev. B 107, L121301 (2023).
  62. M. E. Zhitomirsky and H. Tsunetsugu, Exact low-temperature behavior of a kagomé antiferromagnet at high fields, Phys. Rev. B 70, 100403(R) (2004).
  63. D. L. Bergman, C. Wu, and L. Balents, Band touching from real-space topology in frustrated hopping models, Phys. Rev. B 78, 125104 (2008).
  64. T. Mizoguchi, M. Maruyama, S. Okada, and Y. Hatsugai, Flat bands and higher-order topology in polymerized triptycene: Tight-binding analysis on decorated star lattices, Phys. Rev. Mater. 3, 114201 (2019).
  65. T. Mizoguchi, H. Katsura, I. Maruyama, and Y. Hatsugai, Flat-band solutions in D-dimensional decorated diamond and pyrochlore lattices: Reduction to molecular problem, Phys. Rev. B 104, 035155 (2021).
  66. D. Heo, J. Lee, A. Zhang, and J.-W. Rhim, Flat bands in network superstructures of atomic chains, Photonics 10, 29 (2023).
  67. A. M. Marques, J. Mögerle, G. Pelegrí, S. Flannigan, R. G. Dias, and A. J. Daley, Kaleidoscopes of Hofstadter butterflies and Aharonov-Bohm caging from 2n-root topology in decorated square lattices, Phys. Rev. Res. 5, 023110 (2023).
  68. T. Misumi and H. Aoki, New class of flat-band models on tetragonal and hexagonal lattices: Gapped versus crossing flat bands, Phys. Rev. B 96, 155137 (2017).
  69. C.-C. Lee, A. Fleurence, Y. Yamada-Takamura, and T. Ozaki, Hidden mechanism for embedding the flat bands of Lieb, kagome, and checkerboard lattices in other structures, Phys. Rev. B 100, 045150 (2019).
  70. T. Ogata, M. Kawamura, and T. Ozaki, Methods for constructing parameter-dependent flat-band lattices, Phys. Rev. B 103, 205119 (2021).
  71. B. Pal and G. Bouzerar, Engineering flux-controlled flat bands and topological states in a Stagome lattice, J. Phys.: Condens. Matter 37, 135503 (2025).
  72. M. Maruyama and S. Okada, Two-dimensional sp2 carbon network of fused pentagons: All carbon ferromagnetic sheet, Appl. Phys. Express 6, 095101 (2013).
  73. M. Maruyama and S. Okada, Two-dimensional sp2 carbon networks of fused pentagons, Jpn. J. Appl. Phys. 53, 06JD02 (2014).
  74. Note that the calculated lattice structure (i.e., the optimal position of the carbon atom) is slightly different from Fig. 1, but the connectivity of the bonds is equivalent.
  75. L. Zhu, Q. Xue, X. Li, Y. Jin, H. Zheng, T. Wu, and Q. Guo, Theoretical prediction of hydrogen separation performance of two-dimensional carbon network of fused pentagon, ACS Appl. Mater. Interfaces 7, 28502 (2015).
  76. T. Zeng, Q. Chen, J. Guo, and H. Wang, Fused pentagon carbon network: A new anode material for Li ion batteries, Chem. Phys. Lett. 745, 137225 (2020).
  77. Y. Niu, K. Meng, T. Xu, J. Wang, X. Xiao, J. Rong, X. Yu, Y. Zhang, and Y. Wei, Novel two-dimensional fused pentagon monolayer as a separation membrane with high desalination properties, Diam. Relat. Mater. 140, 110448 (2023).
  78. A. Rüegg, J. Wen, and G. A. Fiete, Topological insulators on the decorated honeycomb lattice, Phys. Rev. B 81, 205115 (2010).
  79. H.-F. Lin, Y.-H. Chen, H.-D. Liu, H.-S. Tao, and W.-M. Liu, Mott transition and antiferromagnetism of cold fermions in the decorated honeycomb lattice, Phys. Rev. A 90, 053627 (2014).
  80. M. F. López and J. Merino, From quantum anomalous Hall phases to topological metals in interacting decorated honeycomb lattices, Phys. Rev. B 100, 075154 (2019).
  81. Y. Hatsugai and I. Maruyama, ZQ topological invariants for polyacetylene, kagome and pyrochlore lattices, Europhys. Lett. 95, 20003 (2011).
  82. T. Mizoguchi and Y. Hatsugai, Molecular-orbital representation of generic flat-band models, Europhys. Lett. 127, 47001 (2019).
  83. T. Mizoguchi, Y. Kuno, and Y. Hatsugai, Flat band, spin-1 Dirac cone, and Hofstadter diagram in the fermionic square kagome model, Phys. Rev. B 104, 035161 (2021).
  84. T. Mizoguchi and M. Udagawa, Flat-band engineering in tight-binding models: Beyond the nearest-neighbor hopping, Phys. Rev. B 99, 235118 (2019).
  85. J. M. Luttinger and L. Tisza, Theory of dipole interaction in crystals, Phys. Rev. 70, 954 (1946).
  86. J. N. Reimers, A. J. Berlinsky, and A.-C. Shi, Mean-field approach to magnetic ordering in highly frustrated pyrochlores, Phys. Rev. B 43, 865 (1991).
  87. D. A. Garanin and B. Canals, Classical spin liquid: Exact solution for the infinite-component antiferromagnetic model on the kagomé lattice, Phys. Rev. B 59, 443 (1999).
  88. S. V. Isakov, K. Gregor, R. Moessner, and S. L. Sondhi, Dipolar spin correlations in classical pyrochlore magnets, Phys. Rev. Lett. 93, 167204 (2004).
  89. K. Essafi, L. D. C. Jaubert, and M. Udagawa, Flat bands and Dirac cones in breathing lattices, J. Phys.: Condens. Matter 29, 315802 (2017).
  90. T. Mizoguchi, L. D. C. Jaubert, R. Moessner, and M. Udagawa, Magnetic clustering, half-moons, and shadow pinch points as signals of a proximate Coulomb phase in frustrated Heisenberg magnets, Phys. Rev. B 98, 144446 (2018).
  91. F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett. 61, 2015 (1988).
  92. T. Fukui, Y. Hatsugai, and H. Suzuki, Chern numbers in discretized Brillouin zone: Efficient method of computing (spin) Hall conductances, J. Phys. Soc. Jpn. 74, 1674 (2005).
  93. F. Wang and Y. Ran, Nearly flat band with Chern number C=2 on the dice lattice, Phys. Rev. B 84, 241103(R) (2011).
  94. D. Sticlet, F. Piéchon, J.-N. Fuchs, P. Kalugin, and P. Simon, Geometrical engineering of a two-band Chern insulator in two dimensions with arbitrary topological index, Phys. Rev. B 85, 165456 (2012).
  95. S. Yang, Z.-C. Gu, K. Sun, and S. Das Sarma, Topological flat band models with arbitrary Chern numbers, Phys. Rev. B 86, 241112(R) (2012).
  96. Y.-F. Wang, H. Yao, C.-D. Gong, and D. N. Sheng, Fractional quantum Hall effect in topological flat bands with Chern number two, Phys. Rev. B 86, 201101(R) (2012).
  97. M. Trescher and E. J. Bergholtz, Flat bands with higher Chern number in pyrochlore slabs, Phys. Rev. B 86, 241111(R) (2012).
  98. W. A. Benalcazar, T. Li, and T. L. Hughes, Quantization of fractional corner charge in Cn-symmetric higher-order topological crystalline insulators, Phys. Rev. B 99, 245151 (2019).
  99. E. Lee, A. Furusaki, and B.-J. Yang, Fractional charge bound to a vortex in two-dimensional topological crystalline insulators, Phys. Rev. B 101, 241109(R) (2020).
  100. F. Zangeneh-Nejad and R. Fleury, Nonlinear second-order topological insulators, Phys. Rev. Lett. 123, 053902 (2019).
  101. T. Mizoguchi, H. Araki, and Y. Hatsugai, Higher-order topological phase in a Honeycomb-Lattice model with anti-Kekulé distortion, J. Phys. Soc. Jpn. 88, 104703 (2019).
  102. S. E. Freeney, J. J. van den Broeke, A. J. J. Harsveld van der Veen, I. Swart, and C. Morais Smith, Edge-dependent topology in Kekulé lattices, Phys. Rev. Lett. 124, 236404 (2020).
  103. M. Bunney, T. Mizoguchi, Y. Hatsugai, and S. Rachel, Competition of first-order and second-order topology on the honeycomb lattice, Phys. Rev. B 105, 045113 (2022).

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Phys. Rev. B 112, 085421– Published 25 August, 2025
  • Received 11 December 2024
  • Revised 9 April 2025
  • Accepted 16 July 2025
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