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Classification of spin-12 fermionic quantum spin liquids on the trillium lattice

Ming-Hao Li1, Sounak Biswas1,2, and S. A. Parameswaran1

  • 1Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford OX1 3PU, United Kingdom
  • 2Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany

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Phys. Rev. B 112, 104429 – Published 19 September, 2025

DOI: https://doi.org/10.1103/3nmp-1vt2

Abstract

We study fermionic quantum spin liquids (QSLs) on the three-dimensional trillium lattice of corner-sharing triangles. We are motivated by recent experimental and theoretical investigations that have explored various classical and quantum spin liquid states on similar networks of triangular motifs with strong geometric frustration. Using the framework of projective symmetry groups (PSG), we obtain a classification of all symmetric Z2 and U(1) QSLs on the trillium lattice. We find two Z2 spin-liquids, and a single U(1) spin-liquid that is proximate to one of the Z2 states. The small number of solutions reflects the constraints imposed by the nonsymmorphic symmetries in the space group of the trillium lattice. Using self-consistency conditions of the mean-field equations, we obtain the spinon band-structure and spin structure factors corresponding to these states. All three of our spin liquids are gapless at their saddle points: one of the two Z2 QSLs is nodal, while the U(1) case hosts a spinon Fermi surface. One of our Z2 spin liquids hosts a stable gapless nodal star that is protected by projective symmetries against additions of further neighbor terms in the mean-field ansatz. We comment on directions for further work.

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References (76)

  1. C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Quantum spin liquids, Science 367, eaay0668 (2020).
  2. L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2017).
  3. R. Moessner, Magnets with strong geometric frustration, Can. J. Phys. 79, 1283 (2001).
  4. J. T. Chalker, Spin Liquids and Frustrated Magnetism (Oxford University Press, Oxford, 2017).
  5. R. Moessner and J. T. Chalker, Low-temperature properties of classical geometrically frustrated antiferromagnets, Phys. Rev. B 58, 12049 (1998).
  6. 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).
  7. S. V. Isakov, K. Gregor, R. Moessner, and S. L. Sondhi, Dipolar spin correlations in classical pyrochlore magnets, Phys. Rev. Lett. 93, 167204 (2004).
  8. B. Canals and C. Lacroix, Mean-field study of the disordered ground state in the β−Mn lattice, Phys. Rev. B 61, 11251 (2000).
  9. S. V. Isakov, J. M. Hopkinson, and H.-Y. Kee, Fate of partial order on trillium and distorted windmill lattices, Phys. Rev. B 78, 014404 (2008).
  10. J. M. Hopkinson, S. V. Isakov, H.-Y. Kee, and Y. B. Kim, Classical antiferromagnet on a hyperkagome lattice, Phys. Rev. Lett. 99, 037201 (2007).
  11. J. Villain, R. Bidaux, J. P. Carton, and C. R, Order as an effect of disorder, J. Phys. (Paris) 41, 1263 (1980).
  12. J. T. Chalker, P. C. W. Holdsworth, and E. F. Shender, Hidden order in a frustrated system: Properties of the Heisenberg kagomé antiferromagnet, Phys. Rev. Lett. 68, 855 (1992).
  13. X.-G. Wen, Quantum orders and symmetric spin liquids, Phys. Rev. B 65, 165113 (2002).
  14. G. Baskaran and P. W. Anderson, Gauge theory of high-temperature superconductors and strongly correlated Fermi systems, Phys. Rev. B 37, 580 (1988).
  15. G. Baskaran, Z. Zou, and P. Anderson, The resonating valence bond state and high-Tc superconductivity—A mean field theory, Solid State Commun. 63, 973 (1987).
  16. I. Affleck, Z. Zou, T. Hsu, and P. W. Anderson, Su(2) gauge symmetry of the large-U limit of the Hubbard model, Phys. Rev. B 38, 745 (1988).
  17. E. Dagotto, E. Fradkin, and A. Moreo, SU(2) gauge invariance and order parameters in strongly coupled electronic systems, Phys. Rev. B 38, 2926 (1988).
  18. X.-G. Wen and P. A. Lee, Theory of underdoped cuprates, Phys. Rev. Lett. 76, 503 (1996).
  19. X. G. Wen, Mean-field theory of spin-liquid states with finite energy gap and topological orders, Phys. Rev. B 44, 2664 (1991).
  20. Y. Zhou and X.-G. Wen, Quantum orders and spin liquids in Cs2CuCl4, arXiv:cond-mat/0210662.
  21. Y.-M. Lu, Symmetric Z2 spin liquids and their neighboring phases on triangular lattice, Phys. Rev. B 93, 165113 (2016).
  22. Y.-M. Lu and Y. Ran, Z2 spin liquid and chiral antiferromagnetic phase in the Hubbard model on a honeycomb lattice, Phys. Rev. B 84, 024420 (2011).
  23. Y.-Z. You, I. Kimchi, and A. Vishwanath, Doping a spin-orbit Mott insulator: Topological superconductivity from the Kitaev-Heisenberg model and possible application to (Na2/Li2)IrO3, Phys. Rev. B 86, 085145 (2012).
  24. Y.-M. Lu, Y. Ran, and P. A. Lee, Z2 spin liquids in the S=12 Heisenberg model on the kagome lattice: A projective symmetry-group study of Schwinger fermion mean-field states, Phys. Rev. B 83, 224413 (2011).
  25. S. Bieri, L. Messio, B. Bernu, and C. Lhuillier, Gapless chiral spin liquid in a kagome Heisenberg model, Phys. Rev. B 92, 060407(R) (2015).
  26. S. Bieri, C. Lhuillier, and L. Messio, Projective symmetry group classification of chiral spin liquids, Phys. Rev. B 93, 094437 (2016).
  27. M. J. Lawler, A. Paramekanti, Y. B. Kim, and L. Balents, Gapless spin liquids on the three-dimensional hyperkagome lattice of Na4Ir3O8, Phys. Rev. Lett. 101, 197202 (2008).
  28. B. Huang, Y. B. Kim, and Y.-M. Lu, Interplay of nonsymmorphic symmetry and spin-orbit coupling in hyperkagome spin liquids: Applications to Na4Ir3O8, Phys. Rev. B 95, 054404 (2017).
  29. L. E. Chern and Y. B. Kim, Theoretical study of quantum spin liquids in S=12 hyper-hyperkagome magnets: Classification, heat capacity, and dynamical spin structure factor, Phys. Rev. B 104, 094413 (2021).
  30. J. Sonnenschein, A. Chauhan, Y. Iqbal, and J. Reuther, Projective symmetry group classifications of quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic lattices, Phys. Rev. B 102, 125140 (2020).
  31. A. Maity, F. Ferrari, R. Thomale, S. Mandal, and Y. Iqbal, Projective symmetry group classification of Abrikosov fermion mean-field ansätze on the square-octagon lattice, Phys. Rev. B 107, 134438 (2023).
  32. A. Chauhan, A. Maity, C. Liu, J. Sonnenschein, F. Ferrari, and Y. Iqbal, Quantum spin liquids on the diamond lattice, Phys. Rev. B 108, 134424 (2023).
  33. L. E. Chern, Y. B. Kim, and C. Castelnovo, Competing quantum spin liquids, gauge fluctuations, and anisotropic interactions in a breathing pyrochlore lattice, Phys. Rev. B 106, 134402 (2022).
  34. C. Liu, G. B. Halász, and L. Balents, Symmetric U(1) and Z2 spin liquids on the pyrochlore lattice, Phys. Rev. B 104, 054401 (2021).
  35. B. Huang, W. Choi, Y. B. Kim, and Y.-M. Lu, Classification and properties of quantum spin liquids on the hyperhoneycomb lattice, Phys. Rev. B 97, 195141 (2018).
  36. L. Capriotti, F. Becca, A. Parola, and S. Sorella, Resonating valence bond wave functions for strongly frustrated spin systems, Phys. Rev. Lett. 87, 097201 (2001).
  37. Y. Iqbal, F. Becca, and D. Poilblanc, Projected wave function study of Z2 spin liquids on the kagome lattice for the spin-12 quantum Heisenberg antiferromagnet, Phys. Rev. B 84, 020407(R) (2011).
  38. Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Gapless spin-liquid phase in the kagome spin-12 Heisenberg antiferromagnet, Phys. Rev. B 87, 060405(R) (2013).
  39. Y. Iqbal, D. Poilblanc, and F. Becca, Vanishing spin gap in a competing spin-liquid phase in the kagome Heisenberg antiferromagnet, Phys. Rev. B 89, 020407(R) (2014).
  40. R. Shindou, S. Yunoki, and T. Momoi, Projective studies of spin nematics in a quantum frustrated ferromagnet, Phys. Rev. B 84, 134414 (2011).
  41. W.-J. Hu, F. Becca, A. Parola, and S. Sorella, Direct evidence for a gapless Z2 spin liquid by frustrating Néel antiferromagnetism, Phys. Rev. B 88, 060402(R) (2013).
  42. F. Ferrari and F. Becca, Spectral signatures of fractionalization in the frustrated Heisenberg model on the square lattice, Phys. Rev. B 98, 100405(R) (2018).
  43. Z.-X. Liu, Y. Zhou, and T.-K. Ng, Fermionic theory for quantum antiferromagnets with spin s>12, Phys. Rev. B 82, 144422 (2010).
  44. S. Bieri, M. Serbyn, T. Senthil, and P. A. Lee, Paired chiral spin liquid with a Fermi surface in S=1 model on the triangular lattice, Phys. Rev. B 86, 224409 (2012).
  45. A. Maity, R. Thomale, and Y. Iqbal, Projective symmetry group classification of quantum spin liquid mean field ansatze in spin-1 diamond lattice (unpublished).
  46. F. Wang and A. Vishwanath, Spin-liquid states on the triangular and kagomé lattices: A projective-symmetry-group analysis of Schwinger boson states, Phys. Rev. B 74, 174423 (2006).
  47. F. Wang, Schwinger boson mean field theories of spin liquid states on a honeycomb lattice: Projective symmetry group analysis and critical field theory, Phys. Rev. B 82, 024419 (2010).
  48. H.-K. Jin and Y. Zhou, Classical and quantum order in hyperkagome antiferromagnets, Phys. Rev. B 101, 054408 (2020).
  49. L. Messio, S. Bieri, C. Lhuillier, and B. Bernu, Chiral spin liquid on a kagome antiferromagnet induced by the Dzyaloshinskii-Moriya interaction, Phys. Rev. Lett. 118, 267201 (2017).
  50. J. M. Hopkinson and H.-Y. Kee, Geometric frustration inherent to the trillium lattice, a sublattice of the B20 structure, Phys. Rev. B 74, 224441 (2006).
  51. I. Živković, V. Favre, C. Salazar Mejia, H. O. Jeschke, A. Magrez, B. Dabholkar, V. Noculak, R. S. Freitas, M. Jeong, N. G. Hegde, L. Testa, P. Babkevich, Y. Su, P. Manuel, H. Luetkens, C. Baines, P. J. Baker, J. Wosnitza, O. Zaharko, Y. Iqbal, J. Reuther, and H. M. Rønnow, Magnetic field induced quantum spin liquid in the two coupled trillium lattices of K2Ni2(SO4)3, Phys. Rev. Lett. 127, 157204 (2021).
  52. M. G. Gonzalez, V. Noculak, A. Sharma, V. Favre, J.-R. Soh, A. Magrez, R. Bewley, H. O. Jeschke, J. Reuther, H. M. Rønnow, Y. Iqbal, and I. Živković, Dynamics of K2Ni2(SO4)3 governed by proximity to a 3D spin liquid model, Nat. Commun. 15, 7191 (2024).
  53. W. Yao, Q. Huang, T. Xie, A. Podlesnyak, A. Brassington, C. Xing, R. S. D. Mudiyanselage, W. Xie, S. Zhang, M. Lee, V. S. Zapf, X. Bai, D. A. Tennant, J. Liu, and H. Zhou, Continuous spin excitations in the three-dimensional frustrated magnet K2Ni2(SO4)3, Phys. Rev. Lett. 131, 146701 (2023).
  54. K. Boya, K. Nam, K. Kargeti, A. Jain, R. Kumar, S. Panda, S. Yusuf, P. Paulose, U. Voma et al., Signatures of spin-liquid state in a 3D frustrated lattice compound KSrFe2 (PO4) 3 with S= 5/2, APL Mater. 10, 101103 (2022).
  55. B. Koteswararao, R. Kumar, P. Khuntia, S. Bhowal, S. K. Panda, M. R. Rahman, A. V. Mahajan, I. Dasgupta, M. Baenitz, K. H. Kim, and F. C. Chou, Magnetic properties and heat capacity of the three-dimensional frustrated S=12 antiferromagnet PbCuTe2O6, Phys. Rev. B 90, 035141 (2014).
  56. S. Chillal, Y. Iqbal, H. O. Jeschke, J. A. Rodriguez-Rivera, R. Bewley, P. Manuel, D. Khalyavin, P. Steffens, R. Thomale, A. T. M. N. Islam, J. Reuther, and B. Lake, Evidence for a three-dimensional quantum spin liquid in PbCuTe2O6, Nat. Commun. 11, 2348 (2020).
  57. P. Khuntia, F. Bert, P. Mendels, B. Koteswararao, A. V. Mahajan, M. Baenitz, F. C. Chou, C. Baines, A. Amato, and Y. Furukawa, Spin liquid state in the 3D frustrated antiferromagnet PbCuTe2O6: NMR and muon spin relaxation studies, Phys. Rev. Lett. 116, 107203 (2016).
  58. F. Wang, A. Vishwanath, and Y. B. Kim, Quantum and classical spins on the spatially distorted kagome lattice: Applications to volborthite Cu3V2O7(OH)2·2H2O, Phys. Rev. B 76, 094421 (2007).
  59. S. Biswas and K. Damle, Semiclassical theory for liquidlike behavior of the frustrated magnet Ca10Cr7O28, Phys. Rev. B 97, 115102 (2018).
  60. S. Biswas, Y. H. Kwan, and S. A. Parameswaran, Beyond the freshman's dream: Classical fractal spin liquids from matrix cellular automata in three-dimensional lattice models, Phys. Rev. B 105, 224410 (2022).
  61. M. J. Lawler, H.-Y. Kee, Y. B. Kim, and A. Vishwanath, Topological spin liquid on the hyperkagome lattice of Na4Ir3O8, Phys. Rev. Lett. 100, 227201 (2008).
  62. Y. Zhou, P. A. Lee, T.-K. Ng, and F.-C. Zhang, Na4Ir3O8 as a 3D spin liquid with fermionic spinons, Phys. Rev. Lett. 101, 197201 (2008).
  63. Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Takagi, Spin-liquid state in the S=1/2 hyperkagome antiferromagnet Na4Ir3O8, Phys. Rev. Lett. 99, 137207 (2007).
  64. Y. Singh, Y. Tokiwa, J. Dong, and P. Gegenwart, Spin liquid close to a quantum critical point in Na4Ir3O8, Phys. Rev. B 88, 220413(R) (2013).
  65. R. Dally, T. Hogan, A. Amato, H. Luetkens, C. Baines, J. Rodriguez-Rivera, M. J. Graf, and S. D. Wilson, Short-range correlations in the magnetic ground state of Na4Ir3O8, Phys. Rev. Lett. 113, 247601 (2014).
  66. A. C. Shockley, F. Bert, J.-C. Orain, Y. Okamoto, and P. Mendels, Frozen state and spin liquid physics in Na4Ir3O8: An NMR study, Phys. Rev. Lett. 115, 047201 (2015).
  67. M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, and H. Wondratschek, Bilbao crystallographic server I: Databases and crystallographic computing programs, Z. Kristallograp. 221, 15 (2006).
  68. M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, and H. Wondratschek, Bilbao crystallographic server II: Representations of crystallographic point groups and space groups, Acta Cryst. A62, 151 (2006).
  69. M. I. Aroyo, ed., International Tables for Crystallography, Volume A: Space-group Symmetry, 6th ed. (Wiley, Hoboken, NJ, 2016).
  70. GAP, GAP – Groups, Algorithms, and Programming, Version 4.12.2, The GAP Group (2022).
  71. L. Messio, C. Lhuillier, and G. Misguich, Time reversal symmetry breaking chiral spin liquids: Projective symmetry group approach of bosonic mean-field theories, Phys. Rev. B 87, 125127 (2013).
  72. B. Schneider, J. C. Halimeh, and M. Punk, Projective symmetry group classification of chiral Z2 spin liquids on the pyrochlore lattice: Application to the spin-12 XXZ Heisenberg model, Phys. Rev. B 105, 125122 (2022).
  73. P. K. Mogensen, K. Carlsson, S. Villemot, S. Lyon, M. Gomez, C. Rackauckas, T. Holy, D. Widmann, T. Kelman, D. Karrasch et al., https://github.com/JuliaNLSolvers/NLsolve.jl (2020).
  74. J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A fresh approach to numerical computing, SIAM Rev. 59, 65 (2017).
  75. F. J. Burnell, S. Chakravarty, and S. L. Sondhi, Monopole flux state on the pyrochlore lattice, Phys. Rev. B 79, 144432 (2009).
  76. R. V. Mishmash, J. R. Garrison, S. Bieri, and C. Xu, Theory of a competitive spin liquid state for weak Mott insulators on the triangular lattice, Phys. Rev. Lett. 111, 157203 (2013).

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Phys. Rev. B 112, 104429– Published 19 September, 2025
  • Received 14 November 2024
  • Revised 9 June 2025
  • Accepted 17 July 2025
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