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Mapping the phase diagram of a frustrated magnet: Flat bands and diabolical locus from irreducible representation collisions on the pyrochlore lattice

Kristian Tyn Kai Chung*

  • *Contact author: ktchung.phys@gmail.com
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Phys. Rev. B 114, 084416 – Published 19 August, 2026

DOI: https://doi.org/10.1103/h7x2-l1z8

Abstract

We map the complete classical phase diagram of the spin Hamiltonian describing pyrochlore rare-earth magnets with all symmetry-allowed nearest-neighbor bond-dependent anisotropic two-spin interactions. We provide a simple derivation of the organization of spins into tensor degrees of freedom describing the multipole moments of a tetrahedron, whose components correspond to irreducible representations (irreps) of the tetrahedral symmetry group Td. By parametrizing the Hamiltonian directly in terms of the energies of the individual irreps, we perform an exhaustive search of all possible irrep degeneracies (collisions) which may host stable classical spin liquids arising at classical multicritical points. We find four one-parameter families of models along which three phases are degenerate, all four of which merge at the Heisenberg antiferromagnetic point and its dual, and we give a complete three-dimensional picture of the phase diagram showing all of the phases and their intersections. The appearance of two copies of a single irrep implies an extra degenerate locus which pierces the phase boundaries at special points, yielding two additional isolated triple points. We demonstrate that one-parameter families of Hamiltonians are characterized by a topological invariant describing how the ground-state spin configuration winds when adiabatically transported around this degenerate locus, analogous to a diabolical locus in quantum field theory. We provide a comprehensive catalog of all flat-band degeneracies in the phase diagram and discuss the mechanisms that may allow for or impede the realization of a variety of classical spin liquids described by tensor gauge fields exhibiting nodal line singularities and concomitant multifold pinch points. Last, we provide a list of all cases where three irreps are degenerate above the ground state, which may lead to interesting features in the spin wave spectrum within each ordered phase.

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References (151)

  1. T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Deconfined quantum critical points, Science 303, 1490 (2004).
  2. T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and Matthew P. A. Fisher, Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm, Phys. Rev. B 70, 144407 (2004).
  3. M. Hermele, T. Senthil, and Matthew P. A. Fisher, Algebraic spin liquid as the mother of many competing orders, Phys. Rev. B 72, 104404 (2005).
  4. L. Balents, L. Bartosch, A. Burkov, S. Sachdev, and K. Sengupta, Competing orders and non-Landau-Ginzburg-Wilson criticality in (Bose) Mott transitions, Prog. Theor. Phys. Suppl. 160, 314 (2005).
  5. T. Senthil and Matthew P. A. Fisher, Competing orders, nonlinear sigma models, and topological terms in quantum magnets, Phys. Rev. B 74, 064405 (2006).
  6. C. Wang, A. Nahum, M. A. Metlitski, C. Xu, and T. Senthil, Deconfined quantum critical points: Symmetries and dualities, Phys. Rev. X 7, 031051 (2017).
  7. X.-Y. Song, C. Wang, A. Vishwanath, and Y.-C. He, Unifying description of competing orders in two-dimensional quantum magnets, Nat. Commun. 10, 4254 (2019).
  8. C. Liu, G. B. Halász, and L. Balents, Competing orders in pyrochlore magnets from a Z2 spin liquid perspective, Phys. Rev. B 100, 075125 (2019).
  9. L. Zou, Y.-C. He, and C. Wang, Stiefel liquids: Possible non-Lagrangian quantum criticality from intertwined orders, Phys. Rev. X 11, 031043 (2021).
  10. in Introduction to Frustrated Magnetism: Materials, Experiments, Theory, Springer Series in Solid-State Sciences, edited by C. Lacroix, P. Mendels, and F. Mila (Springer, Berlin, Heidelberg, 2011), Vol. 164.
  11. in Frustrated Spin Systems, edited by H. T. Diep, 3rd ed. (World Scientific Publishing Co. Pte. Ltd., Singapore, 2020).
  12. L. Balents, Spin liquids in frustrated magnets, Nature (London) 464, 199 (2010).
  13. C. Lhuillier and G. Misguich, Introduction to quantum spin liquids, in Introduction to Frustrated Magnetism: Materials, Experiments, Theory, Springer Series in Solid-State Sciences, edited by C. Lacroix, P. Mendels, and F. Mila (Springer, Berlin, Heidelberg, 2011), pp. 23–41.
  14. L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2017).
  15. Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89, 025003 (2017).
  16. J. Wen, S.-L. Yu, S. Li, W. Yu, and J.-X. Li, Experimental identification of quantum spin liquids, npj Quantum Mater. 4, 12 (2019).
  17. J. Knolle and R. Moessner, A field guide to spin liquids, Annu. Rev. Condens. Matter Phys. 10, 451 (2019).
  18. C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Quantum spin liquids, Science 367, eaay0668 (2020).
  19. M. A. Subramanian, G. Aravamudan, and G. V. Subba Rao, Oxide pyrochlores—A review, Prog. Solid State Chem. 15, 55 (1983).
  20. J. S. Gardner, Michel J. P. Gingras, and J. E. Greedan, Magnetic pyrochlore oxides, Rev. Mod. Phys. 82, 53 (2010).
  21. S.-H. Lee, H. Takagi, D. Louca, M. Matsuda, S. Ji, H. Ueda, Y. Ueda, T. Katsufuji, J.-H. Chung, S. Park, S.-W. Cheong, and C. Broholm, Frustrated magnetism and cooperative phase transitions in spinels, J. Phys. Soc. Jpn. 79, 011004 (2010).
  22. C. R. Wiebe and A. M. Hallas, Frustration under pressure: Exotic magnetism in new pyrochlore oxides, APL Mater. 3, 041519 (2015).
  23. P. Ghosh, Y. Iqbal, T. Müller, R. T. Ponnaganti, R. Thomale, R. Narayanan, J. Reuther, M. J. P. Gingras, and H. O. Jeschke, Breathing chromium spinels: A showcase for a variety of pyrochlore Heisenberg Hamiltonians, npj Quantum Mater. 4, 63 (2019).
  24. D. Reig-i-Plessis and A. M. Hallas, Frustrated magnetism in fluoride and chalcogenide pyrochlore lattice materials, Phys. Rev. Mater. 5, 030301 (2021).
  25. J. G. Rau and M. J. Gingras, Frustrated quantum rare-earth pyrochlores, Annu. Rev. Condens. Matter Phys. 10, 357 (2019).
  26. R. Moessner and J. T. Chalker, Low-temperature properties of classical geometrically frustrated antiferromagnets, Phys. Rev. B 58, 12049 (1998).
  27. C. L. Henley, Power-law spin correlations in pyrochlore antiferromagnets, Phys. Rev. B 71, 014424 (2005).
  28. C. L. Henley, The “Coulomb phase” in frustrated systems, Annu. Rev. Condens. Matter Phys. 1, 179 (2010).
  29. in Spin Ice, edited by M. Udagawa and L. Jaubert, Springer Series in Solid-State Sciences (Springer International Publishing, Cham, 2021), Vol. 197.
  30. C. Castelnovo, R. Moessner, and S. Sondhi, Spin ice, fractionalization, and topological order, Annu. Rev. Condens. Matter Phys. 3, 35 (2012).
  31. M. Hermele, Matthew P. A. Fisher, and L. Balents, Pyrochlore photons: The U(1) spin liquid in a S = 1/2 three-dimensional frustrated magnet, Phys. Rev. B 69, 064404 (2004).
  32. O. Benton, O. Sikora, and N. Shannon, Seeing the light: Experimental signatures of emergent electromagnetism in a quantum spin ice, Phys. Rev. B 86, 075154 (2012).
  33. L. Savary and L. Balents, Coulombic quantum liquids in spin-1/2 pyrochlores, Phys. Rev. Lett. 108, 037202 (2012).
  34. M. J. P. Gingras and P. A. McClarty, Quantum spin ice: A search for gapless quantum spin liquids in pyrochlore magnets, Rep. Prog. Phys. 77, 056501 (2014).
  35. X.-G. Wen, Quantum orders and symmetric spin liquids, Phys. Rev. B 65, 165113 (2002).
  36. E. M. Smith, O. Benton, D. R. Yahne, B. Placke, R. Schäfer, J. Gaudet, J. Dudemaine, A. Fitterman, J. Beare, A. R. Wildes, S. Bhattacharya, T. DeLazzer, C. R. C. Buhariwalla, N. P. Butch, R. Movshovich, J. D. Garrett, C. A. Marjerrison, J. P. Clancy, E. Kermarrec, G. M. Luke, et al., Case for a U(1)-pi quantum spin liquid ground state in the dipole-octupole pyrochlore Ce2Zr2O7, Phys. Rev. X 12, 021015 (2022).
  37. V. Porée, H. Yan, F. Desrochers, S. Petit, E. Lhotel, M. Appel, J. Ollivier, Y. B. Kim, A. H. Nevidomskyy, and R. Sibille, Evidence for fractional matter coupled to an emergent gauge field in a quantum spin ice, Nat. Phys. 21, 83 (2025).
  38. B. Gao, F. Desrochers, D. W. Tam, D. M. Kirschbaum, P. Steffens, A. Hiess, D. H. Nguyen, Y. Su, S.-W. Cheong, S. Paschen, Y. B. Kim, and P. Dai, Neutron scattering and thermodynamic evidence for emergent photons and fractionalization in a pyrochlore spin ice, Nat. Phys. 21, 1203 (2025).
  39. E. M. Smith, E. Lhotel, S. Petit, and B. D. Gaulin, Experimental insights into quantum spin ice physics in dipole-octupole pyrochlore magnets, Annu. Rev. Condens. Matter Phys. 16, 387 (2025).
  40. A. Bertin, Y. Chapuis, P. D. de Réotier, and A. Yaouanc, Crystal electric field in the R2Ti2O7 pyrochlore compounds, J. Phys.: Condens. Matter 24, 256003 (2012).
  41. J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Spin-orbit physics giving rise to novel phases in correlated systems: Iridates and related materials, Annu. Rev. Condens. Matter Phys. 7, 195 (2016).
  42. K. Essafi, O. Benton, and L. D. C. Jaubert, Generic nearest-neighbor kagome model: XYZ and Dzyaloshinskii-Moriya couplings with comparison to the pyrochlore-lattice case, Phys. Rev. B 96, 205126 (2017).
  43. P. A. Maksimov, Z. Zhu, S. R. White, and A. L. Chernyshev, Anisotropic-exchange magnets on a triangular lattice: Spin waves, accidental degeneracies, and dual spin liquids, Phys. Rev. X 9, 021017 (2019).
  44. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. January Special Issue 321, 2 (2006).
  45. 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).
  46. H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Concept and realization of Kitaev quantum spin liquids, Nat. Rev. Phys. 1, 264 (2019).
  47. M. Taillefumier, O. Benton, H. Yan, L. D. C. Jaubert, and N. Shannon, Competing spin liquids and hidden spin-nematic order in spin ice with frustrated transverse exchange, Phys. Rev. X 7, 041057 (2017).
  48. N. Francini, L. Janssen, and D. Lozano-Gómez, Higher-rank spin liquids and spin nematics from competing orders in pyrochlore magnets, Phys. Rev. B 111, 085140 (2025).
  49. O. Benton, L. D. C. Jaubert, H. Yan, and N. Shannon, A spin-liquid with pinch-line singularities on the pyrochlore lattice, Nat. Commun. 7, 11572 (2016).
  50. H. Yan, O. Benton, L. D. C. Jaubert, and N. Shannon, Rank-2 U(1) spin liquid on the breathing pyrochlore lattice, Phys. Rev. Lett. 124, 127203 (2020).
  51. D. Lozano-Gómez, V. Noculak, J. Oitmaa, R. R. P. Singh, Y. Iqbal, J. Reuther, and M. J. P. Gingras, Competing gauge fields and entropically driven spin liquid to spin liquid transition in non-Kramers pyrochlores, Proc. Natl. Acad. Sci. USA 121, e2403487121 (2024).
  52. M. Pretko, Subdimensional particle structure of higher rank U(1) spin liquids, Phys. Rev. B 95, 115139 (2017).
  53. A. Prem, S. Vijay, Y.-Z. Chou, M. Pretko, and R. M. Nandkishore, Pinch point singularities of tensor spin liquids, Phys. Rev. B 98, 165140 (2018).
  54. O. Benton and R. Moessner, Topological route to new and unusual Coulomb spin liquids, Phys. Rev. Lett. 127, 107202 (2021).
  55. H. Yan, O. Benton, R. Moessner, and A. H. Nevidomskyy, Classification of classical spin liquids: Typology and resulting landscape, Phys. Rev. B 110, L020402 (2024).
  56. H. Yan, O. Benton, A. H. Nevidomskyy, and R. Moessner, Classification of classical spin liquids: Detailed formalism and suite of examples, Phys. Rev. B 109, 174421 (2024).
  57. Y. Fang, J. Cano, A. H. Nevidomskyy, and H. Yan, Classification of classical spin liquids: Topological quantum chemistry and crystalline symmetry, Phys. Rev. B 110, 054421 (2024).
  58. R. P. Nutakki, L. D. C. Jaubert, and L. Pollet, The classical Heisenberg model on the centred pyrochlore lattice, SciPost Phys. 15, 040 (2023).
  59. Anson W. C. Wong, Z. Hao, and Michel J. P. Gingras, Ground state phase diagram of generic XY pyrochlore magnets with quantum fluctuations, Phys. Rev. B 88, 144402 (2013).
  60. B. Javanparast, Alexandre G. R. Day, Z. Hao, and Michel J. P. Gingras, Order-by-disorder near criticality in XYpyrochlore magnets, Phys. Rev. B 91, 174424 (2015).
  61. H. Yan, O. Benton, L. Jaubert, and N. Shannon, Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application to Yb2Ti2O7, Er2Ti2O7, Er2Sn2O7, Phys. Rev. B 95, 094422 (2017).
  62. C. Wei and S. H. Curnoe, Exact diagonalization for a 16-site spin-1/2 pyrochlore cluster, J. Phys.: Condens. Matter 35, 295802 (2023).
  63. V. Noculak, D. Lozano-Gómez, J. Oitmaa, Rajiv R. P. Singh, Y. Iqbal, Michel J. P. Gingras, and J. Reuther, Classical and quantum phases of the pyrochlore S = 1/2 magnet with Heisenberg and Dzyaloshinskii-Moriya interactions, Phys. Rev. B 107, 214414 (2023).
  64. P.-S. Hsin, A. Kapustin, and R. Thorngren, Berry phase in quantum field theory: Diabolical points and boundary phenomena, Phys. Rev. B 102, 245113 (2020).
  65. Z. Bi and T. Senthil, Adventure in topological phase transitions in 3 + 1-D: Non-Abelian deconfined quantum criticalities and a possible duality, Phys. Rev. X 9, 021034 (2019).
  66. C.-M. Jian and C. Xu, Generic “unnecessary” quantum critical points with minimal degrees of freedom, Phys. Rev. B 101, 035118 (2020).
  67. A. Prakash, M. Fava, and S. A. Parameswaran, Multiversality and unnecessary criticality in one dimension, Phys. Rev. Lett. 130, 256401 (2023).
  68. A. Prakash and N. G. Jones, Classical origins of Landau-incompatible transitions, Phys. Rev. Lett. 134, 097103 (2025).
  69. J. G. Rau and Michel J. P. Gingras, Frustration and anisotropic exchange in ytterbium magnets with edge-shared octahedra, Phys. Rev. B 98, 054408 (2018).
  70. Y.-P. Huang, G. Chen, and M. Hermele, Quantum spin ices and topological phases from dipolar-octupolar doublets on the pyrochlore lattice, Phys. Rev. Lett. 112, 167203 (2014).
  71. K. A. Ross, L. Savary, B. D. Gaulin, and L. Balents, Quantum excitations in quantum spin ice, Phys. Rev. X 1, 021002 (2011).
  72. K. Remund, R. Pohle, Y. Akagi, J. Romhányi, and N. Shannon, Semi-classical simulation of spin-1 magnets, Phys. Rev. Res. 4, 033106 (2022).
  73. H. Zhang and C. D. Batista, Classical spin dynamics based on SU(N) coherent states, Phys. Rev. B 104, 104409 (2021).
  74. K. T. K. Chung, S. Petit, J. Robert, and P. McClarty, Geometrically frustrated quadrupoles on the pyrochlore lattice and generalized spin liquids, Phys. Rev. X (to be published).
  75. S. Zhang, H. J. Changlani, K. W. Plumb, O. Tchernyshyov, and R. Moessner, Dynamical structure factor of the three-dimensional quantum spin liquid candidate NaCaNi2F7, Phys. Rev. Lett. 122, 167203 (2019).
  76. K. W. Plumb, H. J. Changlani, A. Scheie, S. Zhang, J. W. Krizan, J. A. Rodriguez-Rivera, Y. Qiu, B. Winn, R. J. Cava, and C. L. Broholm, Continuum of quantum fluctuations in a three-dimensional S = 1 Heisenberg magnet, Nat. Phys. 15, 54 (2019).
  77. I. Hagymási, N. Niggemann, and J. Reuther, Phase diagram of the antiferromagnetic J1-J2 spin-1 pyrochlore Heisenberg model, Phys. Rev. B 110, 224416 (2024).
  78. J. M. Luttinger and L. Tisza, Theory of dipole interaction in crystals, Phys. Rev. 70, 954 (1946).
  79. D. B. Litvin, The Luttinger-Tisza method, Physica 77, 205 (1974).
  80. N. Davier, F. A. Gómez Albarracín, H. D. Rosales, and P. Pujol, Combined approach to analyze and classify families of classical spin liquids, Phys. Rev. B 108, 054408 (2023).
  81. S. V. Isakov, K. Gregor, R. Moessner, and S. L. Sondhi, Dipolar spin correlations in classical pyrochlore magnets, Phys. Rev. Lett. 93, 167204 (2004).
  82. A. Pandey, R. Moessner, and C. Castelnovo, Analytical theory of pyrochlore cooperative paramagnets, Phys. Rev. B 101, 115107 (2020).
  83. C. Castelnovo, R. Moessner, and S. L. Sondhi, Magnetic monopoles in spin ice, Nature (London) 451, 42 (2008).
  84. S. E. Palmer and J. T. Chalker, Order induced by dipolar interactions in a geometrically frustrated antiferromagnet, Phys. Rev. B 62, 488 (2000).
  85. N. Francini, L. Schmidt, L. Janssen, and D. Lozano-Gómez, Exact nematic and mixed magnetic phases driven by competing orders on the pyrochlore lattice, arXiv:2510.23704.
  86. A. Scheie, O. Benton, M. Taillefumier, L. D. C. Jaubert, G. Sala, N. Jalarvo, S. M. Koohpayeh, and N. Shannon, Dynamical scaling as a signature of multiple phase competition in Yb2Ti2O7, Phys. Rev. Lett. 129, 217202 (2022).
  87. J. N. Reimers, Absence of long-range order in a three-dimensional geometrically frustrated antiferromagnet, Phys. Rev. B 45, 7287 (1992).
  88. B. Canals and C. Lacroix, Pyrochlore antiferromagnet: A three-dimensional quantum spin liquid, Phys. Rev. Lett. 80, 2933 (1998).
  89. B. Canals and D. A. Garanin, Spin-liquid phase in the pyrochlore anti-ferromagnet, Can. J. Phys. 79, 1323 (2001).
  90. P. H. Conlon and J. T. Chalker, Absent pinch points and emergent clusters: Further neighbor interactions in the pyrochlore Heisenberg antiferromagnet, Phys. Rev. B 81, 224413 (2010).
  91. Y. Iqbal, T. Müller, P. Ghosh, Michel J. P. Gingras, H. O. Jeschke, S. Rachel, J. Reuther, and R. Thomale, Quantum and classical phases of the pyrochlore Heisenberg model with competing interactions, Phys. Rev. X 9, 011005 (2019).
  92. M. V. Berry and M. Wilkinson, Diabolical points in the spectra of triangles, Proc. R. Soc. London A 392, 15 (1984).
  93. M. Elhajal, B. Canals, and C. Lacroix, Ordering in pyrochlore compounds due to Dzyaloshinsky–Moriya interactions: The case of Cu4O3, J. Phys.: Condens. Matter 16, S917 (2004).
  94. M. Elhajal, B. Canals, R. Sunyer, and C. Lacroix, Ordering in the pyrochlore antiferromagnet due to Dzyaloshinsky-Moriya interactions, Phys. Rev. B 71, 094420 (2005).
  95. B. Canals, M. Elhajal, and C. Lacroix, Ising-like order by disorder in the pyrochlore antiferromagnet with Dzyaloshinskii-Moriya interactions, Phys. Rev. B 78, 214431 (2008).
  96. A. Hickey, D. Lozano-Gómez, and Michel J. P. Gingras, Order-by-disorder without quantum zero-point fluctuations in the pyrochlore Heisenberg ferromagnet with Dzyaloshinskii-Moriya interactions, Phys. Rev. B 111, 184434 (2025).
  97. Y. Zhang, X.-Y. Song, and T. Senthil, Dirac spin liquid as an “unnecessary” quantum critical point on square lattice antiferromagnets, SciPost Phys. Core 8, 024 (2025).
  98. A. Kapustin and L. Spodyneiko, Higher-dimensional generalizations of the Thouless charge pump, arXiv:2003.09519.
  99. A. Kapustin and L. Spodyneiko, Higher-dimensional generalizations of Berry curvature, Phys. Rev. B 101, 235130 (2020).
  100. X. Wen, M. Qi, A. Beaudry, J. Moreno, M. J. Pflaum, D. Spiegel, A. Vishwanath, and M. Hermele, Flow of higher Berry curvature and bulk-boundary correspondence in parametrized quantum systems, Phys. Rev. B 108, 125147 (2023).
  101. A. Prakash and S. A. Parameswaran, Charge pumps, boundary modes, and the necessity of unnecessary criticality, Phys. Rev. B 112, L241117 (2025).
  102. G. H. Wannier, Antiferromagnetism. The triangular Ising net, Phys. Rev. 79, 357 (1950).
  103. J. Stephenson, Ising-model spin correlations on the triangular lattice. III. Isotropic antiferromagnetic lattice, J. Math. Phys. 11, 413 (1970).
  104. A. S. Wills, R. Ballou, and C. Lacroix, Model of localized highly frustrated ferromagnetism: The kagome spin ice, Phys. Rev. B 66, 144407 (2002).
  105. A. M. Hallas, J. Gaudet, and B. D. Gaulin, Experimental insights into ground-state selection of quantum XY pyrochlores, Annu. Rev. Condens. Matter Phys. 9, 105 (2018).
  106. G.-W. Chern, Pyrochlore antiferromagnet with antisymmetric exchange interactions: Critical behavior and order from disorder, arXiv:1008.3038.
  107. B. Javanparast, Z. Hao, M. Enjalran, and Michel J. P. Gingras, Fluctuation-driven selection at criticality in a frustrated magnetic system: The case of multiple-k partial order on the pyrochlore lattice, Phys. Rev. Lett. 114, 130601 (2015).
  108. C. L. Henley, Ordering due to disorder in a frustrated vector antiferromagnet, Phys. Rev. Lett. 62, 2056 (1989).
  109. S. Khatua, D. Sen, and R. Ganesh, Effective theories for quantum spin clusters: Geometric phases and state selection by singularity, Phys. Rev. B 100, 134411 (2019).
  110. S. Srinivasan, S. Khatua, G. Baskaran, and R. Ganesh, Order by singularity in Kitaev clusters, Phys. Rev. Res. 2, 023212 (2020).
  111. S. Khatua, S. Srinivasan, and R. Ganesh, State selection in frustrated magnets, Phys. Rev. B 103, 174412 (2021).
  112. O. Cépas, A. P. Young, and B. S. Shastry, Degeneracy and strong fluctuation-induced first-order phase transition in the dipolar pyrochlore antiferromagnet, Phys. Rev. B 72, 184408 (2005).
  113. D. Bergman, J. Alicea, E. Gull, S. Trebst, and L. Balents, Order-by-disorder and spiral spin-liquid in frustrated diamond-lattice antiferromagnets, Nat. Phys. 3, 487 (2007).
  114. N. Niggemann, M. Hering, and J. Reuther, Classical spiral spin liquids as a possible route to quantum spin liquids, J. Phys.: Condens. Matter 32, 024001 (2020).
  115. X.-P. Yao, J. Q. Liu, C.-J. Huang, X. Wang, and G. Chen, Generic spiral spin liquids, Front. Phys. 16, 53303 (2021).
  116. S. Gao, G. Pokharel, A. F. May, Joseph A. M. Paddison, C. Pasco, Y. Liu, K. M. Taddei, S. Calder, D. G. Mandrus, M. B. Stone, and A. D. Christianson, Line-graph approach to spiral spin liquids, Phys. Rev. Lett. 129, 237202 (2022).
  117. H. Yan and J. Reuther, Low-energy structure of spiral spin liquids, Phys. Rev. Res. 4, 023175 (2022).
  118. P. Balla, Y. Iqbal, and K. Penc, Affine lattice construction of spiral surfaces in frustrated Heisenberg models, Phys. Rev. B 100, 140402(R) (2019).
  119. K. T. K. Chung and Michel J. P. Gingras, 2-form U(1) spin liquids: A classical model and quantum aspects, Phys. Rev. B 111, 064417 (2025).
  120. J. N. Hallén, C. Castelnovo, and R. Moessner, Thermodynamics and fractal dynamics of a nematic spin ice: A doubly frustrated pyrochlore Ising magnet, Phys. Rev. B 109, 014438 (2024).
  121. M. E. Brooks-Bartlett, S. T. Banks, L. D. C. Jaubert, A. Harman-Clarke, and P. C. W. Holdsworth, Magnetic-moment fragmentation and monopole crystallization, Phys. Rev. X 4, 011007 (2014).
  122. E. Lhotel, L. D. C. Jaubert, and P. C. Holdsworth, Fragmentation in frustrated magnets: A review, J. Low Temp. Phys. 201, 710 (2020).
  123. Y. Deng, Y. Huang, J. L. Jacobsen, J. Salas, and A. D. Sokal, Finite-temperature phase transition in a class of four-state Potts antiferromagnets, Phys. Rev. Lett. 107, 150601 (2011).
  124. R. Moessner, Spin ice as a Coulomb liquid: From emergent gauge fields to magnetic monopoles, in Spin Ice, Springer Series in Solid-State Sciences, edited by M. Udagawa and L. Jaubert (Springer International Publishing, Cham, 2021), pp. 37–70.
  125. M. Pretko, Generalized electromagnetism of subdimensional particles: A spin liquid story, Phys. Rev. B 96, 035119 (2017).
  126. M. Pretko, X. Chen, and Y. You, Fracton phases of matter, Int. J. Mod. Phys. A 35, 2030003 (2020).
  127. R. M. Nandkishore and M. Hermele, Fractons, Annu. Rev. Condens. Matter Phys. 10, 295 (2019).
  128. N. Sadoune, K. Liu, H. Yan, Ludovic D. C. Jaubert, N. Shannon, and L. Pollet, Human-machine collaboration: Ordering mechanism of rank-2 spin liquid on breathing pyrochlore lattice, Phys. Rev. Res. 7, 033061 (2025).
  129. D. Lozano-Gómez, O. Benton, M. J. P. Gingras, and H. Yan, An atlas of classical pyrochlore spin liquids, arXiv:2411.03547.
  130. D. A. Garanin and B. Canals, Classical spin liquid: Exact solution for the infinite-component antiferromagnetic model on the kagome lattice, Phys. Rev. B 59, 443 (1999).
  131. B. Canals and D. Garanin, Classical spin liquid properties of the infinite-component spin vector model on a fully frustrated two dimensional lattice, Eur. Phys. J. B 26, 439 (2002).
  132. D. L. Bergman, C. Wu, and L. Balents, Band touching from real-space topology in frustrated hopping models, Phys. Rev. B 78, 125104 (2008).
  133. J. Rehn, A. Sen, and R. Moessner, Fractionalized Z2 classical Heisenberg spin liquids, Phys. Rev. Lett. 118, 047201 (2017).
  134. N. Davier, F. A. Gomez Albarracín, H. D. Rosales, P. Pujol, and Ludovic D. C. Jaubert, Pinch line spin liquids as layered Coulomb phases and applications to cubic models, Phys. Rev. B 112, 104425 (2025).
  135. K. Slagle, Foliated quantum field theory of fracton order, Phys. Rev. Lett. 126, 101603 (2021).
  136. M. Stern, M. D. Burke, M. J. P. Gingras, J. Romhányi, and K. T. K. Chung, Fracton spin liquid and exotic frustrated phases in Ising-like octochlore magnets, arXiv:2603.12313.
  137. I. Hagymási, R. Schäfer, R. Moessner, and D. J. Luitz, Possible inversion symmetry breaking in the S = 1/2 pyrochlore Heisenberg magnet, Phys. Rev. Lett. 126, 117204 (2021).
  138. N. Astrakhantsev, T. Westerhout, A. Tiwari, K. Choo, A. Chen, M. H. Fischer, G. Carleo, and T. Neupert, Broken-symmetry ground states of the Heisenberg model on the pyrochlore lattice, Phys. Rev. X 11, 041021 (2021).
  139. R. Pohle, Y. Yamaji, and M. Imada, Ground state of the S = 1/2 pyrochlore Heisenberg antiferromagnet: A quantum spin liquid emergent from dimensional reduction, arXiv:2311.11561.
  140. R. Schäfer, B. Placke, O. Benton, and R. Moessner, Abundance of hard-hexagon crystals in the quantum pyrochlore antiferromagnet, Phys. Rev. Lett. 131, 096702 (2023).
  141. S. E. Han, A. S. Patri, and Y. B. Kim, Realization of fractonic quantum phases in the breathing pyrochlore lattice, Phys. Rev. B 105, 235120 (2022).
  142. 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).
  143. F. Desrochers, L. E. Chern, and Y. B. Kim, Competing U(1) and Z2 dipolar-octupolar quantum spin liquids on the pyrochlore lattice: Application to Ce2Zr2O7, Phys. Rev. B 105, 035149 (2022).
  144. 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).
  145. F. Desrochers, L. E. Chern, and Y. B. Kim, Symmetry fractionalization in the gauge mean-field theory of quantum spin ice, Phys. Rev. B 107, 064404 (2023).
  146. T. Müller, D. Kiese, N. Niggemann, B. Sbierski, J. Reuther, S. Trebst, R. Thomale, and Y. Iqbal, Pseudo-fermion functional renormalization group for spin models, Rep. Prog. Phys. 87, 036501 (2024).
  147. Y. Iqbal, R. Thomale, F. Parisen Toldin, S. Rachel, and J. Reuther, Functional renormalization group for three-dimensional quantum magnetism, Phys. Rev. B 94, 140408(R) (2016).
  148. N. Niggemann, Y. Iqbal, and J. Reuther, Quantum effects on unconventional pinch point singularities, Phys. Rev. Lett. 130, 196601 (2023).
  149. L. E. Chern, F. Desrochers, Y. B. Kim, and C. Castelnovo, Pseudofermion functional renormalization group study of dipolar-octupolar pyrochlore magnets, Phys. Rev. B 109, 184421 (2024).
  150. L. Gresista, D. Lozano-Gómez, M. Vojta, S. Trebst, and Y. Iqbal, Quantum effects on pyrochlore higher-rank U(1) spin liquids: Pinch-line singularities, spin nematics, and connections to oxide materials, Phys. Rev. Res. 7, 033109 (2025).
  151. K. Essafi, L. D. C. Jaubert, and M. Udagawa, Flat bands and Dirac cones in breathing lattices, J. Phys.: Condens. Matter 29, 315802 (2017).

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Phys. Rev. B 114, 084416– Published 19 August, 2026
  • Received 1 July 2025
  • Revised 15 July 2026
  • Accepted 17 July 2026
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