Classical and quantum spin liquids

Sylvain Capponi

doi:10.5802/crphys.228

Intervention en colloque

Classical and quantum spin liquids
[Liquides de spins classiques et quantiques]

Sylvain Capponi ¹

¹ Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, Toulouse, France

Comptes Rendus. Physique, Volume 26 (2025), pp. 91-111.

Cet article fait partie du numéro thématique Questions ouvertes dans le problème quantique à N corps coordonné par Yvan Castin et al..

Résumé
Abstract

Lorsque l’on considère des systèmes magnétiques dans la limite thermodynamique et à suffisamment basse température, on trouve des phases généralement ordonnées sur le plan magnétique. Au contraire, à haute température, les interactions entre les degrés de liberté de spin deviennent moins pertinentes et le système perd son ordre : c’est une phase paramagnétique. Ce phénomène de transition de phase a été bien compris grâce à la mécanique statistique et à des modèles simples.

Dans ces brèves note de cours, nous examinerons la possibilité qu’un système magnétique puisse rester désordonné magnétiquement jusqu’à la température nulle, à la fois pour des spins classiques ou quantiques. Ces phases exotiques de la matière sont connues respectivement sous le nom de liquides de spin classiques et quantiques.

Nous aborderons en particulier la question de la classification de ces phases désordonnées classiques ou quantiques. En effet, bien qu’elles n’aient pas de paramètre d’ordre local par définition, elles peuvent néanmoins posséder différentes caractéristiques qualitatives liées par exemple à la nature de leurs corrélations ou de leurs excitations élémentaires, qui peuvent être sondées expérimentalement.

When considering magnetic systems in the thermodynamic limit and at low enough temperature, one finds typically magnetically ordered phases. In contrast, in the high-temperature regime, the interactions between the spin degrees of freedom become less relevant and the system loses its order: this is a paramagnet. This phenomenon of phase transition has been well understood using statistical mechanics and simple modelling.

In this short lecture notes, we will review the possibility that a many-body magnetic system may remain magnetically disordered down to zero-temperature, both for classical or quantum spins. These exotic phases of matter are known, respectively, as classical and quantum spin liquids.

We will address in particular the question of classification of these classical or quantum disordered phases. Indeed, while they have no local order parameter by definition, they can still possess different qualitative features related e.g. to the nature of their correlations or elementary excitations, which could be probed experimentally.

Reçu le : 2024-10-17
Révisé le : 2024-12-13
Accepté le : 2024-12-18
Publié le : 2025-01-24

DOI : 10.5802/crphys.228

Keywords: Magnetism, Spin liquids, Condensed matter
Mots-clés : Magnétisme, Liquides de spins, Matière condensée

Affiliations des auteurs :

Sylvain Capponi ¹

¹ Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, Toulouse, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRPHYS_2025__26_G1_91_0,
     author = {Sylvain Capponi},
     title = {Classical and quantum spin liquids},
     journal = {Comptes Rendus. Physique},
     pages = {91--111},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {26},
     year = {2025},
     doi = {10.5802/crphys.228},
     language = {en},
}

TY  - JOUR
AU  - Sylvain Capponi
TI  - Classical and quantum spin liquids
JO  - Comptes Rendus. Physique
PY  - 2025
SP  - 91
EP  - 111
VL  - 26
PB  - Académie des sciences, Paris
DO  - 10.5802/crphys.228
LA  - en
ID  - CRPHYS_2025__26_G1_91_0
ER  -

%0 Journal Article
%A Sylvain Capponi
%T Classical and quantum spin liquids
%J Comptes Rendus. Physique
%D 2025
%P 91-111
%V 26
%I Académie des sciences, Paris
%R 10.5802/crphys.228
%G en
%F CRPHYS_2025__26_G1_91_0

Sylvain Capponi. Classical and quantum spin liquids. Comptes Rendus. Physique, Volume 26 (2025), pp. 91-111. doi : 10.5802/crphys.228. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.228/

Bibliographie
Cité par

[1] P. W. Anderson More is different, Science, Volume 177 (1972) no. 4047, 180408, pp. 393-396 | DOI

[2] A. P. Ramirez Strongly geometrically frustrated magnets, Annu. Rev. Mater. Res., Volume 24 (1994) no. 24, pp. 453-480 | DOI

[3] C. Lacroix; F. Mila; P. Mendels Introduction to Frustrated Magnetism: Materials, Experiments, Theory, Springer Series in Solid-State Sciences, 164, Springer, Berlin, 2011 | DOI

[4] Y. Zhou; K. Kanoda; T.-K. Ng Quantum spin liquid states, Rev. Mod. Phys., Volume 89 (2017), 025003 | DOI

[5] G. H. Wannier Antiferromagnetism. The triangular Ising net, Phys. Rev., Volume 79 (1950), pp. 357-364 | DOI

[6] J. Villain; R. Bidaux; J.-P. Carton; R. Conte Order as an effect of disorder, J. Phys. France, Volume 41 (1980) no. 11, 115147, pp. 1263-1272 | DOI

[7] J. Villain Insulating spin glasses, Z. Phys. B: Condens. Matter., Volume 33 (1979) no. 1, 5137, pp. 31-42 | DOI

[8] J. T. Chalker Spin liquids and frustrated magnetism, Topological Aspects of Condensed Matter Physics: Lecture Notes of the Les Houches Summer School: Volume 103, August 2014, Oxford University Press, 2017, 020402 | DOI

[9] H. Duminil-Copin 100 years of the (critical) Ising model on the hypercubic lattice, preprint, 2022, 224412 | arXiv | DOI

[10] W. Shirley; K. Slagle; Z. Wang; X. Chen Fracton models on general three-dimensional manifolds, Phys. Rev. X, Volume 8 (2018), 031051, 374010 | DOI

[11] D. Chowdhury; A. Georges; O. Parcollet; S. Sachdev Sachdev–Ye–Kitaev models and beyond: Window into non-Fermi liquids, Rev. Mod. Phys., Volume 94 (2022), 035004, 094437 | DOI

[12] J. Colbois; B. Vanhecke; L. Vanderstraeten; A. Smerald; F. Verstraete; F. Mila Partial lifting of degeneracy in the J $_{1}$ –J $_{2}$ –J $_{3}$ Ising antiferromagnet on the kagome lattice, Phys. Rev. B, Volume 106 (2022), 174403, 144416 | DOI

[13] R. J. Baxter Exactly Solved Models in Statistical Mechanics, Academic, London, 1982, 067205 (Dover, New York, 2007) | DOI

[14] L. Pauling The structure and entropy of ice and of other crystals with some randomness of atomic arrangement, J. Am. Chem. Soc., Volume 57 (1935) no. 12, 035004, pp. 2680-2684 | DOI

[15] E. H. Lieb Residual entropy of square ice, Phys. Rev., Volume 162 (1967), pp. 162-172 | DOI

[16] W. F. Giauque; J. W. Stout The entropy of water and the third law of thermodynamics. The heat capacity of ice from 15 to 273 K, J. Am. Chem. Soc., Volume 58 (1936) no. 7, pp. 1144-1150 | DOI

[17] S. T. Bramwell; M. J. Harris The history of spin ice, J. Phys.: Condens. Matter., Volume 32 (2020) no. 37, 374010, 235104 | DOI

[18] C. Castelnovo; R. Moessner; S. L. Sondhi Magnetic monopoles in spin ice, Nature, Volume 451 (2008) no. 7174, 235141, pp. 42-45 | DOI

[19] P. W. Kasteleyn The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice, Physica, Volume 27 (1961) no. 12, 134424, pp. 1209-1225 | DOI

[20] H. N. V. Temperley; M. E. Fisher Dimer problem in statistical mechanics-an exact result, Philos. Mag., Volume 6 (1961) no. 68, pp. 1061-1063 | DOI

[21] J. M. Luttinger; L. Tisza Theory of dipole interaction in crystals, Phys. Rev., Volume 70 (1946), pp. 954-964 | DOI

[22] J. C. Maxwell XLV. On reciprocal figures and diagrams of forces, Lond. Edinb. Dublin Philos. Mag. J. Sci., Volume 27 (1864) no. 182, 184409, pp. 250-261 | DOI

[23] R. Moessner; J. T. Chalker Low-temperature properties of classical geometrically frustrated antiferromagnets, Phys. Rev. B, Volume 58 (1998), 174403, pp. 12049-12062 | DOI

[24] R. Moessner; J. T. Chalker Properties of a classical spin liquid: The Heisenberg pyrochlore antiferromagnet, Phys. Rev. Lett., Volume 80 (1998), 054408, pp. 2929-2932 | DOI

[25] J. T. Chalker; P. C. W. Holdsworth; E. F. Shender Hidden order in a frustrated system: Properties of the Heisenberg Kagomé antiferromagnet, Phys. Rev. Lett., Volume 68 (1992), pp. 855-858 | arXiv | DOI

[26] N. Davier; F. A. Gómez Albarracín; H. D. Rosales; P. Pujol Combined approach to analyze and classify families of classical spin liquids, Phys. Rev. B, Volume 108 (2023), 054408, 104406 | DOI

[27] H. Yan; O. Benton; R. Moessner; A. H. Nevidomskyy Classification of classical spin liquids: Typology and resulting landscape, Phys. Rev. B, Volume 110 (2024), L020402, 054411 | DOI

[28] L. Savary; L. Balents Quantum spin liquids: a review, Rep. Prog. Phys., Volume 80 (2016) no. 1, 016502 | DOI

[29] P. W. Anderson Resonating valence bonds: A new kind of insulator?, Mater. Res. Bull., Volume 8 (1973) no. 2, 056501, pp. 153-160 | DOI

[30] B. S. Shastry; B. Sutherland Excitation spectrum of a dimerized next-neighbor antiferromagnetic chain, Phys. Rev. Lett., Volume 47 (1981), pp. 964-967 | DOI

[31] H. Li; F. D. M. Haldane Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-Abelian fractional quantum Hall effect states, Phys. Rev. Lett., Volume 101 (2008), 010504 | DOI

[32] X.-G. Wen Quantum orders and symmetric spin liquids, Phys. Rev. B, Volume 65 (2002), 165113, 075112 | DOI

[33] X.-G. Wen Quantum Field Theory of Many-body Systems, Oxford University Press, New Delhi, 2004, 6317 (Includes bibliographical references and index) | DOI

[34] M. Oshikawa; T. Senthil Fractionalization, topological order, and quasiparticle statistics, Phys. Rev. Lett., Volume 96 (2006), 060601 | DOI

[35] V. Kalmeyer; R. B. Laughlin Equivalence of the resonating-valence-bond and fractional quantum Hall states, Phys. Rev. Lett., Volume 59 (1987), 104431, pp. 2095-2098 | DOI

[36] T. S. Cubitt; D. Perez-Garcia; M. M. Wolf Undecidability of the spectral gap, Nature, Volume 528 (2015) no. 7581, 060402, pp. 207-211 | DOI

[37] E. Lieb; T. Schultz; D. Mattis Two soluble models of an antiferromagnetic chain, Ann. Phys., Volume 16 (1961) no. 3, 064404, pp. 407-466 | DOI

[38] M. Oshikawa Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice, Phys. Rev. Lett., Volume 84 (2000), 137202, pp. 1535-1538 | DOI

[39] M. B. Hastings Lieb–Schultz–Mattis in higher dimensions, Phys. Rev. B, Volume 69 (2004) no. 10, 104431, 214437 | DOI

[40] C. Liu; W. Ye Crystallography, group cohomology, and Lieb–Schultz–Mattis constraints, preprint, 2024, 207203 | arXiv | DOI

[41] S. A. Parameswaran; A. M. Turner; D. P. Arovas; A. Vishwanath Topological order and absence of band insulators at integer filling in non-symmorphic crystals, Nat. Phys., Volume 9 (2013) no. 5, 144411, pp. 299-303 | DOI

[42] H. C. Po; H. Watanabe; C.-M. Jian; M. P. Zaletel Lattice homotopy constraints on phases of quantum magnets, Phys. Rev. Lett., Volume 119 (2017), 127202 | DOI

[43] W. Ye; M. Guo; Y.-C. He; C. Wang; L. Zou Topological characterization of Lieb–Schultz–Mattis constraints and applications to symmetry-enriched quantum criticality, SciPost Phys., Volume 13 (2022), 066 | DOI

[44] S. A. Parameswaran; I. Kimchi; A. M. Turner; D. M. Stamper-Kurn; A. Vishwanath Wannier permanent wave functions for featureless bosonic Mott insulators on the 1/3-filled Kagome lattice, Phys. Rev. Lett., Volume 110 (2013), 125301 | DOI

[45] I. Kimchi; S. A. Parameswaran; A. M. Turner; F. Wang; A. Vishwanath Featureless and nonfractionalized Mott insulators on the honeycomb lattice at 1/2 site filling, Proc. Natl. Acad. Sci. USA, Volume 110 (2013) no. 41, pp. 16378-16383 | DOI

[46] P. Kim; H. Lee; S. Jiang; B. Ware; C.-M. Jian; M. Zaletel; J. H. Han; Y. Ran Featureless quantum insulator on the honeycomb lattice, Phys. Rev. B, Volume 94 (2016), 064432 | DOI

[47] P. A. Lee; N. Nagaosa; X.-G. Wen Doping a Mott insulator: Physics of high-temperature superconductivity, Rev. Mod. Phys., Volume 78 (2006), 110404, pp. 17-85 | DOI

[48] M. C. Gutzwiller Effect of correlation on the ferromagnetism of transition metals, Phys. Rev. Lett., Volume 10 (1963), pp. 159-162 | DOI

[49] L. Messio; C. Lhuillier; G. Misguich Time reversal symmetry breaking chiral spin liquids: Projective symmetry group approach of bosonic mean-field theories, Phys. Rev. B, Volume 87 (2013), 125127, 010504 | DOI

[50] S. Bieri; C. Lhuillier; L. Messio Projective symmetry group classification of chiral spin liquids, Phys. Rev. B, Volume 93 (2016), 094437, 035130 | DOI

[51] A. M. Polyakov Quark confinement and topology of gauge theories, Nuclear Phys. B, Volume 120 (1977) no. 3, pp. 429-458 | DOI

[52] M. Hermele; T. Senthil; M. P. A. Fisher; P. A. Lee; N. Nagaosa; X.-G. Wen Stability of U(1) spin liquids in two dimensions, Phys. Rev. B, Volume 70 (2004), 214437 | DOI

[53] M. Barkeshli; P. Bonderson; M. Cheng; Z. Wang Symmetry fractionalization, defects, and gauging of topological phases, Phys. Rev. B, Volume 100 (2019), 115147 | DOI

[54] A. M. Essin; M. Hermele Classifying fractionalization: Symmetry classification of gapped $ℤ_{2}$ spin liquids in two dimensions, Phys. Rev. B, Volume 87 (2013), 104406 | DOI

[55] A. Chauhan; A. Maity; C. Liu; J. Sonnenschein; F. Ferrari; Y. Iqbal Quantum spin liquids on the diamond lattice, Phys. Rev. B, Volume 108 (2023), 134424, 177204 | DOI

[56] F. D. M. Haldane Exact Jastrow-Gutzwiller resonating-valence-bond ground state of the spin- $\frac{1}{2}$ antiferromagnetic Heisenberg chain with 1/r $^{2}$ exchange, Phys. Rev. Lett., Volume 60 (1988), pp. 635-638 | DOI

[57] I. Affleck; T. Kennedy; E. H. Lieb; H. Tasaki Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett., Volume 59 (1987) no. 7, 195158, pp. 799-802 | DOI

[58] F. Pollmann; E. Berg; A. M. Turner; M. Oshikawa Symmetry protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B, Volume 85 (2012), 075125, 045110 | DOI

[59] M. Lemm; A. W. Sandvik; L. Wang Existence of a spectral gap in the Affleck–Kennedy–Lieb–Tasaki model on the hexagonal lattice, Phys. Rev. Lett., Volume 124 (2020), 177204, 110405 | DOI

[60] X. Chen; Z.-X. Liu; X.-G. Wen Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations, Phys. Rev. B, Volume 84 (2011), 235141 | arXiv | DOI

[61] C. K. Majumdar; D. K. Ghosh On next-nearest-neighbor interaction in linear chain. I, J. Math. Phys., Volume 10 (1969) no. 8, pp. 1388-1398 | DOI

[62] J. B. Fouet; P. Sindzingre; C. Lhuillier An investigation of the quantum J $_{1}$ –J $_{2}$ –J $_{3}$ model on the honeycomb lattice, Eur. Phys. J. B, Volume 20 (2001), pp. 241-254 | DOI

[63] J.-B. Fouet; M. Mambrini; P. Sindzingre; C. Lhuillier Planar pyrochlore: A valence-bond crystal, Phys. Rev. B, Volume 67 (2003), 054411 | DOI

[64] J. Schulenburg; A. Honecker; J. Schnack; J. Richter; H.-J. Schmidt Macroscopic magnetization jumps due to independent magnons in frustrated quantum spin lattices, Phys. Rev. Lett., Volume 88 (2002), 167207, 134413 | DOI

[65] S. Capponi; O. Derzhko; A. Honecker; A. M. Läuchli; J. Richter Numerical study of magnetization plateaus in the spin- $\frac{1}{2}$ kagome Heisenberg antiferromagnet, Phys. Rev. B, Volume 88 (2013), 144416 | DOI

[66] X. G. Wen Mean-field theory of spin-liquid states with finite energy gap and topological orders, Phys. Rev. B, Volume 44 (1991), 157203, pp. 2664-2672 | DOI

[67] M. Levin; X.-G. Wen Detecting topological order in a ground state wave function, Phys. Rev. Lett., Volume 96 (2006), 110405, 125127 | DOI

[68] A. Kitaev; J. Preskill Topological entanglement entropy, Phys. Rev. Lett., Volume 96 (2006), 110404, 045105 | DOI

[69] A. Y. Kitaev Fault-tolerant quantum computation by anyons, Ann. Phys., Volume 303 (2003) no. 1, pp. 2-30 | DOI

[70] A. Kitaev Anyons in an exactly solved model and beyond, Ann. Phys., Volume 321 (2006) no. 1, 2864, pp. 2-111 | DOI

[71] M. A. Levin; X.-G. Wen String-net condensation: A physical mechanism for topological phases, Phys. Rev. B, Volume 71 (2005), 045110, 060601 | DOI

[72] D. S. Rokhsar; S. A. Kivelson Superconductivity and the quantum hard-core dimer gas, Phys. Rev. Lett., Volume 61 (1988), pp. 2376-2379 | DOI

[73] R. Moessner; S. L. Sondhi Resonating valence bond phase in the triangular lattice quantum dimer model, Phys. Rev. Lett., Volume 86 (2001), pp. 1881-1884 | DOI

[74] L. Balents; M. P. A. Fisher; S. M. Girvin Fractionalization in an easy-axis Kagome antiferromagnet, Phys. Rev. B, Volume 65 (2002), 224412, 075125 | DOI

[75] D. N. Sheng; L. Balents Numerical evidences of fractionalization in an easy-axis two-spin Heisenberg antiferromagnet, Phys. Rev. Lett., Volume 94 (2005), 146805, 125301 | DOI

[76] M. S. Block; J. D’Emidio; R. K. Kaul Kagome model for a $ℤ_{2}$ quantum spin liquid, Phys. Rev. B, Volume 101 (2020), 020402 | DOI

[77] K. S. Raman; R. Moessner; S. L. Sondhi SU(2)-invariant spin- $\frac{1}{2}$ Hamiltonians with resonating and other valence bond phases, Phys. Rev. B, Volume 72 (2005), 064413 | DOI

[78] J. Cano; P. Fendley Spin Hamiltonians with resonating-valence-bond ground states, Phys. Rev. Lett., Volume 105 (2010), 067205, 127202 | DOI

[79] M. Mambrini; S. Capponi; F. Alet Engineering SU(2) invariant spin models to mimic quantum dimer physics on the square lattice, Phys. Rev. B, Volume 92 (2015), 134413 | DOI

[80] B. Bauer; L. Cincio; B. P. Keller; M. Dolfi; G. Vidal; S. Trebst; A. W. W. Ludwig Chiral spin liquid and emergent anyons in a Kagome lattice Mott insulator, Nat. Commun., Volume 5 (2014) no. 1, 5137 | DOI

[81] S.-S. Gong; W. Zhu; D. N. Sheng Emergent chiral spin liquid: Fractional quantum Hall effect in a Kagome Heisenberg model, Sci. Rep., Volume 4 (2014) no. 1, 6317, 064413 | DOI

[82] Y.-C. He; D. N. Sheng; Y. Chen Chiral spin liquid in a frustrated anisotropic Kagome Heisenberg model, Phys. Rev. Lett., Volume 112 (2014), 137202, 047201 | DOI

[83] S.-S. Gong; W. Zhu; L. Balents; D. N. Sheng Global phase diagram of competing ordered and quantum spin-liquid phases on the kagome lattice, Phys. Rev. B, Volume 91 (2015), 075112, 146805 | DOI

[84] A. Wietek; A. M. Läuchli Chiral spin liquid and quantum criticality in extended $S = \frac{1}{2}$ Heisenberg models on the triangular lattice, Phys. Rev. B, Volume 95 (2017), 035141, 016502 | DOI

[85] A. Szasz; J. Motruk; M. P. Zaletel; J. E. Moore Chiral spin liquid phase of the triangular lattice Hubbard model: A density matrix renormalization group study, Phys. Rev. X, Volume 10 (2020), 021042 | DOI

[86] D. F. Schroeter; E. Kapit; R. Thomale; M. Greiter Spin Hamiltonian for which the chiral spin liquid is the exact ground state, Phys. Rev. Lett., Volume 99 (2007), 097202 | DOI

[87] A. E. B. Nielsen; G. Sierra; J. I. Cirac Local models of fractional quantum Hall states in lattices and physical implementation, Nat. Commun., Volume 4 (2013) no. 1, 2864, 167207 | DOI

[88] J.-Y. Chen; L. Vanderstraeten; S. Capponi; D. Poilblanc Non-Abelian chiral spin liquid in a quantum antiferromagnet revealed by an iPEPS study, Phys. Rev. B, Volume 98 (2018), 184409, 011033 | DOI

[89] Z.-X. Liu; H.-H. Tu; Y.-H. Wu; R.-Q. He; X.-J. Liu; Y. Zhou; T.-K. Ng Non-Abelian S = 1 chiral spin liquid on the kagome lattice, Phys. Rev. B, Volume 97 (2018), 195158, 097202 | DOI

[90] W.-W. Luo; Y. Huang; D. N. Sheng; W. Zhu Global quantum phase diagram and non-Abelian chiral spin liquid in a spin- $\frac{3}{2}$ square-lattice antiferromagnet, Phys. Rev. B, Volume 108 (2023), 035130, 021042 | DOI

[91] J.-Y. Chen; J.-W. Li; P. Nataf et al. Abelian SU(N) $_{1}$ chiral spin liquids on the square lattice, Phys. Rev. B, Volume 104 (2021), 235104, 003 | DOI

[92] Z. Zhu; I. Kimchi; D. N. Sheng; L. Fu Robust non-Abelian spin liquid and a possible intermediate phase in the antiferromagnetic Kitaev model with magnetic field, Phys. Rev. B, Volume 97 (2018), 241110 | DOI

[93] H. Yao; S. A. Kivelson Exact chiral spin liquid with non-Abelian anyons, Phys. Rev. Lett., Volume 99 (2007), 247203, 031051 | DOI

[94] Y. Iqbal; W.-J. Hu; R. Thomale; D. Poilblanc; F. Becca Spin liquid nature in the Heisenberg J $_{1}$ –J $_{2}$ triangular antiferromagnet, Phys. Rev. B, Volume 93 (2016), 144411 | DOI

[95] S. Hu; W. Zhu; S. Eggert; Y.-C. He Dirac spin liquid on the spin-1/2 triangular Heisenberg antiferromagnet, Phys. Rev. Lett., Volume 123 (2019), 207203, 174427 | DOI

[96] X.-Y. Song; Y.-C. He; A. Vishwanath; C. Wang From spinon band topology to the symmetry quantum numbers of monopoles in dirac spin liquids, Phys. Rev. X, Volume 10 (2020), 011033 | arXiv | DOI

[97] A. Wietek; S. Capponi; A. M. Läuchli Quantum electrodynamics in 2 + 1 dimensions as the organizing principle of a triangular lattice antiferromagnet, Phys. Rev. X, Volume 14 (2024), 021010 | DOI

[98] M. J. P. Gingras; P. A. McClarty Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets, Rep. Prog. Phys., Volume 77 (2014) no. 5, 056501 | DOI

[99] W.-J. Hu; F. Becca; A. Parola; S. Sorella Direct evidence for a gapless Z $_{2}$ spin liquid by frustrating Néel antiferromagnetism, Phys. Rev. B, Volume 88 (2013), 060402 | arXiv | DOI

[100] J. Yang; A. W. Sandvik; L. Wang Quantum criticality and spin liquid phase in the Shastry–Sutherland model, Phys. Rev. B, Volume 105 (2022), L060409 | DOI

[101] L. L. Viteritti; R. Rende; A. Parola; S. Goldt; F. Becca Transformer wave function for the Shastry–Sutherland model: emergence of a spin-liquid phase, preprint, 2024, 021010 | arXiv | DOI

[102] R. V. Mishmash; J. R. Garrison; S. Bieri; C. Xu Theory of a competitive spin liquid state for weak Mott insulators on the triangular lattice, Phys. Rev. Lett., Volume 111 (2013), 157203, 165113 | DOI

[103] B. S. Shastry Exact solution of an S = 1/2 Heisenberg antiferromagnetic chain with long-ranged interactions, Phys. Rev. Lett., Volume 60 (1988), pp. 639-642 | DOI

[104] O. I. Motrunich Variational study of triangular lattice spin-1/2 model with ring exchanges and spin liquid state in $κ$ –(ET) $_{2}$ Cu $_{2}$ (CN) $_{3}$ , Phys. Rev. B, Volume 72 (2005), 045105 | DOI

[105] A. F. Albuquerque; F. Alet Critical correlations for short-range valence-bond wave functions on the square lattice, Phys. Rev. B, Volume 82 (2010), 180408, 035141 | DOI

[106] Y. Tang; A. W. Sandvik; C. L. Henley Properties of resonating-valence-bond spin liquids and critical dimer models, Phys. Rev. B, Volume 84 (2011), 174427, L020402 | DOI

[107] M. Hermele; M. P. A. Fisher; L. Balents Pyrochlore photons: The U(1) spin liquid in a S = (1/2) three-dimensional frustrated magnet, Phys. Rev. B, Volume 69 (2004) no. 6, 064404, 066 | DOI

[108] N. Seiberg; S.-H. Shao Exotic $ℤ_{N}$ symmetries, duality, and fractons in 3 + 1-dimensional quantum field theory, SciPost Phys., Volume 10 (2021), 003, 247203 | DOI

[109] J. Rehn; A. Sen; R. Moessner Fractionalized $ℤ_{2}$ classical Heisenberg spin liquids, Phys. Rev. Lett., Volume 118 (2017), 047201, L060409 | DOI

[110] T. Senthil Deconfined quantum critical points: a review, 50 Years of the Renormalization Group, Dedicated to the Memory of Michael E. Fisher, World Scientific, Singapore, 2024, 031043 | DOI

[111] J. Takahashi; H. Shao; B. Zhao; W. Guo; A. W. Sandvik SO(5) multicriticality in two-dimensional quantum magnets, preprint, 2024, 025003 | arXiv | DOI

[112] L. Zou; Y.-C. He; C. Wang Stiefel liquids: Possible non-Lagrangian quantum criticality from intertwined orders, Phys. Rev. X, Volume 11 (2021), 031043, 241110 | DOI

Cité par Sources :

Commentaires - Politique