Haldane, F. D. M. ‘Luttinger liquid theory’ of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas. J. Phys. C 14, 2585–2610 (1981).
Lake, B., Tennant, D. A., Frost, C. D. & Nagler, S. E. Quantum criticality and universal scaling of a quantum antiferromagnet. Nat. Mater. 4, 329–334 (2005).
Coldea, R. et al. Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry. Science 327, 177–180 (2010).
Lake, B., Tennant, D. A., Frost, C. D. & Nagler, S. E. Quantum criticality and universal scaling of a quantum antiferromagnet. Phys. Rev. Lett. 111, 137205 (2013).
Mourigal, M. et al. Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain. Nat. Phys. 9, 435–441 (2013).
Shao, H. et al. Nearly deconfined spinon excitations in the square-lattice spin-1/2 Heisenberg antiferromagnet. Phys. Rev. X 7, 041072 (2017).
Haegeman, J., Lubich, C., Oseledets, I., Vandereycken, B. & Verstraete, F. Unifying time evolution and optimization with matrix product states. Phys. Rev. B 94, 165116 (2016).
Savary, L. & Balents, L. Quantum spin liquids: a review. Rep. Prog. Phys. 80, 016502 (2017).
Knolle, J. & Moessner, R. A field guide to spin liquids. Annu. Rev. Condens. Matter Phys. 10, 451–472 (2019).
Villanova, J. W., Scheie, A. O., Tennant, D. A., Okamoto, S. & Berlijn, T. First-principles derivation of magnetic interactions in the triangular quantum spin liquid candidates KYbCh2 (Ch = S, Se, Te) and AYbSe2 (A = Na, Rb). Phys. Rev. Res. 5, 033050 (2023).
Liu, J. et al. Gapless spin liquid behavior in a kagome Heisenberg antiferromagnet with randomly distributed hexagons of alternate bonds. Phys. Rev. B 105, 024418 (2022).
Hering, M. et al. Phase diagram of a distorted kagome antiferromagnet and application to Y-kapellasite. npj Comput. Mater. 8, 10 (2022).
Li, H. et al. Kosterlitz–Thouless melting of magnetic order in the triangular quantum Ising material TmMgGaO4. Nat. Commun. 11, 1111 (2020).
Cevallos, F. A., Stolze, K., Kong, T. & Cava, R. Anisotropic magnetic properties of the triangular plane lattice material TmMgGaO4. Mater. Res. Bull. 105, 154–158 (2018).
Li, Y. et al. Partial up–up–down order with the continuously distributed order parameter in the triangular antiferromagnet TmMgGaO4. Phys. Rev. X 10, 011007 (2020).
Isakov, S. V. & Moessner, R. Interplay of quantum and thermal fluctuations in a frustrated magnet. Phys. Rev. B 68, 104409 (2003).
Da Liao, Y. et al. Phase diagram of the quantum Ising model on a triangular lattice under external field. Phys. Rev. B 103, 104416 (2021).
Shen, Y. et al. Intertwined dipolar and multipolar order in the triangular-lattice magnet TmMgGaO4. Nat. Commun. 10, 4530 (2019).
Hu, Z. et al. Evidence of the Berezinskii–Kosterlitz–Thouless phase in a frustrated magnet. Nat. Commun. 11, 5631 (2020).
Anderson, P. Resonating valence bonds: a new kind of insulator?. Mater. Res. Bull. 8, 153–160 (1973).
Zheng, W., Fjærestad, J. O., Singh, R. R. P., McKenzie, R. H. & Coldea, R. Excitation spectra of the spin-½ triangular-lattice Heisenberg antiferromagnet. Phys. Rev. B 74, 224420 (2006).
Huse, D. A. & Elser, V. Simple variational wave functions for two-dimensional Heisenberg spin-s = ½ J1–J2 antiferromagnets. Phys. Rev. Lett. 60, 2531–2534 (1988).
Bernu, B., Lhuillier, C. & Pierre, L. Signature of Néel order in exact spectra of quantum antiferromagnets on finite lattices. Phys. Rev. Lett. 69, 2590–2593 (1992).
Capriotti, L., Trumper, A. E. & Sorella, S. Long-range Néel order in the triangular Heisenberg model. Phys. Rev. Lett. 82, 3899–3902 (1999).
Starykh, O. A., Chubukov, A. V. & Abanov, A. G. Flat spin-wave dispersion in a triangular antiferromagnet. Phys. Rev. B 74, 180403 (2006).
Zhitomirsky, M. E. & Chernyshev, A. L. Colloquium: Spontaneous magnon decays. Rev. Mod. Phys. 85, 219–242 (2013).
Read, N. & Sachdev, S. Large-n expansion for frustrated quantum antiferromagnets. Phys. Rev. Lett. 66, 1773–1776 (1991).
Chubukov, A. V., Sachdev, S. & Senthil, T. Quantum phase transitions in frustrated quantum antiferromagnets. Nucl. Phys. B 426, 601–643 (1994).
Kaneko, R., Morita, S. & Imada, M. Gapless spin-liquid phase in an extended spin 1/2 triangular Heisenberg model. J. Phys. Soc. Jpn. 83, 093707 (2014).
Zhu, Z. & White, S. R. Spin liquid phase of the S = ½ J1–J2 Heisenberg model on the triangular lattice. Phys. Rev. B 92, 041105 (2015).
Hu, W.-J., Gong, S.-S., Zhu, W. & Sheng, D. N. Competing spin-liquid states in the spin-½ Heisenberg model on the triangular lattice. Phys. Rev. B 92, 140403 (2015).
Iqbal, Y., Hu, W.-J., Thomale, R., Poilblanc, D. & Becca, F. Spin liquid nature in the Heisenberg J1–J2 triangular antiferromagnet. Phys. Rev. B 93, 144411 (2016).
Saadatmand, S. N. & McCulloch, I. P. Symmetry fractionalization in the topological phase of the spin-½ J1–J2 triangular Heisenberg model. Phys. Rev. B 94, 121111 (2016).
Zhu, Z., Maksimov, P. A., White, S. R. & Chernyshev, A. L. Topography of spin liquids on a triangular lattice. Phys. Rev. Lett. 120, 207203 (2018).
Ma, J. et al. Static and dynamical properties of the spin-1/2 equilateral triangular-lattice antiferromagnet Ba3CoSb2O9. Phys. Rev. Lett. 116, 087201 (2016).
Ito, S. et al. Structure of the magnetic excitations in the spin-1/2 triangular-lattice Heisenberg antiferromagnet Ba3CoSb2O9. Nat. Commun. 8, 235 (2017).
Kamiya, Y. et al. The nature of spin excitations in the one-third magnetization plateau phase of Ba3CoSb2O9. Nat. Commun. 9, 2666 (2018).
Macdougal, D. et al. Avoided quasiparticle decay and enhanced excitation continuum in the spin-½ near-Heisenberg triangular antiferromagnet Ba3CoSb2O9. Phys. Rev. B 102, 064421 (2020).
Arovas, D. P. & Auerbach, A. Functional integral theories of low-dimensional quantum Heisenberg models. Phys. Rev. B 38, 316–332 (1988).
Ghioldi, E. A. et al. Dynamical structure factor of the triangular antiferromagnet: Schwinger boson theory beyond mean field. Phys. Rev. B 98, 184403 (2018).
Ghioldi, E. A. et al. Evidence of two-spinon bound states in the magnetic spectrum of Ba3CoSb2O9. Phys. Rev. B 106, 064418 (2022).
Xie, T. et al. Complete field-induced spectral response of the spin-1/2 triangular-lattice antiferromagnet CsYbSe2. npj Quantum Mater. 8, 48 (2023).
Scheie, A. O. et al. Proximate spin liquid and fractionalization in the triangular antiferromagnet KYbSe2. Nat. Phys. 20, 74–81 (2024).
Scheie, A. O. et al. Spectrum and low-energy gap in triangular quantum spin liquid NaYbSe2. Preprint at https://arxiv.org/html/2406.17773 (2024).
Lyu, Y. et al. Entanglement randomness and gapped itinerant carriers in a frustrated quantum magnet. Phys. Rev. X 15, 041035 (2025).
Sachdev, S. Kagomé- and triangular-lattice Heisenberg antiferromagnets: ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinons. Phys. Rev. B 45, 12377–12396 (1992).
Wang, F. & Vishwanath, A. Spin-liquid states on the triangular and kagomé lattices: a projective-symmetry-group analysis of Schwinger boson states. Phys. Rev. B 74, 174423 (2006).
Shackleton, H. & Sachdev, S. Sign-problem-free effective models of triangular lattice quantum antiferromagnets. Phys. Rev. B 111, 075101 (2025).
Miksch, B. et al. Gapped magnetic ground state in quantum spin liquid candidate κ-(BEDT-TTF)2Cu2(CN)3. Science 372, 276–279 (2021).
Seifert, U. F. P., Willsher, J., Drescher, M., Pollmann, F. & Knolle, J. Spin-Peierls instability of the U(1) Dirac spin liquid. Nat. Commun. 15, 7110 (2024).
Szasz, A., Motruk, J., Zaletel, M. P. & Moore, J. E. Chiral spin liquid phase of the triangular lattice Hubbard model: a density matrix renormalization group study. Phys. Rev. X 10, 021042 (2020).
Ma, Z. et al. Spin-glass ground state in a triangular-lattice compound YbZnGaO4. Phys. Rev. Lett. 120, 087201 (2018).
Norman, M. R. Colloquium: Herbertsmithite and the search for the quantum spin liquid. Rev. Mod. Phys. 88, 041002 (2016).
Han, T.-H. et al. Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet. Nature 492, 406–410 (2012).
Feng, Z. et al. Gapped spin-1/2 spinon excitations in a new kagome quantum spin liquid compound Cu3Zn(OH)6FBr. Chin. Phys. Lett. 34, 077502 (2017).
Khuntia, P. et al. Gapless ground state in the archetypal quantum kagome antiferromagnet ZnCu3(OH)6Cl2. Nat. Phys. 16, 469–474 (2020).
Nilsen, G. J., de Vries, M. A., Stewart, J. R., Harrison, A. & Rønnow, H. M. Low-energy spin dynamics of the s = 1/2 kagome system herbertsmithite. J. Phys.: Condens. Matter 25, 106001 (2013).
Wei, Y. et al. Nonlocal effects of low-energy excitations in quantum-spin-liquid candidate Cu3Zn(OH)6FBr. Chin. Phys. Lett. 38, 097501 (2021).
Zeng, Z. et al. Possible Dirac quantum spin liquid in the kagome quantum antiferromagnet YCu3(OH)6Br2[Brx(OH)1−x]. Phys. Rev. B 105, L121109 (2022).
Zeng, Z. et al. Spectral evidence for Dirac spinons in a kagome lattice antiferromagnet. Nat. Phys. 20, 1097–1102 (2024).
Jeon, S. et al. One-ninth magnetization plateau stabilized by spin entanglement in a kagome antiferromagnet. Nat. Phys. 20, 435–441 (2024).
Chen, X.-H., Huang, Y.-X., Pan, Y. & Mi, J.-X. Quantum spin liquid candidate YCu3(OH)6Br2[Brx(OH)1−x] (x ≈ 0.51): with an almost perfect kagomé layer. J. Magn. Magn. Mater. 512, 167066 (2020).
Ran, Y., Hermele, M., Lee, P. A. & Wen, X.-G. Projected-wave-function study of the spin-1/2 Heisenberg model on the kagomé lattice. Phys. Rev. Lett. 98, 117205 (2007).
Chatterjee, D. et al. From spin liquid to magnetic ordering in the anisotropic kagome Y-kapellasite Y3Cu9(OH)19Cl8: a single-crystal study. Phys. Rev. B 107, 125156 (2023).
Han, L. et al. Spin excitations arising from anisotropic Dirac spinons in YCu3(OD)6Br2[Br0.33(OD)0.67]. Phys. Rev. B 112, 045114 (2025).
Xu, A. et al. Magnetic ground states in the kagome system YCu3(OH)6[(ClxBr1−x)3−y(OH)y]. Phys. Rev. B 110, 085146 (2024).
Yan, S., Huse, D. A. & White, S. R. Spin-liquid ground state of the s = 1/2 kagome Heisenberg antiferromagnet. Science 332, 1173–1176 (2011).
Depenbrock, S., McCulloch, I. P. & Schollwöck, U. Nature of the spin-liquid ground state of the s = 1/2 Heisenberg model on the kagome lattice. Phys. Rev. Lett. 109, 067201 (2012).
Liao, H. J. et al. Gapless spin-liquid ground state in the s = 1/2 kagome antiferromagnet. Phys. Rev. Lett. 118, 137202 (2017).
Li, Z. et al. Antiferromagnetic order and possible quantum spin liquid in kagome antiferromagnet LuCu3(OH)6Br2[Brx(OH)1−x]. Chin. Phys. Lett. 42, 027504 (2025).
Xu, X. Y. et al. Monte Carlo study of lattice compact quantum electrodynamics with fermionic matter: the parent state of quantum phases. Phys. Rev. X 9, 021022 (2019).
Willsher, J. & Knolle, J. Dynamics and stability of U(1) spin liquids beyond mean-field theory: triangular-lattice J1−J2 Heisenberg model. Preprint at https://arxiv.org/abs/2503.13831 (2025).
Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. (N. Y.) 321, 2–111 (2006).
Jackeli, G. & Khaliullin, G. Mott insulators in the strong spin–orbit coupling limit: from Heisenberg to a quantum compass and Kitaev models. Phys. Rev. Lett. 102, 017205 (2009).
Trebst, S. & Hickey, C. Kitaev materials. Phys. Rep. 950, 1–37 (2022). Comprehensive theoretical and experimental review of Kitaev materials.
Möller, M. et al. Rethinking α-RuCl3: parameters, models, and phase diagram. Phys. Rev. B 112, 104403 (2025).
Matsuda, Y., Shibauchi, T. & Kee, H.-Y. Kitaev quantum spin liquids. Rev. Mod. Phys. 97, 045003 (2025).
Kogut, J. B. An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys. 51, 659–713 (1979).