An integrated theoretical and numerical approach to understand modern experiments on quantum magnetism

Nature Physics (2026) Cite this article In recent decades, the study of quantum magnets, which feature unconventional behaviour such as exotic quantum phase transitions and quantum spin liquids, and unconventional magnetic states of matter, has made remarkable progress. However, each of the three foundational pillars—numerical simulations, analytical methods and, to a lesser extent, materials synthesis and experiments—often tends to view itself as the primary driver of the field. Even though the need for collaboration among theory, numerics and experiment to understand the complex phases of quantum magnets is well established, in our view there remains a persistent perception from experts in one area that the other two serve merely as supporting tools, primarily useful for validating the dominant ideas of one speciality, and less relevant to shaping the underlying scientific narrative. In this Perspective, we advocate for a different, more integrated approach to overcome the challenges faced by quantum magnetism researchers. We argue that this alternative mindset has already started to advance the understanding of several important quantum magnetic models and their materials realizations.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutHaldane, 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).Article ADS Google Scholar 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).Article Google Scholar Coldea, R. et al. Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry. Science 327, 177–180 (2010).Article ADS Google Scholar 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).Article ADS Google Scholar Mourigal, M. et al. Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain. Nat. Phys. 9, 435–441 (2013).Article Google Scholar Shao, H. et al. Nearly deconfined spinon excitations in the square-lattice spin-1/2 Heisenberg antiferromagnet. Phys. Rev. X 7, 041072 (2017).

Google Scholar 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).Article ADS Google Scholar Savary, L. & Balents, L. Quantum spin liquids: a review. Rep. Prog. Phys. 80, 016502 (2017).Article ADS Google Scholar Knolle, J. & Moessner, R. A field guide to spin liquids. Annu. Rev. Condens. Matter Phys. 10, 451–472 (2019).Article ADS Google Scholar 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).Article Google Scholar 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).Article ADS Google Scholar Hering, M. et al. Phase diagram of a distorted kagome antiferromagnet and application to Y-kapellasite. npj Comput. Mater. 8, 10 (2022).Article Google Scholar Li, H. et al. Kosterlitz–Thouless melting of magnetic order in the triangular quantum Ising material TmMgGaO4. Nat. Commun. 11, 1111 (2020).Article ADS Google Scholar 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).Article Google Scholar 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).ADS Google Scholar Isakov, S. V. & Moessner, R. Interplay of quantum and thermal fluctuations in a frustrated magnet. Phys. Rev. B 68, 104409 (2003).Article ADS Google Scholar 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).Article ADS Google Scholar Shen, Y. et al. Intertwined dipolar and multipolar order in the triangular-lattice magnet TmMgGaO4. Nat. Commun. 10, 4530 (2019).Article ADS Google Scholar Hu, Z. et al. Evidence of the Berezinskii–Kosterlitz–Thouless phase in a frustrated magnet. Nat. Commun. 11, 5631 (2020).Article ADS Google Scholar Anderson, P. Resonating valence bonds: a new kind of insulator?. Mater. Res. Bull. 8, 153–160 (1973).Article Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar Capriotti, L., Trumper, A. E. & Sorella, S. Long-range Néel order in the triangular Heisenberg model. Phys. Rev. Lett. 82, 3899–3902 (1999).Article ADS Google Scholar Starykh, O. A., Chubukov, A. V. & Abanov, A. G. Flat spin-wave dispersion in a triangular antiferromagnet. Phys. Rev. B 74, 180403 (2006).Article ADS Google Scholar Zhitomirsky, M. E. & Chernyshev, A. L. Colloquium: Spontaneous magnon decays. Rev. Mod. Phys. 85, 219–242 (2013).Article ADS Google Scholar Read, N. & Sachdev, S. Large-n expansion for frustrated quantum antiferromagnets. Phys. Rev. Lett. 66, 1773–1776 (1991).Article ADS Google Scholar Chubukov, A. V., Sachdev, S. & Senthil, T. Quantum phase transitions in frustrated quantum antiferromagnets. Nucl. Phys. B 426, 601–643 (1994).Article ADS Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar Ma, J. et al. Static and dynamical properties of the spin-1/2 equilateral triangular-lattice antiferromagnet Ba3CoSb2O9. Phys. Rev. Lett. 116, 087201 (2016).Article ADS Google Scholar Ito, S. et al. Structure of the magnetic excitations in the spin-1/2 triangular-lattice Heisenberg antiferromagnet Ba3CoSb2O9. Nat. Commun. 8, 235 (2017).Article ADS Google Scholar Kamiya, Y. et al. The nature of spin excitations in the one-third magnetization plateau phase of Ba3CoSb2O9. Nat. Commun. 9, 2666 (2018).Article ADS Google Scholar 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).Article ADS Google Scholar Arovas, D. P. & Auerbach, A. Functional integral theories of low-dimensional quantum Heisenberg models. Phys. Rev. B 38, 316–332 (1988).Article ADS Google Scholar Ghioldi, E. A. et al. Dynamical structure factor of the triangular antiferromagnet: Schwinger boson theory beyond mean field. Phys. Rev. B 98, 184403 (2018).Article ADS Google Scholar Ghioldi, E. A. et al. Evidence of two-spinon bound states in the magnetic spectrum of Ba3CoSb2O9. Phys. Rev. B 106, 064418 (2022).Article ADS Google Scholar Xie, T. et al. Complete field-induced spectral response of the spin-1/2 triangular-lattice antiferromagnet CsYbSe2. npj Quantum Mater. 8, 48 (2023).Article ADS Google Scholar Scheie, A. O. et al. Proximate spin liquid and fractionalization in the triangular antiferromagnet KYbSe2. Nat. Phys. 20, 74–81 (2024).Article Google Scholar 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).

Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar Shackleton, H. & Sachdev, S. Sign-problem-free effective models of triangular lattice quantum antiferromagnets. Phys. Rev. B 111, 075101 (2025).Article ADS Google Scholar Miksch, B. et al. Gapped magnetic ground state in quantum spin liquid candidate κ-(BEDT-TTF)2Cu2(CN)3. Science 372, 276–279 (2021).Article ADS Google Scholar 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).Article ADS Google Scholar 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).

Google Scholar Ma, Z. et al. Spin-glass ground state in a triangular-lattice compound YbZnGaO4. Phys. Rev. Lett. 120, 087201 (2018).Article ADS Google Scholar Norman, M. R. Colloquium: Herbertsmithite and the search for the quantum spin liquid. Rev. Mod. Phys. 88, 041002 (2016).Article ADS MathSciNet Google Scholar Han, T.-H. et al. Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet. Nature 492, 406–410 (2012).Article ADS Google Scholar 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).Article ADS Google Scholar Khuntia, P. et al. Gapless ground state in the archetypal quantum kagome antiferromagnet ZnCu3(OH)6Cl2. Nat. Phys. 16, 469–474 (2020).Article Google Scholar 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).ADS Google Scholar Wei, Y. et al. Nonlocal effects of low-energy excitations in quantum-spin-liquid candidate Cu3Zn(OH)6FBr. Chin. Phys. Lett. 38, 097501 (2021).Article ADS Google Scholar 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).Article ADS Google Scholar Zeng, Z. et al. Spectral evidence for Dirac spinons in a kagome lattice antiferromagnet. Nat. Phys. 20, 1097–1102 (2024).Article Google Scholar Jeon, S. et al. One-ninth magnetization plateau stabilized by spin entanglement in a kagome antiferromagnet. Nat. Phys. 20, 435–441 (2024).Article Google Scholar 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).Article Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar 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).Article ADS Google Scholar Liao, H. J. et al. Gapless spin-liquid ground state in the s = 1/2 kagome antiferromagnet. Phys. Rev. Lett. 118, 137202 (2017).Article ADS Google Scholar 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).Article ADS Google Scholar 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).

Google Scholar 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).Article ADS MathSciNet Google Scholar 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).Article ADS Google Scholar Trebst, S. & Hickey, C. Kitaev materials. Phys. Rep. 950, 1–37 (2022). Comprehensive theoretical and experimental review of Kitaev materials.Article ADS Google Scholar 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).Article ADS Google Scholar Kogut, J. B. An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys. 51, 659–713 (1979).Article ADS MathSciNet Google Scholar Download referencesZ.Y.M. thanks C. Zhou for help with creating the figures and acknowledges support from the Research Grants Council (RGC) of Hong Kong (projects AoE/P701/20, C7037-22GF, 17302223, 17301924, 17301725), the ANR/RGC Joint Research Scheme sponsored by the RGC of Hong Kong and French National Research Agency (project A_HKU703/22) and the State Key Laboratory of Optical Quantum Materials at HKU. We thank the HPC2021 system under Information Technology Services and the Blackbody high-performance computing system at the Department of Physics, University of Hong Kong, as well as the Beijing PARATERA Tech Co., Ltd. (https://cloud.paratera.com), for providing high-performance computing resources that have contributed to the research results reported within this paper. C.D.B. acknowledges support by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center. S.L. acknowledges support from the National Key Research and Development Program of China (grants 2022YFA1403400, 2021YFA1400400) and the Chinese Academy of Sciences (grants XDB33000000, GJTD-2020-01). Z.Y.M. and C.D.B. are grateful to the Pollica Physics Centre for hospitality during the stimulating workshop ‘Exotic quantum matter: from quantum spin liquids to novel field theories’ (2024).Department of Physics, The University of Hong Kong, Hong Kong, ChinaZi Yang Meng (孟子楊)State Key Laboratory of Optical Quantum Materials, The University of Hong Kong, Hong Kong, ChinaZi Yang Meng (孟子楊)Department of Physics and Astronomy, University of Tennessee, Knoxville, TN, USACristian D. BatistaNeutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN, USACristian D. BatistaBeijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, ChinaShiliang Li (李世亮)School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, ChinaShiliang Li (李世亮)Search author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarZ.Y.M. and C.D.B. conceived the content and structure of the paper and all three authors contributed to the writing and discussion of the paper.Correspondence to Zi Yang Meng (孟子楊), Cristian D. Batista or Shiliang Li (李世亮).The authors declare no competing interests.Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsMeng, Z.Y., Batista, C.D. & Li, S. An integrated theoretical and numerical approach to understand modern experiments on quantum magnetism. Nat. Phys. (2026). https://doi.org/10.1038/s41567-026-03304-5Download citationReceived: 18 November 2024Accepted: 17 April 2026Published: 29 June 2026Version of record: 29 June 2026DOI: https://doi.org/10.1038/s41567-026-03304-5Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative