Six-state clock physics in an atomically thin antiferromagnet

Nature Materials (2026)Cite this article The study of collective behaviour driven by spontaneous symmetry breaking and topology is crucial for understanding phase transitions in quantum matter. The two-dimensional (2D) XY model, describing spins with continuous in-plane rotational symmetry, hosts the topological Berezinskii–Kosterlitz–Thouless (BKT) transition, where vortex–antivortex binding induces quasi-long-range order. This model was later extended to include anisotropy fields, leading to the six-state clock model, which predicts the instability of the BKT phase toward true long-range order at low temperatures. Here we investigate this physics in the van der Waals antiferromagnet NiPS3 using nonlinear optical micropolarimetry. As the material is thinned to a monolayer, its magnetic response switches abruptly from the 3D XXZ behaviour of multilayers to a distinct 2D regime consistent with a BKT state. Upon further cooling, the monolayer BKT phase becomes unstable and transforms into a pinned state with long-range order. These results, corroborated by Monte Carlo simulations, open pathways to explore spin vortices and topological dynamics in 2D antiferromagnets.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 checkoutThe data required to assess the conclusions of this study are provided as source data files accompanying the article. Source data are provided with this paper.Mermin, N. D. The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591 (1979).Article CAS Google Scholar Paoletti, M. S. & Lathrop, D. P. Quantum turbulence. Annu. Rev. Condens. Matter Phys. 2, 213–234 (2011).Article CAS Google Scholar Landau, L. et al. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion 8, 101–114 (1935).

Google Scholar Nelson, D. R. & Halperin, B. I. Dislocation-mediated melting in two dimensions. Phys. Rev. B 19, 2457 (1979).Article CAS Google Scholar Mross, D. F. & Senthil, T. Theory of a continuous stripe melting transition in a two-dimensional metal: a possible application to cuprate superconductors. Phys. Rev. Lett. 108, 267001 (2012).Article PubMed Google Scholar Mermin, N. D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133 (1966).Article CAS Google Scholar Berezinsky, V. L. Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. Quantum systems. Sov. Phys. JETP 34, 610 (1972).

Google Scholar Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181 (1973).Article CAS Google Scholar Kosterlitz, J. M. The critical properties of the two-dimensional XY model. J. Phys. C 7, 1046 (1974).Article Google Scholar José, J. V., Kadanoff, L. P., Kirkpatrick, S. & Nelson, D. R. Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model. Phys. Rev. B 16, 1217–1241 (1977).Article Google Scholar Chatterjee, S., Puri, S. & Paul, R. Ordering kinetics in the q-state clock model: scaling properties and growth laws. Phys. Rev. E 98, 032109 (2018).Article CAS Google Scholar Li, Z.-Q. et al. Critical properties of the two-dimensional q-state clock model. Phys. Rev. E 101, 060105 (2020).Article CAS PubMed Google Scholar Wu, F.-Y. The Potts model. Rev. Mod. Phys. 54, 235 (1982).Article Google Scholar Fernandes, R. M., Orth, P. P. & Schmalian, J. Intertwined vestigial order in quantum materials: mematicity and beyond. Annu. Rev. Condens. Matter Phys. 10, 133–154 (2019).Article Google Scholar Little, A. et al. Three-state nematicity in the triangular lattice antiferromagnet Fe1/3NbS2. Nat. Mater. 19, 1062–1067 (2020).Article CAS PubMed Google Scholar Ni, Z., Huang, N., Haglund, A. V., Mandrus, D. G. & Wu, L. Observation of giant surface second-harmonic generation coupled to nematic orders in the van der Waals antiferromagnet FePS3. Nano Lett. 22, 3283–3288 (2022).Article CAS PubMed Google Scholar Elitzur, S., Pearson, R. & Shigemitsu, J. Phase structure of discrete Abelian spin and gauge systems. Phys. Rev. D 19, 3698 (1979).Article Google Scholar Tobochnik, J. Properties of the q-state clock model for q = 4, 5, and 6. Phys. Rev. B 26, 6201 (1982).Article CAS Google Scholar Challa, M. S. S. & Landau, D. P. Critical behavior of the six-state clock model in two dimensions. Phys. Rev. B 33, 437–443 (1986).Article CAS Google Scholar Ueda, H. et al. Finite-m scaling analysis of Berezinskii–Kosterlitz–Thouless phase transitions and entanglement spectrum for the six-state clock model. Phys. Rev. E 101, 062111 (2020).Article CAS PubMed Google Scholar Okabe, Y. & Otsuka, H. BKT transitions of the XY and six-state clock models on the various two-dimensional lattices. J. Phys. A 58, 065003 (2025).Article Google Scholar Taroni, A., Bramwell, S. T. & Holdsworth, P. C. W. Universal window for two-dimensional critical exponents. J. Phys. Condens. Matter 20, 275233 (2008).Article CAS PubMed Google Scholar Kim, S. Y. et al. Charge–spin correlation in van der Waals antiferromagnet NiPS3. Phys. Rev. Lett. 120, 136402 (2018).Article CAS PubMed Google Scholar Sun, Z. et al. Dimensionality crossover to a two-dimensional vestigial nematic state from a three-dimensional antiferromagnet in a honeycomb van der Waals magnet. Nat. Phys. 20, 1764–1771 (2024).Article CAS Google Scholar Lee, Y. et al. Imaging thermally fluctuating Néel vectors in van der Waals antiferromagnet NiPS3. Nano Lett. 24, 6043–6050 (2024).Article CAS PubMed Google Scholar Kim, K. et al. Suppression of magnetic ordering in XXZ-type antiferromagnetic monolayer NiPS3. Nat. Commun. 10, 345 (2019).Article PubMed PubMed Central Google Scholar Wildes, A. R. et al. Magnetic structure of the quasi-two-dimensional antiferromagnet NiPS3. Phys. Rev. B 92, 224408 (2015).Article Google Scholar Tan, Q. et al. Observation of three-state nematicity and domain evolution in atomically thin antiferromagnetic NiPS3. Nano Lett. 24, 7166–7172 (2024).Article CAS Google Scholar Cuccoli, A., Roscilde, T., Tognetti, V., Vaia, R. & Verrucchi, P.

Quantum Monte Carlo study of \(S=\frac{1}{2}\) weakly anisotropic antiferromagnets on the square lattice. Phys. Rev. B 67, 104414 (2003).Article Google Scholar Chu, H. et al. Linear magnetoelectric phase in ultrathin MnPS3 probed by optical second harmonic generation. Phys. Rev. Lett. 124, 027601 (2020).Article CAS PubMed Google Scholar Lançon, D., Ewings, R. A., Guidi, T., Formisano, F. & Wildes, A. R. Magnetic exchange parameters and anisotropy of the quasi-two-dimensional antiferromagnet NiPS3. Phys. Rev. B 98, 134414 (2018).Article Google Scholar Belvin, C. A. et al. Exciton-driven antiferromagnetic metal in a correlated van der Waals insulator. Nat. Commun. 12, 4837 (2021).Article CAS PubMed PubMed Central Google Scholar Plumley, R. et al. 3D Heisenberg universality in the van der Waals antiferromagnet NiPS3. npj Quantum Mater. 9, 95 (2024).Article CAS Google Scholar Hu, L. et al. Observation of a magnetic phase transition in monolayer NiPS3. Phys. Rev. B 107, L220407 (2023).Article CAS Google Scholar Kim, D. S. et al. Anisotropic excitons reveal local spin chain directions in a van der Waals antiferromagnet. Adv. Mater. 35, 2206585 (2023).Article CAS Google Scholar Liu, Q. et al. Magnetic order in XY-type antiferromagnetic monolayer CoPS3 revealed by Raman spectroscopy. Phys. Rev. B 103, 235411 (2021).Article CAS Google Scholar Na, W. et al. Direct observation and analysis of low-energy magnons with Raman spectroscopy in atomically thin NiPS3. ACS Nano 18, 20482–20492 (2024).Article CAS Google Scholar Nguyen, M. H., Park, G., Park, J.-G. & Cheong, H. Interlayer interaction and Davydov splitting in antiferromagnetic few-layer NiPS3. npj Quantum Mater. 10, 65 (2025).Article Google Scholar Kang, S. et al. Coherent many-body exciton in van der Waals antiferromagnet NiPS3. Nature 583, 785–789 (2020).Article CAS PubMed Google Scholar Hwangbo, K. et al. Highly anisotropic excitons and multiple phonon bound states in a van der Waals antiferromagnetic insulator. Nat. Nanotechnol. 16, 655–660 (2021).Article CAS PubMed Google Scholar Houmes, M. J. A. et al. Magnetic order in 2D antiferromagnets revealed by spontaneous anisotropic magnetostriction. Nat. Commun. 14, 8503 (2023).Article CAS PubMed PubMed Central Google Scholar Fiebig, M. et al. Second harmonic generation in the centrosymmetric antiferromagnet NiO. Phys. Rev. Lett. 87, 137202 (2001).Article CAS PubMed Google Scholar Wang, X. et al. Electronic Raman scattering in the 2D antiferromagnet NiPS3. Sci. Adv. 8, eabl7707 (2022).Article CAS PubMed PubMed Central Google Scholar Wang, X. et al. Spin-induced linear polarization of photoluminescence in antiferromagnetic van der Waals crystals. Nat. Mater. 20, 964–970 (2021).Article CAS PubMed Google Scholar Afanasiev, D. et al. Controlling the anisotropy of a van der Waals antiferromagnet with light. Sci. Adv. 7, eabf3096 (2021).Article CAS PubMed PubMed Central Google Scholar Bramwell, S. T. & Holdsworth, P. C. W. Magnetization and universal sub-critical behaviour in two-dimensional XY magnets. J. Phys. Condens. Matter 5, L53 (1993).Article Google Scholar Bedoya-Pinto, A. et al. Intrinsic 2D-XY ferromagnetism in a van der Waals monolayer. Science 374, 616–620 (2021).Article CAS PubMed Google Scholar Szeto, K. Y., Chen, S. T. & Dresselhaus, G. Temperature dependence of the magnetic susceptibility of CoCl2–graphite intercalation compounds. Phys. Rev. B 32, 4628–4638 (1985).Article CAS Google Scholar Fiory, A., Hebard, A. F. & Glaberson, W. Superconducting phase transitions in indium/indium-oxide thin-film composites. Phys. Rev. B 28, 5075 (1983).Article CAS Google Scholar Minnhagen, P. The two-dimensional Coulomb gas, vortex unbinding, and superfluid-superconducting films. Rev. Mod. Phys. 59, 1001 (1987).Article CAS Google Scholar Caputo, D. et al. Topological order and thermal equilibrium in polariton condensates. Nat. Mater. 17, 145–151 (2018).Article CAS PubMed Google Scholar Sunami, S. et al. Universal scaling of the dynamic BKT transition in quenched 2D Bose gases. Science 382, 443–447 (2023).Article CAS PubMed Google Scholar Cheon, C.-Y. et al. Nature of 2D XY antiferromagnetism in a van der Waals monolayer. Nat. Commun. 17, 60 (2026).Article CAS Google Scholar Meier, D. et al. Second harmonic generation on incommensurate structures: the case of multiferroic MnWO4. Phys. Rev. B 82, 155112 (2010).Article Google Scholar Denev, S. A., Lummen, T. T., Barnes, E., Kumar, A. & Gopalan, V. Probing ferroelectrics using optical second harmonic generation. J. Amer. Ceram. Soc. 94, 2699–2727 (2011).Article CAS Google Scholar Lee, K. et al. Magnetic order and symmetry in the 2D semiconductor CrSBr. Nano Lett. 21, 3511–3517 (2021).Article CAS PubMed Google Scholar Jin, W. et al. Observation of a ferro-rotational order coupled with second-order nonlinear optical fields. Nat. Phys. 16, 42–46 (2020).Article CAS Google Scholar Gao, F. Y. et al. Giant chiral magnetoelectric oscillations in a van der Waals multiferroic. Nature 632, 273–279 (2024).Article CAS PubMed PubMed Central Google Scholar Seifert, U. F. P., Ye, M. & Balents, L. Ultrafast optical excitation of magnetic dynamics in van der Waals magnets: coherent magnons and BKT dynamics in NiPS3. Phys. Rev. B 105, 155138 (2022).Article CAS Google Scholar DiScala, M. F. et al. Elucidating the role of dimensionality on the electronic structure of the van der Waals antiferromagnet NiPS3. Adv. Phys. Res. 3, 2300096 (2024).Article Google Scholar Evans, R. F. L. et al. Atomistic spin model simulations of magnetic nanomaterials. J. Phys. Condens. Matter 26, 103202 (2014).Article CAS PubMed Google Scholar Download referencesThis research was primarily supported by the National Science Foundation through the Center for Dynamics and Control of Materials: an NSF MRSEC under cooperative agreement number DMR-2308817. Part of the experiments were performed at the user facility supported by the National Science Foundation through the Center for Dynamics and Control of Materials under cooperative agreement number DMR-2308817. We also gratefully acknowledge financial support from Love, Tito’s for the sample preparation activities carried out in the Baldini group at the University of Texas at Austin. Work in the Baldini group was additionally supported by the Robert A. Welch Foundation under grant F-2092-20250403 (F.Y.G. for set-up development, data taking, data analysis and paper writing), the W. M. Keck Foundation under grant 996588 (X.P., for set-up development and data taking), the National Science Foundation under the NSF CAREER award 2441874 (F.B. for paper writing), the Air Force Office of Scientific Research under Young Investigator Program award FA9550-24-1-0097 (to S.Z. for optimization of the SHG set-up), and the United States Army Research Office (W911NF-23-1-0394) (E.B. for data interpretation, paper writing and project supervision). F.Y.G. acknowledges additional support from the Texas Quantum Institute, while F.B. acknowledges additional support from the Swiss NSF under fellowship P500PT_214437. D.S.K. and K.P.L. are partially supported by the NSF Designing Materials to Revolutionize and Engineer our Future (DMREF) program via grant DMR-2118806 and NSF EPM programme grant DMR-2225645. D.S.K. also acknowledges support from the Glenn Focht Memorial Fellowship Program. X. Li gratefully acknowledges the support from the Center for Energy Efficient Magnonics, an Energy Frontier Research Center funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award number DE-AC02-76SF00515, for support on layered magnetic materials and from the Robert A.

Welch Foundation Chair F-0014 for sample preparation. C.L. and A.H.M. were supported by the Robert A. Welch Foundation under grant TBF1473. A.K. was supported by the Gordon and Betty Moore Foundation EPiQS Initiative through grant GBMF8684 at the Massachusetts Institute of Technology. S.-F.L. acknowledges financial support provided by Academia Sinica, Project AS-iMATE-111-11. The collaboration between S.-F.L. and X. Li is facilitated by AFOSR grant FA2386-21-1-4067 and the National Science and Technology Council of Taiwan under project number MOST110-2124-M-001-009-MY3. R.S. acknowledges the financial support provided by the National Science and Technology Council in Taiwan under project numbers NSTC-114-2124-M-001-009 and NSTC-113-2112M001-045-MY3, and support from Academia Sinica under project AS-iMATE-114-12.These authors contributed equally: Frank Y. Gao, Dong Seob Kim.Department of Physics and Center for Complex Quantum Systems, The University of Texas at Austin, Austin, TX, USAFrank Y. Gao, Dong Seob Kim, Chao Lei, Xinyue Peng, Xiaohui Liu, Francesco Barantani, Shangjie Zhang, Kyoung Pyo Lee, David Lujan, Saba Arash, Xiaoqin Li, Allan H. MacDonald & Edoardo BaldiniDepartment of Physics, Massachusetts Institute of Technology, Cambridge, MA, USAAjesh KumarInstitute of Physics, Academia Sinica, Taipei, TaiwanKalaivanan Raju, Sankar Raman & Shang-Fan LeeDepartment of Physics & Astronomy, The University of Utah, Salt Lake City, UT, USAMengxing YeSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarE.B. conceived the study. F.Y.G., D.S.K., X. Li and E.B. designed the research. F.Y.G., D.S.K. and X.P. performed the low-temperature optical experiments with support from S.Z., D.L. and S.A. and under the supervision of X. Li and E.B. D.S.K. and K.P.L. exfoliated the few-layer samples and characterized with atomic force microscopy, with support from X. Liu and under the supervision of X. Li. C.L. performed Monte Carlo simulations, and A.K. performed group theory analysis of the SHG signal under the supervision of A.H.M. M.Y. performed theoretical analysis of the anisotropic SHG signal and the q-clock model. R.K., R.S. and S.-F.L. grew the NiPS3 samples and performed X-ray diffraction and magnetic susceptibility characterization. F.Y.G., D.S.K., C.L., A.K., M.Y., F.B. and E.B. prepared the paper with input from all other authors. E.B. supervised the project.Correspondence to Edoardo Baldini.The authors declare no competing interests.Nature Materials thanks Istvan Kezsmarki, José Lorenzana and the other, anonymous, reviewer(s) 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.Photoluminescence spectra of NiPS3 with various thicknesses from bulk to monolayer at 3 K. A sharp exciton emission dominates the response in the bulk compound and disappears in the bilayer and monolayer flakes. The curves are vertically separated for clarity.Source dataSupplementary Notes 1–7, Figs. 1–21 and Tables 1–4.Fig2b: AFM image source data Fig2c: AFM linecut source data Fig2d: 2L Raman spectra source data Fig2e: 1L Raman spectra source data Fig2f: Raman Peak Splitting source data Fig2de_Raman_Tdependence_1L(2L, 3L, 4L)_0de(45de): All Raman spectra source data.Fig3c: 2L-4L SHG temperature dependence source data Fig3d: 1L SHG temperature dependence source data Fig3e_MSF_K_Dependence: Monte Carlo Calculation source data for K6 dependence Fig3f_MSF_L_Dependence: Monte Carlo Calculation source data for L dependence Fig3f_MSF_L_Dependence_fit: Fits to Monte Carlo Calculation source data for L dependence.Fig4a: 4K SHG Polar Plots 1L-4L Fig4b: Temperature-dependent SHG Polar Plots 1L Fig4c: SHG Anisotropy Source Data Fig4ab README - Fig4ab SHG Polarimetry: Explanation Fig4ab_SHG_Polar_1L(2L, 3L, QL)_XX(XY): Temperature dependent SHG Polarimetry source data Fig4ab_SHG_Polar_ML(BL, TL, QL)_XX(XY: Fitting source data.FigE1: PL spectrum source data.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 permissionsGao, F.Y., Kim, D.S., Lei, C. et al. Six-state clock physics in an atomically thin antiferromagnet. Nat. Mater. (2026). https://doi.org/10.1038/s41563-026-02516-7Download citationReceived: 12 May 2025Accepted: 26 January 2026Published: 23 February 2026Version of record: 23 February 2026DOI: https://doi.org/10.1038/s41563-026-02516-7Anyone 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