Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Frustration- and doping-induced magnetism in a Fermi–Hubbard simulator


Geometrical frustration in strongly correlated systems can give rise to a plethora of novel ordered states and intriguing magnetic phases, such as quantum spin liquids1,2,3. Promising candidate materials for such phases4,5,6 can be described by the Hubbard model on an anisotropic triangular lattice, a paradigmatic model capturing the interplay between strong correlations and magnetic frustration7,8,9,10,11. However, the fate of frustrated magnetism in the presence of itinerant dopants remains unclear, as well as its connection to the doped phases of the square Hubbard model12. Here we investigate the local spin order of a Hubbard model with controllable frustration and doping, using ultracold fermions in anisotropic optical lattices continuously tunable from a square to a triangular geometry. At half-filling and strong interactions U/t ≈ 9, we observe at the single-site level how frustration reduces the range of magnetic correlations and drives a transition from a collinear Néel antiferromagnet to a short-range correlated 120° spiral phase. Away from half-filling, the triangular limit shows enhanced antiferromagnetic correlations on the hole-doped side and a reversal to ferromagnetic correlations at particle dopings above 20%, hinting at the role of kinetic magnetism in frustrated systems. This work paves the way towards exploring possible chiral ordered or superconducting phases in triangular lattices8,13 and realizing tt′ square lattice Hubbard models that may be essential to describe superconductivity in cuprate materials14.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Investigating frustration on a tunable triangular lattice with a quantum gas microscope.
Fig. 2: Frustrating short-range antiferromagnetic order in the square-to-triangular lattice transition.
Fig. 3: Particle–hole asymmetry of magnetic correlations and particle-doping-induced ferromagnetism.
Fig. 4: Next-nearest-neighbour spin correlations at finite doping.

Data availability

The datasets generated and analysed during this study are available from the corresponding author on reasonable request. Source data are provided with this paper.

Code availability

The code used for the analysis are available from the corresponding author on reasonable request.


  1. Anderson, P. W. Resonating valence bonds: a new kind of insulator? Mater. Res. Bull. 8, 153–160 (1973).

    Article  CAS  Google Scholar 

  2. Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).

    Article  ADS  CAS  PubMed  Google Scholar 

  3. Zhou, Y., Kanoda, K. & Ng, T.-K. Quantum spin liquid states. Rev. Mod. Phys. 89, 025003 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  4. Williams, J. M. et al. Organic superconductors—new benchmarks. Science 252, 1501–1508 (1991).

    Article  ADS  CAS  PubMed  Google Scholar 

  5. Kino, H. & Fukuyama, H. Phase diagram of two-dimensional organic conductors: (BEDT-TTF) 2X. J. Phys. Soc. Jpn 65, 2158–2169 (1996).

    Article  ADS  CAS  Google Scholar 

  6. Shimizu, Y., Miyagawa, K., Kanoda, K., Maesato, M. & Saito, G. Spin liquid state in an organic Mott insulator with a triangular lattice. Phys. Rev. Lett. 91, 107001 (2003).

    Article  ADS  CAS  PubMed  Google Scholar 

  7. Laubach, M., Thomale, R., Platt, C., Hanke, W. & Li, G. Phase diagram of the Hubbard model on the anisotropic triangular lattice. Phys. Rev. B 91, 245125 (2015).

    Article  ADS  Google Scholar 

  8. 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).

    CAS  Google Scholar 

  9. Motrunich, O. I. Variational study of triangular lattice spin 1/2 model with ring exchanges and spin liquid state in κ−(ET)2Cu2(CN)3. Phys. Rev. B 72, 045105 (2005).

    Article  ADS  Google Scholar 

  10. Wietek, A. et al. Mott insulating states with competing orders in the triangular lattice Hubbard model. Phys. Rev. X 11, 041013 (2021).

    CAS  Google Scholar 

  11. Zhu, Z., Sheng, D. N. & Vishwanath, A. Doped Mott insulators in the triangular-lattice Hubbard model. Phys. Rev. B 105, 205110 (2022).

    Article  ADS  CAS  Google Scholar 

  12. Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).

    Article  ADS  CAS  Google Scholar 

  13. Song, X.-Y., Vishwanath, A. & Zhang, Y.-H. Doping the chiral spin liquid: topological superconductor or chiral metal. Phys. Rev. B 103, 165138 (2021).

    Article  ADS  CAS  Google Scholar 

  14. Pavarini, E., Dasgupta, I., Saha-Dasgupta, T., Jepsen, O. & Andersen, O. K. Band-structure trend in hole-doped cuprates and correlation with Tcmax. Phys. Rev. Lett. 87, 047003 (2001).

    Article  ADS  CAS  PubMed  Google Scholar 

  15. Wannier, G. H. Antiferromagnetism. The triangular Ising net. Phys. Rev. 79, 357–364 (1950).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Lee, P. A. An end to the drought of quantum spin liquids. Science 321, 1306–1307 (2008).

    Article  CAS  PubMed  Google Scholar 

  17. Savary, L. & Balents, L. Quantum spin liquids: a review. Rep. Prog. Phys. 80, 016502 (2016).

    Article  ADS  PubMed  Google Scholar 

  18. Yang, J., Liu, L., Mongkolkiattichai, J. & Schauss, P. Site-resolved imaging of ultracold fermions in a triangular-lattice quantum gas microscope. PRX Quantum 2, 020344 (2021).

    Article  ADS  Google Scholar 

  19. Mongkolkiattichai, J., Liu, L., Garwood, D., Yang, J. & Schauss, P. Quantum gas microscopy of a geometrically frustrated Hubbard system. Preprint at (2022).

  20. Struck, J. et al. Quantum simulation of frustrated classical magnetism in triangular optical lattices. Science 333, 996–999 (2011).

    Article  ADS  CAS  PubMed  Google Scholar 

  21. Tarruell, L., Greif, D., Uehlinger, T., Jotzu, G. & Esslinger, T. Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice. Nature 483, 302–305 (2012).

    Article  ADS  CAS  PubMed  Google Scholar 

  22. Sebby-Strabley, J. et al. Preparing and probing atomic number states with an atom interferometer. Phys. Rev. Lett. 98, 200405 (2007).

    Article  ADS  CAS  PubMed  Google Scholar 

  23. Jo, G.-B. et al. Ultracold atoms in a tunable optical kagome lattice. Phys. Rev. Lett. 108, 045305 (2012).

    Article  ADS  PubMed  Google Scholar 

  24. Yamamoto, R., Ozawa, H., Nak, D. C., Nakamura, I. & Fukuhara, T. Single-site-resolved imaging of ultracold atoms in a triangular optical lattice. New J. Phys. 22, 123028 (2020).

    Article  ADS  CAS  Google Scholar 

  25. Trisnadi, J., Zhang, M., Weiss, L. & Chin, C. Design and construction of a quantum matter synthesizer. Rev. Sci. Instrum. 93, 083203 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  26. Hirsch, J. E. & Tang, S. Antiferromagnetism in the two-dimensional Hubbard model. Phys. Rev. Lett. 62, 591–594 (1989).

    Article  ADS  CAS  PubMed  Google Scholar 

  27. Singh, R. R. P. & Huse, D. A. Three-sublattice order in triangular- and Kagomé-lattice spin-half antiferromagnets. Phys. Rev. Lett. 68, 1766–1769 (1992).

    Article  ADS  CAS  PubMed  Google Scholar 

  28. Huse, D. A. & Elser, V. Simple variational wave functions for two-dimensional Heisenberg spin-½ antiferromagnets. Phys. Rev. Lett. 60, 2531–2534 (1988).

    Article  ADS  CAS  PubMed  Google Scholar 

  29. Jolicoeur, T. & Le Guillou, J. C. Spin-wave results for the triangular Heisenberg antiferromagnet. Phys. Rev. B 40, 2727–2729 (1989).

    Article  ADS  CAS  Google Scholar 

  30. 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  CAS  Google Scholar 

  31. Trumper, A. E. Spin-wave analysis to the spatially anisotropic Heisenberg antiferromagnet on a triangular lattice. Phys. Rev. B 60, 2987–2989 (1999).

    Article  ADS  CAS  Google Scholar 

  32. Merino, J., McKenzie, R. H., Marston, J. B. & Chung, C. H. The Heisenberg antiferromagnet on an anisotropic triangular lattice: linear spin-wave theory. J. Phys. Condens. Matter 11, 2965–2975 (1999).

    Article  ADS  CAS  Google Scholar 

  33. Weihong, Z., McKenzie, R. H. & Singh, R. R. P. Phase diagram for a class of spin-½ Heisenberg models interpolating between the square-lattice, the triangular-lattice, and the linear-chain limits. Phys. Rev. B 59, 14367–14375 (1999).

    Article  ADS  Google Scholar 

  34. Parsons, M. F. et al. Site-resolved measurement of the spin-correlation function in the Fermi-Hubbard model. Science 353, 1253–1256 (2016).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  35. Chang, C.-C., Scalettar, R. T., Gorelik, E. V. & Blümer, N. Discriminating antiferromagnetic signatures in systems of ultracold fermions by tunable geometric frustration. Phys. Rev. B 88, 195121 (2013).

    Article  ADS  Google Scholar 

  36. Tasaki, H. The Hubbard model - an introduction and selected rigorous results. J. Phys. Condens. Matter 10, 4353 (1998).

    Article  ADS  CAS  Google Scholar 

  37. Morera, I. et al. High-temperature kinetic magnetism in triangular lattices. Phys. Rev. Res. 5, L022048 (2023).

    Article  CAS  Google Scholar 

  38. Nagaoka, Y. Ferromagnetism in a narrow, almost half-filled s band. Phys. Rev. 147, 392–405 (1966).

    Article  ADS  CAS  Google Scholar 

  39. Hirsch, J. E. Two-dimensional Hubbard model: numerical simulation study. Phys. Rev. B 31, 4403–4419 (1985).

    Article  ADS  CAS  Google Scholar 

  40. Haerter, J. O. & Shastry, B. S. Kinetic antiferromagnetism in the triangular lattice. Phys. Rev. Lett. 95, 087202 (2005).

    Article  ADS  PubMed  Google Scholar 

  41. Hanisch, T., Kleine, B., Ritzl, A. & Müller-Hartmann, E. Ferromagnetism in the Hubbard model: instability of the Nagaoka state on the triangular, honeycomb and kagome lattices. Ann. Phys. 507, 303–328 (1995).

    Article  Google Scholar 

  42. Martin, I. & Batista, C. D. Itinerant electron-driven chiral magnetic ordering and spontaneous quantum Hall effect in triangular lattice models. Phys. Rev. Lett. 101, 156402 (2008).

    Article  ADS  PubMed  Google Scholar 

  43. Merino, J., Powell, B. J. & McKenzie, R. H. Ferromagnetism, paramagnetism, and a Curie-Weiss metal in an electron-doped Hubbard model on a triangular lattice. Phys. Rev. B 73, 235107 (2006).

    Article  ADS  Google Scholar 

  44. Weber, C., Läuchli, A., Mila, F. & Giamarchi, T. Magnetism and superconductivity of strongly correlated electrons on the triangular lattice. Phys. Rev. B 73, 014519 (2006).

    Article  ADS  Google Scholar 

  45. Lee, K., Sharma, P., Vafek, O. & Changlani, H. J. Triangular lattice Hubbard model physics at intermediate temperatures. Phys. Rev. B 107, 235105 (2023).

    Article  ADS  CAS  Google Scholar 

  46. Tang, Y. et al. Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices. Nature 579, 353–358 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  47. Koepsell, J. et al. Imaging magnetic polarons in the doped Fermi–Hubbard model. Nature 572, 358–362 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  48. Garwood, D., Mongkolkiattichai, J., Liu, L., Yang, J. & Schauss, P. Site-resolved observables in the doped spin-imbalanced triangular Hubbard model. Phys. Rev. A 106, 013310 (2022).

    Article  ADS  CAS  Google Scholar 

  49. Brown, P. T. et al. Angle-resolved photoemission spectroscopy of a Fermi–Hubbard system. Nat. Phys. 16, 26–31 (2020).

    Article  CAS  Google Scholar 

  50. Chen, B.-B. et al. Quantum spin liquid with emergent chiral order in the triangular-lattice Hubbard model. Phys. Rev. B 106, 094420 (2022).

    Article  ADS  CAS  Google Scholar 

  51. Greif, D. et al. Site-resolved imaging of a fermionic Mott insulator. Science 351, 953–957 (2016).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  52. Kale, A. et al. Schrieffer-Wolff transformations for experiments: dynamically suppressing virtual doublon-hole excitations in a Fermi-Hubbard simulator. Phys. Rev. A 106, 012428 (2022).

    Article  ADS  CAS  Google Scholar 

  53. Mazurenko, A. et al. A cold-atom Fermi–Hubbard antiferromagnet. Nature 545, 462–466 (2017).

    Article  ADS  CAS  PubMed  Google Scholar 

  54. Chakravarty, S., Halperin, B. I. & Nelson, D. R. Two-dimensional quantum Heisenberg antiferromagnet at low temperatures. Phys. Rev. B 39, 2344 (1989).

    Article  ADS  CAS  Google Scholar 

  55. Varney, C. N. et al. Quantum Monte Carlo study of the two-dimensional fermion Hubbard model. Phys. Rev. B 80, 075116 (2009).

    Article  ADS  Google Scholar 

  56. Iglovikov, V. I., Khatami, E. & Scalettar, R. T. Geometry dependence of the sign problem in quantum Monte Carlo simulations. Phys. Rev. B 92, 045110 (2015).

    Article  ADS  Google Scholar 

  57. Müller, T. et al. Local observation of antibunching in a trapped Fermi gas. Phys. Rev. Lett. 105, 040401 (2010).

    Article  ADS  PubMed  Google Scholar 

  58. Sanner, C. et al. Suppression of density fluctuations in a quantum degenerate Fermi gas. Phys. Rev. Lett. 105, 040402 (2010).

    Article  ADS  PubMed  Google Scholar 

  59. Cheuk, L. W. et al. Observation of spatial charge and spin correlations in the 2D Fermi-Hubbard model. Science 353, 1260–1264 (2016).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

Download references


We thank A. Bohrdt, E. Demler, A. Georges, D. Greif, F. Grusdt, E. Khatami, I. Morera, A. Vishwanath and S. Sachdev for insightful discussions. We acknowledge support from NSF grant nos. PHY-1734011 and OAC-1934598; ONR grant no. N00014-18-1-2863; DOE contract no. DE-AC02-05CH11231; QuEra grant no. A44440; ARO/AFOSR/ONR DURIP grant no. W911NF2010104; the NSF Graduate Research Fellowship Program (L.H.K. and A.K.); the DoD through the NDSEG programme (G.J.); the grant DOE DE-SC0014671 funded by the US Department of Energy, Office of Science (R.T.S.); the Swiss National Science Foundation and the Max Planck Harvard Research Center for Quantum Optics (M.L.).

Author information

Authors and Affiliations



M.X., L.H.K., A.K., Y.G., G.J. and M.L. performed the experiment and collected and analysed data. M.X. performed the numerical DQMC simulations based on code curated by and under the guidance of R.T.S. M.G. supervised the study. All authors contributed extensively to the interpretation of the results and production of the manuscript.

Corresponding author

Correspondence to Markus Greiner.

Ethics declarations

Competing interests

M.G. is co-founder and shareholder of QuEra Computing.

Peer review

Peer review information

Nature thanks Thomas Schäfer and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Schematic of the lattice ramps used in the experimental protocol.

We linearly ramp up the two physics lattice beams X and Y to experimental powers within 160 ms and quench them to freeze out dynamics. The X beam is handed over to an intermediate beam \(\bar{X}\) by ramping down X and ramping up \(\bar{X}\) simultaneously and then both \(\bar{X}\) and Y beams are handed over to imaging beams by first ramping up the imaging beams and then ramping down the \(\bar{X}\) and Y beams. All ramps use a 20-ms linear ramp. Optionally, one spin species can be removed with a resonant laser in the imaging lattice.

Extended Data Fig. 2 Band structure for the full lattice potential from equation (3).

Contour lines show the Fermi surface for different density levels in steps of Δn = 1/4. The dashed black line indicates half-filling. Hole-doped regions are shown in purple and particle-doped regions in brown.

Extended Data Fig. 3 Calibration of the interference phase.

Atom number imbalance \({\mathcal{I}}\) between the two sublattices associated with potential (equation (3)), averaged over the whole cloud, as the interference phase ϕ is scanned using the electronic phase-shifter phase ϕp. We perform a linear regression to find out the phase ϕp at which the imbalance cancels, which corresponds to ϕ = π/2 (mod π). The maximum interference phase ϕ = 0 (mod π) is then obtained by increasing the phase-shifter phase ϕp by π.

Extended Data Fig. 4 Correlations at half-filling from the DQMC simulation.

a, We plot the spin correlation functions from DQMC simulations on an 8 × 8 lattice as in Fig. 2a, at the same temperature T/t and interaction U/t as in experiments for each anisotropy t′/t. b The spin structure factors from DQMC are computed with the same interpolation method as in Fig. 2. The broadening of the spin structure factor peaks and its splitting in the isotropic triangular lattice agree quantitatively with the experiment.

Extended Data Fig. 5 Magnetic correlation length at half-filling.

The correlation length is obtained from experimental data shown in Fig. 2 at different lattice anisotropies t′/t, by fitting the real-space spin–spin correlations Cd in the square lattice (square symbol) or the spin structure factor Szz(q) with an Ornstein–Zernike form at the M point (circles, isotropic form; diamonds, anisotropic form) or at the K and K′ points (triangle). See text for details.

Extended Data Fig. 6 Interaction dependence of the simulated spin correlations.

The nearest-neighbour spin correlations C(1,0) are computed using DQMC for t′/t = 1, temperature T/t = 0.4 and for different interaction strengths U/t = 0, 2, 4 and 6. In the non-interacting case, the spin correlation is antiferromagnetic at all densities and decays to zero with a steeper slope on the hole-doped side than on the particle-doped side. As interaction U increases, the peak of correlation shifts towards half-filling and the slope of correlation is steeper on the particle-doped side. A sign reversal to ferromagnetic correlations is clearly visible at U/t = 6. Statistical error bars are smaller than the symbol size.

Source data

Extended Data Fig. 7 Temperature dependence of the simulated spin correlations.

The nearest-neighbour spin correlations C(1,0) are computed using DQMC for U/t = 10, t′/t = 1 and different temperatures T/t = 0.5–0.9 and show a clear particle–hole asymmetry. Statistical error bars are smaller than the symbol size.

Source data

Extended Data Fig. 8 Comparing experimental spin correlations with DQMC simulations at constant entropy.

Nearest-neighbour spin correlations across the t-bonds C(1,0) (blue), across the t′-bonds C(1,1) (purple) and next-nearest-neighbour correlation C(1,−1) (orange) are shown, along with simulations at fixed entropy per particle S = 0.5644kB (to be compared with Fig. 2b). The difference between experimental and simulated data hint at a larger entropy increase when preparing the system in a triangular lattice compared with a square lattice.

Source data

Extended Data Table 1 Summary of tunnelling and Hubbard parameters
Extended Data Table 2 Summary of experimental temperatures and number of experiment realizations for each dataset

Source data

Rights and permissions

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 Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, M., Kendrick, L.H., Kale, A. et al. Frustration- and doping-induced magnetism in a Fermi–Hubbard simulator. Nature 620, 971–976 (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing