Abstract
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 t–t′ square lattice Hubbard models that may be essential to describe superconductivity in cuprate materials14.
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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.
Change history
29 January 2024
A Correction to this paper has been published: https://doi.org/10.1038/s41586-024-07072-1
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Acknowledgements
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, OAC-1934598 and OAC-2118310; 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.).
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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.
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M.G. is co-founder and shareholder of QuEra Computing.
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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.
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.
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.
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Xu, M., Kendrick, L.H., Kale, A. et al. Frustration- and doping-induced magnetism in a Fermi–Hubbard simulator. Nature 620, 971–976 (2023). https://doi.org/10.1038/s41586-023-06280-5
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DOI: https://doi.org/10.1038/s41586-023-06280-5
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