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Observation of Nagaoka polarons in a Fermi–Hubbard quantum simulator

Nature volume 629, pages 317–322 (2024)Cite this article

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Abstract

Quantum interference can deeply alter the nature of many-body phases of matter1. In the case of the Hubbard model, Nagaoka proved that introducing a single itinerant charge can transform a paramagnetic insulator into a ferromagnet through path interference2,3,4. However, a microscopic observation of this kinetic magnetism induced by individually imaged dopants has been so far elusive. Here we demonstrate the emergence of Nagaoka polarons in a Hubbard system realized with strongly interacting fermions in a triangular optical lattice5,6. Using quantum gas microscopy, we image these polarons as extended ferromagnetic bubbles around particle dopants arising from the local interplay of coherent dopant motion and spin exchange. By contrast, kinetic frustration due to the triangular geometry promotes antiferromagnetic polarons around hole dopants7. Our work augurs the exploration of exotic quantum phases driven by charge motion in strongly correlated systems and over sizes that are challenging for numerical simulation8,9,10.

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Fig. 1: Nagaoka polarons in a Fermi–Hubbard quantum simulator.
Fig. 2: Emergence of Nagaoka polarons around particle dopants.
Fig. 3: Antiferromagnetic polarons around holes and doping dependence of correlations.
Fig. 4: Critical doping for ferromagnetic correlations.

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

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Acknowledgements

We thank W. Bakr, T. Esslinger, B. S. Shastry, R. T. Scalettar, A. Bohrdt, F. Grusdt, H. Schlömer and R. Samajdar for their discussions. We acknowledge support from the Gordon and Betty Moore Foundation, grant no. GBMF-11521; the National Science Foundation (NSF), grants nos. PHY-1734011, OAC-1934598 and OAC-2118310; the ONR, grant no. N00014-18-1-2863; the DOE, QSA Lawrence Berkeley Lab award no. DE-AC02-05CH11231; QuEra, grant no. A44440; the ARO/AFOSR/ONR DURIP, grants nos. W911NF-20-1-0104 and W911NF-20-1-0163. M.L. acknowledges support from the Swiss National Science Foundation (SNSF) and the Max Planck/Harvard Research Center for Quantum Optics. L.H.K. and A.K. acknowledge support from the NSF Graduate Research Fellowship Program. Y.G. acknowledges support from the AWS Generation Q Fund at the Harvard Quantum Initiative. I.M. acknowledges support from grant no. PID2020-114626GB-I00 from the MICIN/AEI/10.13039/501100011033, Secretaria d’Universitats i Recerca del Departament d’Empresa i Coneixement de la Generalitat de Catalunya, cofunded by the European Union Regional Development Fund within the ERDF Operational Program of Catalunya (project no. QuantumCat, ref. 001-P-001644). E.K. and P.S. acknowledge support from the NSF under grant no. DMR-1918572. E.D. and I.M. acknowledge support from the SNSF project 200021_212899 and the NCCR SPIN of the SNSF. NLCE calculations were done on the Spartan high-performance computing facility at San José State University.

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Author notes

  1. These authors contributed equally: Martin Lebrat, Muqing Xu

Authors and Affiliations

  1. Department of Physics, Harvard University, Cambridge, MA, USA

    Martin Lebrat, Muqing Xu, Lev Haldar Kendrick, Anant Kale, Youqi Gang & Markus Greiner

  2. Department of Physics and Astronomy, San José State University, San Jose, CA, USA

    Pranav Seetharaman & Ehsan Khatami

  3. Departament de Física Quàntica i Astrofísica, Universitat de Barcelona, Barcelona, Spain

    Ivan Morera

  4. Institut de Ciències del Cosmos, Universitat de Barcelona, Barcelona, Spain

    Ivan Morera

  5. Institute for Theoretical Physics, ETH Zurich, Zurich, Switzerland

    Ivan Morera & Eugene Demler

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Contributions

M.L., M.X., L.H.K., A.K. and Y.G. performed the experiment and analysed the data. The numerical simulations were performed by M.X. (DQMC), L.H.K. (non-interacting), A.K. (FTLM), P.S. (NLCE), I.M. (DMRG) and E.K. (NLCE). I.M., E.K. and E.D. developed the theoretical framework. M.G. supervised the study. All authors contributed to the interpretation of the results and production of the paper.

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Correspondence to Markus Greiner.

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

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Extended data figures and tables

Extended Data Fig. 1 Schematic of experimental sequence.

A degenerate Fermi gas is loaded into a lattice formed by beams X and Y with a linear ramp of the lattice power. The lattice power is quenched to freeze tunneling. Radiofrequency Landau-Zener transfers are used in some shots to change the spin states on singly-occupied sites. Handing off from X + Y to \(\bar{X}+Y\) adiabatically doubles the unit cell, converting doubly-occupied sites to pairs of singly-occupied sites. Atoms are handed off to a separate imaging lattice, where a resonant laser is used in some shots to selectively remove one spin state.

Extended Data Fig. 2 Effect of potential gradients.

Numerical simulation (FTLM) of the nearest-neighbour non-normalized spin-spin and hole-spin-spin (doublon-spin-spin) correlation functions in a 4 × 3t − J cluster as a function of doping δ and gradient strength Δ, at fixed U/t = 30 and T/t = 0.5.

Extended Data Fig. 3 Numerical simulation of doublon-spin-spin correlation map at different densities.

We compute the connected doublon-spin-spin correlation function a, with DQMC at U/t = 5 and T/t = 0.5; b, with DQMC at U/t = 12 and T/t = 0.5; c, with NLCE at U/t = 38 and T/t = 0.5; d, with NLCE at U/t = 72 and T/t = 0.52729; e, with NLCE at U/t = 100 and T/t = 0.52729. f, Definition of bonds averaged together in NLCE simulations. Bonds beyond fifth nearest-neighbor are not computed and set to zero in the plot.

Extended Data Fig. 4 NLCE closest and second-closest doublon-spin-spin correlations.

Connected doublon-spin-spin correlator as a function of interaction strength, obtained from NLCE simulations at T/t = 0.7; a, at half-filling and b, at particle doping δ = 0.05. See Fig. 2 for a definition of the correlators.

Extended Data Fig. 5 Comparison of numerically computed three-point correlators as a function of doping and interaction strength.

a to d, Comparison between a, b, finite-temperature, T/t = 1 correlators and c, d, ground-state correlators between nearest neighbors, normalized according to Eqs. (3), (4), (6) and (7) (see Fig. 3d). a, c, doublon-spin-spin correlators \({C}_{{\rm{dss}}}^{(1)}\), showing an almost universal behavior above half-filling (δ > 0) for the various interaction strengths. b, d, hole-spin-spin correlators \({C}_{{\rm{hss}}}^{(1)}\). The U/t = 0 numerics are computed using Wick’s contraction, U/t = 12 using DQMC, U/t = 11, 38, 100 using NLCE, U/t = ∞ using FTLM, and U/t = 5, 10, 20 using DMRG. e to h, Comparison between e, f, bare correlators \({C}_{{\rm{dss}},{\rm{hss}}}^{{\rm{bare}}}\) and g, h, non-normalized correlators \({C}_{{\rm{dss}},{\rm{hss}}}^{{\rm{tot}}}\) (as defined in Eq. (8) and Fig. 3c). The U/t = 0 numerics are computed using Wick’s contraction, U/t = 6, 12 using DQMC at T/t = 0.5, and U/t > 20 using FTLM. The errors in FTLM and DQMC are statistical while in DMRG they indicate the spatial variation of the correlators over the simulated system.

Extended Data Fig. 6 Ferromagnetic state in ground-state simulations.

a, DMRG simulation of the net total spin 〈S〉 normalized by maximal spin as a function of doping δ, at U/t = 20 and zero temperature, showing the emergence of long-range ferromagnetism with doublon doping. b, Spectrum of the Hubbard Hamiltonian on a triangular plaquette. Eigenenergies are shown as a function of interaction strength U/t for one particle dopant (left) and one hole dopant (right). Labels show the nature of the state at infinite interaction U/t = ±∞ (S: singlet; T: triplet; H: one hole; D: one doublon) and its angular momentum ℓ = 0, ±1 (see text for definitions). Colors indicate the sign and magnitude of the spin correlations.

Extended Data Fig. 7 Extended data on doping dependence of three-point correlators.

a, Doublon-spin-spin and b, hole-spin-spin correlation function on a triangular plaquette without normalization factor, see Eq. (8) for definition and Fig. 3 for details.

Extended Data Table 1 Summary of experimental datasets
Full size table
Extended Data Table 2 Summary of the density resolved and spin-removal imaging technique
Full size table
Extended Data Table 3 Trap curvature in a representative subset of datasets
Full size table

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Lebrat, M., Xu, M., Kendrick, L.H. et al. Observation of Nagaoka polarons in a Fermi–Hubbard quantum simulator. Nature 629, 317–322 (2024). https://doi.org/10.1038/s41586-024-07272-9

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