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Directly imaging spin polarons in a kinetically frustrated Hubbard system


The emergence of quasiparticles in quantum many-body systems underlies the rich phenomenology in many strongly interacting materials. In the context of doped Mott insulators, magnetic polarons are quasiparticles that usually arise from an interplay between the kinetic energy of doped charge carriers and superexchange spin interactions1,2,3,4,5,6,7,8. However, in kinetically frustrated lattices, itinerant spin polarons—bound states of a dopant and a spin flip—have been theoretically predicted even in the absence of superexchange coupling9,10,11,12,13,14. Despite their important role in the theory of kinetic magnetism, a microscopic observation of these polarons is lacking. Here we directly image itinerant spin polarons in a triangular-lattice Hubbard system realized with ultracold atoms, revealing enhanced antiferromagnetic correlations in the local environment of a hole dopant. In contrast, around a charge dopant, we find ferromagnetic correlations, a manifestation of the elusive Nagaoka effect15,16. We study the evolution of these correlations with interactions and doping, and use higher-order correlation functions to further elucidate the relative contributions of superexchange and kinetic mechanisms. The robustness of itinerant spin polarons at high temperature paves the way for exploring potential mechanisms for hole pairing and superconductivity in frustrated systems10,11. Furthermore, our work provides microscopic insights into related phenomena in triangular-lattice moiré materials17,18,19,20.

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Fig. 1: Itinerant spin polaron.
Fig. 2: Polaron internal structure versus doping.
Fig. 3: Evolution of three-point correlations with doping and interactions.
Fig. 4: Comparing AFM correlations due to kinetic and superexchange mechanisms.

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Data availability

Source data are provided with this paper and can be found in the Harvard Dataverse60. All other supporting data are available from the corresponding author upon request.

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The code used in this paper is available from the corresponding author upon request.


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We acknowledge M. Greiner, D. Huse, R. Samajdar, L. Cheuk, E.-A. Kim, D. Khomskii, A. Bohrdt, F. Grusdt, H. Schlömer and G. Refael for discussions. We thank S. Dandavate for early assistance in performing the DQMC simulations. The experimental work was supported by the NSF (grant no. 2110475), the David and Lucile Packard Foundation (grant no. 2016-65128) and the ONR (grant no. N00014-21-1-2646). M.L.P. acknowledges support from the NSF Graduate Research Fellowship Program. E.D. acknowledges support from the ARO (grant no. W911NF-20-1-0163) and the SNSF (project 200021_212899). I.M. acknowledges support from grant no. PID2020-114626GB-I00 from the MICIN/AEI/10.13039/501100011033 and Secretaria d’Universitats i Recerca del Departament d’Empresa i Coneixement de la Generalitat de Catalunya, co-funded by the European Union Regional Development Fund within the ERDF Operational Program of Catalunya (project no. QuantumCat, Ref. 001-P-001644). I.M. and E.D. acknowledge support by the NCCR SPIN of the Swiss NSF.

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Authors and Affiliations



E.D., I.M. and W.S.B. conceived the study and supervised the experiment. M.L.P., B.M.S. and Z.Z.Y. performed the experiments and analysed the data. All authors contributed to writing the paper.

Corresponding author

Correspondence to Waseem S. Bakr.

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Nature thanks Georg Bruun, Jae-yoon Choi and Zheng Zhu for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Superlattice Phase Calibration.

P-band spectroscopy used to calibrate the superlattice phase and stability. The dips at 292 MHz and 340 MHz superlattice detuning correspond to ϕ = 3π/2 and ϕ = π/2. Fitted peaks (solid lines) are at [292.4(6) MHz, 339.8(8) MHz] and [292.6(8) MHz, 341.0(8) MHz] for the red and blue data respectively. Callout fluoresence images show the expansion of a Mott insulator after 1 second for superlattice phase ϕ = π (top) and ϕ = 0 (bottom) at V752 = 42.0(3)ER,752 and V532 = 3.7(1)ER,532. Inset: Experimental setup used for realizing the optical superlattice.

Source Data

Extended Data Fig. 2 Bilayer Image Reconstruction.

Sample deconvolved experimental images and occupation histograms for state \(\left|3\right\rangle \) (top, red) and state \(\left|2\right\rangle \) (bottom, blue) atoms. We use the Lucy-Richardson algorithm with five iterations for the deconvolution.

Source Data

Extended Data Fig. 3 Multi-point Correlations at Different Global Chemical Potentials.

a, \({C}_{h}^{(3)}\) (red data) and \({C}_{d}^{(3)}\) (green data) evaluated at the bond closest to the dopant, and b, C(4) for two different datasets with distinct global chemical potentials. As these two datasets track each other well, particularly at low filling, we conclude spatial gradients do not appreciably suppress resonant tunneling to affect measured correlations. Filled data points have mean atom number 799(35) while empty data points have mean atom number 622(27).

Source Data

Extended Data Fig. 4 Interaction Dependence of Four-point Correlations.

Four-point correlations vs. doping for different interaction strengths U/t = 8.0(2) (open circles) and U/t = 11.8(4) (filled circles).

Source Data

Extended Data Fig. 5 Unsymmetrised Correlations versus Distance.

\({C}_{h}^{(3)}\) for δ = −0.10(2) and \({C}_{d}^{(3)}\) for δ = 0.15(2) out to d = (2, 0) without averaging over the 120 degree rotational symmetry.

Source Data

Extended Data Fig. 6 Unsymmetrised Correlations versus Doping.

\({C}_{h}^{(3)}\) and \({C}_{d}^{(3)}\) for different dopings at a U/t = 11.8 without averaging over all six individual plaquettes. We see that unsymmetrised correlations are largely consistent between different orientations. Error bars are 1 s.e.m.

Source Data

Extended Data Fig. 7 Doublon Density versus Doping.

Number of measured doubles (red points) and theoretical expected number of doubles from DQMC (red band) with imaging fidelity of 0.96 accounted for at U/t = 11.8(4), T/t = 0.94(4). The highest doping bin (δ = 0.15) has around 20 percent more, the second highest doping bin (δ = 0.12 has around 10 percent more, and the third highest doping bin (δ = 0.08) has around 5 percent more doubles than predicted. We believe this is caused by image reconstruction errors and leads to a systematic underestimate of certain three and four point correlators above zero doping. This qualitative trend appears for all three interaction strengths. Error bars are 1 s.e.m.

Source Data

Extended Data Fig. 8 Three-point Correlations Temperature Dependence.

DQMC results for C(3) evaluated at the closest bond to the dopant versus doping at a fixed U/t = 12 and different temperatures. All temperature curves have the same qualitative trend with the \({C}_{h}^{(3)}\) minimum at a doping of ~ −0.3 and the \({C}_{d}^{(3)}\) maximum at a doping of ~0.15 with a linear region near zero doping. Decreasing T/t of the gas from 0.95 to 0.65 causes the magnitude of the peak values of C(3) to increase by roughly fifty percent. An imaging fidelity of 0.96 is assumed.

Source Data

Extended Data Fig. 9 Doublon Correlations versus Temperature.

\({C}_{d}^{(3)}\) calculated at δ = 0.02 for U/t = 12 at three different temperatures using DQMC. As the temperature decreases, the magnitude of the farther correlations increases slightly while there are no major qualitative differences in the structure of the correlations; the size of the polaron does not have a strong dependence on the temperature.

Source Data

Extended Data Fig. 10 Three-point Correlations Interaction Dependence.

DQMC results for C(3) evaluated at the closed bond to the dopant versus doping at fixed T/t = 0.85 and different interaction strengths. We see that for all interaction strengths the \({C}_{h}^{(3)}\) minimum is at a doping of ~ −0.3 and the \({C}_{d}^{(3)}\) maximum is at a doping of ~0.15 for the two higher interaction strengths, while for U/t = 4 the peak appears slightly closer to zero doping. The two higher doping curves have roughly the same peak C(3) magnitudes, while for U/t = 4 the peaks are roughly 10 percent lower. However, we see that qualitatively the curves are quite different close to half filling, where as U/t increases the onset of \({C}_{h}^{(3)}\) and \({C}_{d}^{(3)}\) with doping becomes sharper, leading to a region where the \({C}_{h}^{(3)}\) is linear below zero doping and \({C}_{d}^{(3)}\) is linear above zero doping. An imaging fidelity of 0.96 is assumed.

Source Data

Extended Data Table 1 Single plaquette correlation functions

Source data

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Prichard, M.L., Spar, B.M., Morera, I. et al. Directly imaging spin polarons in a kinetically frustrated Hubbard system. Nature 629, 323–328 (2024).

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