Imaging magnetic polarons in the doped Fermi–Hubbard model

Article metrics


Polarons—electronic charge carriers ‘dressed’ by a local polarization of the background environment—are among the most fundamental quasiparticles in interacting many-body systems, and emerge even at the level of a single dopant1. In the context of the two-dimensional Fermi–Hubbard model, polarons are predicted to form around charged dopants in an antiferromagnetic background in the low-doping regime, close to the Mott insulating state2,3,4,5,6,7; this prediction is supported by macroscopic transport and spectroscopy measurements in materials related to high-temperature superconductivity8. Nonetheless, a direct experimental observation of the internal structure of magnetic polarons is lacking. Here we report the microscopic real-space characterization of magnetic polarons in a doped Fermi–Hubbard system, enabled by the single-site spin and density resolution of our ultracold-atom quantum simulator. We reveal the dressing of doublons by a local reduction—and even sign reversal—of magnetic correlations, which originates from the competition between kinetic and magnetic energy in the system. The experimentally observed polaron signatures are found to be consistent with an effective string model at finite temperature7. We demonstrate that delocalization of the doublon is a necessary condition for polaron formation, by comparing this setting with a scenario in which a doublon is pinned to a lattice site. Our work could facilitate the study of interactions between polarons, which may lead to collective behaviour, such as stripe formation, as well as the microscopic exploration of the fate of polarons in the pseudogap and ‘bad metal’ phases.

Access optionsAccess options

Fig. 1: Mobile and immobile dopants with ultracold atoms.
Fig. 2: Mobile doublons dressed by local spin disturbance.
Fig. 3: Spin correlations around trapped doublons.
Fig. 4: Spin correlations as a function of bond distance from doublons.

Data availability

The datasets generated and analysed during this study are available from the corresponding author upon reasonable request.


  1. 1.

    Alexandrov, S. & Devreese, J. T. Advances in Polaron Physics (Springer, 2010).

  2. 2.

    Schmitt-Rink, S., Varma, C. M. & Ruckenstein, A. E. Spectral function of holes in a quantum antiferromagnet. Phys. Rev. Lett. 60, 2793–2796 (1988).

  3. 3.

    Shraiman, B. I. & Siggia, E. D. Mobile vacancies in a quantum Heisenberg antiferromagnet. Phys. Rev. Lett. 61, 467–470 (1988).

  4. 4.

    Sachdev, S. Hole motion in a quantum Néel state. Phys. Rev. B 39, 12232–12247 (1989).

  5. 5.

    Kane, C. L., Lee, P. A. & Read, N. Motion of a single hole in a quantum antiferromagnet. Phys. Rev. B 39, 6880–6897 (1989).

  6. 6.

    Dagotto, E., Moreo, A. & Barnes, T. Hubbard model with one hole: ground-state properties. Phys. Rev. B 40, 6721–6725 (1989).

  7. 7.

    Grusdt, F. et al. Parton theory of magnetic polarons: mesonic resonances and signatures in dynamics. Phys. Rev. X 8, 011046 (2018).

  8. 8.

    Schrieffer, J. R. Handbook of High-Temperature Superconductivity (Springer, 2007).

  9. 9.

    Watanabe, S. et al. Polaron spin current transport in organic semiconductors. Nat. Phys. 10, 308–313 (2014).

  10. 10.

    Ramirez, A. P. Colossal magnetoresistance. J. Phys. Condens. Matter 9, 8171–8199 (1997).

  11. 11.

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

  12. 12.

    Trugman, S. A. Interaction of holes in a Hubbard antiferromagnet and high-temperature superconductivity. Phys. Rev. B 37, 1597–1603 (1988).

  13. 13.

    Schrieffer, J. R., Wen, X. & Zhang, S. C. Dynamic spin fluctuations and the bag mechanism of high-T c superconductivity. Phys. Rev. B 39, 11663–11679 (1989).

  14. 14.

    Anderson, P. W. The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 1196–1198 (1987).

  15. 15.

    Auerbach, A. & Larson, B. E. Small-polaron theory of doped antiferromagnets. Phys. Rev. Lett. 66, 2262–2265 (1991).

  16. 16.

    Punk, M., Allais, A. & Sachdev, S. Quantum dimer model for the pseudogap metal. Proc. Natl Acad. Sci. USA 112, 9552–9557 (2015).

  17. 17.

    Gross, C. & Bloch, I. Quantum simulations with ultracold atoms in optical lattices. Science 357, 995–1001 (2017).

  18. 18.

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

  19. 19.

    Brown, P. T. et al. Bad metallic transport in a cold atom Fermi–Hubbard system. Science 363, 379–382 (2019).

  20. 20.

    Nichols, M. A. et al. Spin transport in a Mott insulator of ultracold fermions. Science 363, 383–387 (2019).

  21. 21.

    Salomon, G. et al. Direct observation of incommensurate magnetism in Hubbard chains. Nature 565, 56–60 (2019); correction 566, E5 (2019).

  22. 22.

    Boll, M. et al. Spin- and density-resolved microscopy of antiferromagnetic correlations in Fermi–Hubbard chains. Science 353, 1257–1260 (2016).

  23. 23.

    White, S. R. & Scalapino, D. Hole and pair structures in the t–J model. Phys. Rev. B 55, 6504–6517 (1997).

  24. 24.

    Martins, G. B., Eder, R. & Dagotto, E. Indications of spin-charge separation in the two-dimensional t–J model. Phys. Rev. B 73, 170–173 (1999).

  25. 25.

    Martins, G. B., Gazza, C., Xavier, J. C., Feiguin, A. & Dagotto, E. Doped stripes in models for the cuprates emerging from the one-hole properties of the insulator. Phys. Rev. Lett. 84, 5844–5847 (2000).

  26. 26.

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

  27. 27.

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

  28. 28.

    Chiu, C. S. et al. String patterns in the doped Hubbard model. Preprint at (2018).

  29. 29.

    Devreese, J. T. & Alexandrov, A. S. Froehlich polaron and bipolaron: recent developments. Rep. Prog. Phys. 72, 066501 (2009).

  30. 30.

    Radzihovsky, L. & Sheehy, D. E. Imbalanced Feshbach-resonant Fermi gases. Rep. Prog. Phys. 73, 076501 (2010).

  31. 31.

    Stewart, J. T., Gaebler, J. P. & Jin, D. S. Using photoemission spectroscopy to probe a strongly interacting Fermi gas. Nature 454, 744–747 (2008).

  32. 32.

    Lubasch, M., Murg, V., Schneider, U., Cirac, J. I. & Bañuls, M.-C. Adiabatic preparation of a Heisenberg antiferromagnet using an optical superlattice. Phys. Rev. Lett. 107, 165301 (2011).

  33. 33.

    Kantian, A., Langer, S. & Daley, A. J. Dynamical disentangling and cooling of atoms in bilayer optical lattices. Phys. Rev. Lett. 120, 060401 (2018).

  34. 34.

    Omran, A. et al. Microscopic observation of Pauli blocking in degenerate fermionic lattice gases. Phys. Rev. Lett. 115, 263001 (2015).

  35. 35.

    Trotzky, S. et al. Time-resolved observation and control of superexchange interactions with ultracold atoms in optical lattices. Science 319, 295–299 (2008).

  36. 36.

    Khatami, E. & Rigol, M. Thermodynamics of strongly interacting fermions in two-dimensional optical lattices. Phys. Rev. A 84, 053611 (2011).

  37. 37.

    Rigol, M., Bryant, T. & Singh, R. R. P. Numerical linked-cluster algorithms. I. Spin systems on square, triangular, and kagomé lattices. Phys. Rev. E 75, 061118 (2007).

  38. 38.

    Büchler, H. P. Microscopic derivation of Hubbard parameters for cold atomic gases. Phys. Rev. Lett. 104, 090402 (2010).

Download references


We thank D. Stamper-Kurn, Y. Wang, E. Manousakis, S. Sachdev and T. Giamarchi for discussions. This work benefited from financial support by the Max Planck Society (MPG), the European Union (UQUAM, FET-Flag 817482, PASQUANS) and the Max Planck Harvard Research Center for Quantum Optics (MPHQ). J.K. gratefully acknowledges funding from Hector Fellow Academy. F.G. and E.D. acknowledge support from Harvard-MIT CUA, NSF grant number DMR-1308435 and AFOSR-MURI Quantum Phases of Matter (grant FA9550-14-1-0035). F.G. acknowledges financial support from the Gordon and Betty Moore Foundation through the EPiQS programme and from the Technical University of Munich – Institute for Advanced Study, funded by the German Excellence Initiative and the European Union FP7 under grant agreement 291763, from DFG grant number KN 1254/1-1 and DFG TRR80 (Project F8).

Author information

J.K., J.V., P.S., T.A.H., G.S., I.B. and C.G. planned the experiment and analysed and discussed the data. F.G. and E.D. performed the theoretical simulations. All authors contributed to the interpretation of the data and writing of the manuscript.

Correspondence to Joannis Koepsell.

Ethics declarations

Competing interests

The authors declare no competing interests.

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 Doublon–hole correlation.

Two-point correlation function g2 between double occupations and holes, showing a strong bunching effect at NN distances. This motivates us to neglect double occupations with holes as NNs in the analysis of mobile doublons.

Extended Data Fig. 2 Dataset statistics for the measurement of mobile doublons.

ac, Distribution of the number of atoms (a), spins (b) and holes (c) in the region with density greater than 0.7. Red bars in c indicate shots discarded by the applied hole filter (see text). d, Number of mobile doublons (doublon–hole fluctuations subtracted) in the doped 5 × 3-site region.

Extended Data Fig. 3 Doping calibration.

a, b, When scanning our final evaporation parameters, we measured the fraction of double occupations (a)and the number of doped doublons (b; excluding doublon–hole fluctuations) in the system as a function of the mean total atom number, N. Statistical error bars are smaller than the marker size. Pinned doublon measurements were taken in an undoped system (pink bar). For the mobile doublon dataset, settings for weak doping were used (purple bar). The bar width represents the standard deviation, obtained from atom number fluctuations.

Extended Data Fig. 4 Calibration of tweezer power.

Density of the lattice site on which the tweezer is focused as a function of final tweezer power. Error bars denote one s.e.m. For the realization of pinned doublons the power was set to 0.11 (arbitrary units).

Extended Data Fig. 5 Temperature estimation.

Two-point NN spin correlations as a function of binned density. Error bars denote one s.e.m. Upper (lower) values of the purple band correspond to temperatures of T/t = 0.43 (0.46) in numerical linked-cluster expansions at U/t = 13.

Extended Data Fig. 6 Radially averaged doublon–doublon correlation function, g2.

In our system, excess doublons appear anti-correlated at short distances and quickly become uncorrelated at longer distances. Error bars denote one s.e.m.

Extended Data Fig. 7 Extended polaron analysis.

a, Comparison of the local spin environment around a lattice site r0 occupied by a doublon (double black circle) or a singlon (single black circle). To simplify the notation, site positions in the doped region of our system are labelled from 0 to 15 (see inset at right). At any position in the 5 × 3-site system, spin distortion is present in NN (b), diagonal (c) and NNN (d) correlations whenever a doublon is detected (blue), and absent whenever a singlon is detected (red). Error bars denote one s.e.m. NNN correlations are measured across doublons and have a high signal-to-noise ratio, which we attribute to their short bond distance of 0, compared to, for example, NN correlations (bond distance of 1.1).

Extended Data Fig. 8 Diagonal two-point spin correlations.

Spin correlations in the central region (see lattice site positions in Fig. 2), represented by bonds connecting the two sites (black dots). At the centre, a clear reduction of correlations from the positive antiferromagnetic background value is visible. In the area of highest doublon density, correlations even flip sign and become negative.

Extended Data Fig. 9 NN correlations around mobile or pinned doublons.

a, Two-point NN correlations in the central region for the mobile doublon setting, represented by colored bonds between lattice sites (black dots). b, NN spin correlations as a function of distance from mobile doublons. c, NN correlations around pinned doublons. The enhancement effect of correlations is visible in the strong bonds surrounding the trapping site.

Supplementary information

Supplementary Information

This file contains information on numerical exact diagonalization and string model results, including two Supplementary Figures. Correlations around mobile doublons obtained from exact diagonalization are shown as a function of bond distance as well as a map of diagonal correlations in the string model.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Further reading


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.