Skip to main content

Thank you for visiting nature.com. 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.

  • Letter
  • Published:

A cold-atom Fermi–Hubbard antiferromagnet

Abstract

Exotic phenomena in systems with strongly correlated electrons emerge from the interplay between spin and motional degrees of freedom. For example, doping an antiferromagnet is expected to give rise to pseudogap states and high-temperature superconductors1. Quantum simulation2,3,4,5,6,7,8 using ultracold fermions in optical lattices could help to answer open questions about the doped Hubbard Hamiltonian9,10,11,12,13,14, and has recently been advanced by quantum gas microscopy15,16,17,18,19,20. Here we report the realization of an antiferromagnet in a repulsively interacting Fermi gas on a two-dimensional square lattice of about 80 sites at a temperature of 0.25 times the tunnelling energy. The antiferromagnetic long-range order manifests through the divergence of the correlation length, which reaches the size of the system, the development of a peak in the spin structure factor and a staggered magnetization that is close to the ground-state value. We hole-dope the system away from half-filling, towards a regime in which complex many-body states are expected, and find that strong magnetic correlations persist at the antiferromagnetic ordering vector up to dopings of about 15 per cent. In this regime, numerical simulations are challenging21 and so experiments provide a valuable benchmark. Our results demonstrate that microscopy of cold atoms in optical lattices can help us to understand the low-temperature Fermi–Hubbard model.

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

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

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

Figure 1: Probing antiferromagnetism in the Hubbard model with a quantum gas microscope.
Figure 2: Observing antiferromagnetic long-range order.
Figure 3: Full counting statistics of the staggered magnetization operator.
Figure 4: Doping the antiferromagnet.

Similar content being viewed by others

References

  1. Lee, P., 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 

  2. Friedenauer, A., Schmitz, H., Glueckert, J. T., Porras, D. & Schaetz, T. Simulating a quantum magnet with trapped ions. Nat. Phys. 4, 757–761 (2008)

    Article  CAS  Google Scholar 

  3. Kim, K. et al. Quantum simulation of frustrated Ising spins with trapped ions. Nature 465, 590–593 (2010)

    Article  ADS  CAS  PubMed  Google Scholar 

  4. 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 

  5. Simon, J. et al. Quantum simulation of antiferromagnetic spin chains in an optical lattice. Nature 472, 307–312 (2011)

    Article  ADS  CAS  PubMed  Google Scholar 

  6. Yan, B. et al. Observation of dipolar spin-exchange interactions with lattice-confined polar molecules. Nature 501, 521–525 (2013)

    Article  ADS  CAS  PubMed  Google Scholar 

  7. Drewes, J. H. et al. Antiferromagnetic correlations in two-dimensional fermionic Mott-insulating and metallic phases. Phys. Rev. Lett. 118, 170401 (2017)

    Article  ADS  CAS  PubMed  Google Scholar 

  8. Murmann, S. et al. Antiferromagnetic Heisenberg spin chain of a few cold atoms in a one-dimensional trap. Phys. Rev. Lett. 115, 215301 (2015)

    Article  ADS  CAS  PubMed  Google Scholar 

  9. Hofstetter, W., Cirac, J. I., Zoller, P., Demler, E. & Lukin, M. D. High-temperature superfluidity of fermionic atoms in optical lattices. Phys. Rev. Lett. 89, 220407 (2002)

    Article  ADS  CAS  PubMed  Google Scholar 

  10. Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008)

    Article  ADS  CAS  Google Scholar 

  11. Jördens, R., Strohmaier, N., Günter, K., Moritz, H. & Esslinger, T. A Mott insulator of fermionic atoms in an optical lattice. Nature 455, 204–207 (2008)

    Article  ADS  PubMed  CAS  Google Scholar 

  12. Schneider, U. et al. Metallic and insulating phases of repulsively interacting fermions in a 3D optical lattice. Science 322, 1520–1525 (2008)

    Article  ADS  CAS  PubMed  Google Scholar 

  13. Greif, D., Uehlinger, T., Jotzu, G., Tarruell, L. & Esslinger, T. Short-range quantum magnetism of ultracold fermions in an optical lattice. Science 340, 1307–1310 (2013)

    Article  ADS  CAS  PubMed  Google Scholar 

  14. Hart, R. A. et al. Observation of antiferromagnetic correlations in the Hubbard model with ultracold atoms. Nature 519, 211–214 (2015)

    Article  ADS  CAS  PubMed  Google Scholar 

  15. Haller, E. et al. Single-atom imaging of fermions in a quantum-gas microscope. Nat. Phys. 11, 738–742 (2015)

    Article  CAS  Google Scholar 

  16. Edge, G. J. A. et al. Imaging and addressing of individual fermionic atoms in an optical lattice. Phys. Rev. A 92, 063406 (2015)

    Article  ADS  CAS  Google Scholar 

  17. 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 

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

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  19. 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 

  20. Brown, P. T. et al. Observation of canted antiferromagnetism with ultracold fermions in an optical lattice. Preprint at https://arxiv.org/abs/1612.07746 (2016)

  21. Staar, P., Maier, T. & Schulthess, T. C. Dynamical cluster approximation with continuous lattice self energy. Phys. Rev. B 88, 115101 (2013)

    Article  ADS  CAS  Google Scholar 

  22. Manousakis, E. The spin-1/2 Heisenberg antiferromagnet on a square lattice and its application to the cuprous oxides. Rev. Mod. Phys. 63, 1–62 (1991)

    Article  ADS  CAS  Google Scholar 

  23. Sandvik, A. W. Finite-size scaling of the ground-state parameters of the two-dimensional Heisenberg model. Phys. Rev. B 56, 11678–11690 (1997)

    Article  ADS  CAS  Google Scholar 

  24. Liang, J., Kohn, R. N., Becker, M. F. & Heinzen, D. J. High-precision laser beam shaping using a binary-amplitude spatial light modulator. Appl. Opt. 49, 1323–1330 (2010)

    Article  ADS  PubMed  Google Scholar 

  25. Ho, T.-L. & Zhou, Q. Universal cooling scheme for quantum simulation. Preprint at https://arxiv.org/abs/0911.5506 (2009)

  26. Gorelik, E. V. et al. Universal probes for antiferromagnetic correlations and entropy in cold fermions on optical lattices. Phys. Rev. A 85, 061602 (2012)

    Article  ADS  CAS  Google Scholar 

  27. Chakravarty, S., Halperin, B. I. & Nelson, D. R. Two-dimensional quantum Heisenberg antiferromagnet at low temperatures. Phys. Rev. B 39, 2344–2371 (1989)

    Article  ADS  CAS  Google Scholar 

  28. Denteneer, P. J. H. & Van Leeuwen, J. M. J. Spin waves in the half-filled Hubbard Model beyond the random phase approximation. Europhys. Lett. 22, 413–418 (1993)

    Article  ADS  CAS  Google Scholar 

  29. Hofferberth, S. et al. Probing quantum and thermal noise in an interacting many-body system. Nat. Phys. 4, 489–495 (2008)

    Article  CAS  Google Scholar 

  30. Yamada, K. et al. Doping dependence of the spatially modulated dynamical spin correlations and the superconducting-transition temperature in La2−xSrxCuO4 . Phys. Rev. B 57, 6165–6172 (1998)

    Article  ADS  CAS  Google Scholar 

  31. Zupancic, P. et al. Ultra-precise holographic beam shaping for microscopic quantum control. Opt. Express 24, 13881–13893 (2016)

    Article  ADS  CAS  PubMed  Google Scholar 

  32. Gaunt, A. L. et al. Robust digital holography for ultracold atom trapping. Sci. Rep. 2, 721 (2012)

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  33. Hueck, K., Mazurenko, A., Luick, N., Lompe, T. & Moritz, H. Suppression of kHz-frequency switching noise in digital micro-mirror devices. Rev. Sci. Instrum. 88, 016103 (2017)

    Article  ADS  PubMed  CAS  Google Scholar 

  34. Parsons, M. F. et al. Site-resolved imaging of fermionic 6Li in an optical lattice. Phys. Rev. Lett. 114, 213002 (2015)

    Article  ADS  PubMed  CAS  Google Scholar 

  35. Paiva, T., Scalettar, R., Randeria, M. & Trivedi, N. Fermions in 2D optical lattices: temperature and entropy scales for observing antiferromagnetism and superfluidity. Phys. Rev. Lett. 104, 066406 (2010)

    Article  ADS  PubMed  CAS  Google Scholar 

  36. Hubbard, J. Electron correlations in narrow energy bands. Proc. R. Soc. Lond. A 276, 238–257 (1963)

    Article  ADS  Google Scholar 

  37. LeBlanc, J. P. F. & Gull, E. Equation of state of the fermionic two-dimensional Hubbard model. Phys. Rev. B 88, 155108 (2013)

    Article  ADS  CAS  Google Scholar 

  38. LeBlanc, J. P. F. et al. Solutions of the two-dimensional Hubbard model: benchmarks and results from a wide range of numerical algorithms. Phys. Rev. X 5, 041041 (2015)

    Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  40. Machida, K. Magnetism in La2CuO4 based compounds. Physica C 158, 192–196 (1989)

    Article  ADS  CAS  Google Scholar 

  41. Schulz, H. J. Incommensurate antiferromagnetism in the two-dimensional Hubbard model. Phys. Rev. Lett. 64, 1445–1448 (1990)

    Article  ADS  CAS  PubMed  Google Scholar 

  42. Schulz, H. J. in The Hubbard Model: Its Physics and Mathematical Physics (eds Baeriswyl, D. et al.) 89–102 (Springer, 1995)

  43. Chubukov, A. V., Sachdev, S. & Ye, J. Theory of two-dimensional quantum Heisenberg antiferromagnets with a nearly critical ground state. Phys. Rev. B 49, 11919–11961 (1994)

    Article  ADS  CAS  Google Scholar 

  44. Caffarel, M. et al. Monte Carlo calculation of the spin stiffness of the two-dimensional Heisenberg model. Europhys. Lett. 26, 493–498 (1994)

    Article  ADS  CAS  Google Scholar 

  45. Chang, C.-C., Gogolenko, S., Perez, J., Bai, Z. & Scalettar, R. T. Recent advances in determinant quantum Monte Carlo. Phil. Mag. 95, 1260–1281 (2015)

    Article  ADS  CAS  Google Scholar 

  46. Gogolenko, S., Bai, Z. & Scalettar, R. Structured orthogonal inversion of block p-cyclic matrices on multicores with GPU accelerators. In Euro-Par 2014 Parallel Processing (eds Silva, F. et al.) 524–535 (Lecture Notes in Computer Science Vol. 8632, Springer, 2014)

    Chapter  Google Scholar 

  47. Jiang, C ., Bai, Z . & Scalettar, R. T. A fast selected inversion algorithm for Green’s function calculations in many-body quantum Monte Carlo simulations. In Proc. 30th IEEE International Parallel and Distributed Processing Symposium https://doi.org/10.1109/IPDPS.2016.69 (IEEE, 2016)

  48. Cherng, R. W. & Demler, E. Quantum noise analysis of spin systems realized with cold atoms. New J. Phys. 9, 7 (2007)

    Article  ADS  Google Scholar 

  49. Braungardt, S., Sen(De), A., Sen, U., Glauber, R. J. & Lewenstein, M. Fermion and spin counting in strongly correlated systems. Phys. Rev. A 78, 063613 (2008)

    Article  ADS  CAS  Google Scholar 

  50. Braungardt, S. et al. Counting of fermions and spins in strongly correlated systems in and out of thermal equilibrium. Phys. Rev. A 83, 013601 (2011)

    Article  ADS  CAS  Google Scholar 

  51. Lamacraft, A. Noise and counting statistics of insulating phases in one-dimensional optical lattices. Phys. Rev. A 76, 011603 (2007)

    Article  ADS  CAS  Google Scholar 

  52. Blanter, Y. & Büttiker, M. Shot noise in mesoscopic conductors. Phys. Rep. 336, 1–166 (2000)

    Article  ADS  CAS  Google Scholar 

  53. de-Picciotto, R. et al. Direct observation of a fractional charge. Nature 389, 162–164 (1997)

    Article  ADS  CAS  Google Scholar 

  54. Saminadayar, L., Glattli, D. C., Jin, Y. & Etienne, B. Observation of the e/3 fractionally charged Laughlin quasiparticle. Phys. Rev. Lett. 79, 2526–2529 (1997)

    Article  ADS  CAS  Google Scholar 

  55. Goldman, V. J. & Su, B. Resonant tunneling in the quantum Hall regime: measurement of fractional charge. Science 267, 1010–1012 (1995)

    Article  ADS  CAS  PubMed  Google Scholar 

  56. Gritsev, V., Altman, E., Demler, E. & Polkovnikov, A. Full quantum distribution of contrast in interference experiments between interacting one-dimensional Bose liquids. Nat. Phys. 2, 705–709 (2006)

    Article  CAS  Google Scholar 

  57. Polkovnikov, A., Altman, E. & Demler, E. Interference between independent fluctuating condensates. Proc. Natl Acad. Sci. USA 103, 6125–6129 (2006)

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  58. Mermin, N. D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133–1136 (1966)

    Article  ADS  CAS  Google Scholar 

  59. Hohenberg, P. C. Existence of long-range order in one and two dimensions. Phys. Rev. 158, 383–386 (1967)

    Article  ADS  CAS  Google Scholar 

  60. Sandvik, A. W. Stochastic series expansion method with operator-loop update. Phys. Rev. B 59, R14157(R) (1999)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank R. Desbuquois, S. Dickerson, A. Eberlein, A. Kaufman, M. Messer, N. Prokov’ev, S. Sachdev, R. Scalettar, B. Svistunov, W. Zwerger, and M. Zwierlein and his research group for discussions. We thank S. Blatt, D. Cotta, S. Fölling, F. Huber, W. Setiawan and K. Wooley-Brown for early-stage contributions to the experiment. We acknowledge support from AFOSR (MURI), ARO (MURI, NDSEG), the Gordon and Betty Moore foundation EPiQS initiative, HQOC, NSF (CUA, ITAMP, GRFP, SAO) and SNSF.

Author information

Authors and Affiliations

Authors

Contributions

A.M., C.S.C., G.J., M.F.P. and D.G. performed the experiment and analysed the data. G.J. carried out the determinant quantum Monte Carlo calculations for Fig. 2e using the QUEST package. M.K.-N. developed the QMC code for the full-counting statistics and analysed the results together with R.S., F.G. and E.D. M.G. supervised the work. All authors contributed extensively to the writing of the manuscript and to discussions.

Corresponding author

Correspondence to Markus Greiner.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks T. Giamarchi and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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 Figure 1 Amplitude of light fields applied to atoms.

a, The computed light field generated by the DMD, applied to the atoms for half-filled samples. A gradient compensates residual gradients in the lattice. The rim of the doughnut provides sharp walls for the inner subsystem. A small peak in the centre flattens the potential when combined with the optical lattice. The plot on the right shows a schematic of a radial cut of the potential, including the contribution of the lattice. b, The amplitude of the light field with an offset in the centre of the trap, used to dope the system with a finite population of holes.

Extended Data Figure 2 Average density profile in the system.

a, The average single-particle density map for a sample at half-filling shows a central region of uniform density, surrounded by a doughnut-shaped ring of low density. The dotted white circle indicates our system size, excluding edge effects. b, The azimuthal average of the single-particle density ns shown in a, for the system and for the inner edge of the doughnut where the density drops off to the reservoir density. The vertical dotted lines denote the boundary of the system. c, Azimuthal average of the single-particle density ns for three values of the hole doping δ used in the experiment, indicating uniformity of atom number across our system to within 4%. The horizontal lines are at the system-wide average densities. Error bars in c are one standard deviation of the sample mean. The figure is based on 2,105 experimental realizations.

Extended Data Figure 3 Comparison of staggered magnetizations obtained directly through single-spin images and from spin correlations.

We calculate the corrected staggered magnetization from images with one spin state removed (main text). It can also be calculated from the spin correlator (Methods), with the two methods being identical in the limit of no noise and exactly one particle per site. Plotting these two quantities against each other, we find very good agreement with the line y = x (dotted line), indicating that any error due to deviation from one particle per site is small. The comparison is performed for the datasets used in Fig. 2 (labelled temperature) and Fig. 4 (labelled density). Error bars are computed as described in Methods.

Extended Data Figure 4 Alternative basis measurement.

We optionally apply a π/2 or π microwave pulse before the spin removal pulse and correlation measurement. The sign-corrected spin correlation functions (−1)iCd are insensitive to the presence and duration of this microwave pulse, consistent with an SU(2) symmetry of the state. The error bars are computed as described in Methods. This figure is based on 667 experimental realizations.

Extended Data Figure 5 Staggered magnetization obtained from spin correlations, with and without the nearest-neighbour contribution included.

To investigate the contributions to the corrected staggered magnetization at high dopings δ, we consider the value calculated from the spin correlator (blue circles). We then omit the longest-range correlations, which have the greatest level of noise owing to the low number of pairs of sites extending across the cloud, as well as the nearest-neighbour correlations, which are essentially the only non-zero correlator outside of the antiferromagnetic phase (red circles). In the high-doping regime, we see that the greatest contribution to the staggered magnetization is the nearest-neighbour correlation, followed by the noisy longest-range correlations. Error bars are one standard deviation of the sample mean.

PowerPoint slides

Source data

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mazurenko, A., Chiu, C., Ji, G. et al. A cold-atom Fermi–Hubbard antiferromagnet. Nature 545, 462–466 (2017). https://doi.org/10.1038/nature22362

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature22362

This article is cited by

Comments

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.

Search

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