Competing magnetic orders in a bilayer Hubbard model with ultracold atoms

Abstract

Fermionic atoms in optical lattices have served as a useful model system in which to study and emulate the physics of strongly correlated matter. Driven by the advances of high-resolution microscopy, the current research focus is on two-dimensional systems1,2,3, in which several quantum phases—such as antiferromagnetic Mott insulators for repulsive interactions4,5,6,7 and charge-density waves for attractive interactions8—have been observed. However, the lattice structure of real materials, such as bilayer graphene, is composed of coupled layers and is therefore not strictly two-dimensional, which must be taken into account in simulations. Here we realize a bilayer Fermi–Hubbard model using ultracold atoms in an optical lattice, and demonstrate that the interlayer coupling controls a crossover between a planar antiferromagnetically ordered Mott insulator and a band insulator of spin-singlets along the bonds between the layers. We probe the competition of the magnetic ordering by measuring spin–spin correlations both within and between the two-dimensional layers. Our work will enable the exploration of further properties of coupled-layer Hubbard models, such as theoretically predicted superconducting pairing mechanisms9,10.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Illustration of the bilayer Hubbard model.
Fig. 2: Detection of intralayer correlations.
Fig. 3: Magnetic correlations between the layers.
Fig. 4: Crossover from the antiferromagnetic Mott insulator to a band insulator of singlets.

Data availability

The data presented in the figures are available at https://osf.io/u9wj6. More detailed data and information of this study are available from the corresponding author upon request.

Code availability

The DQMC theory simulations were performed using the QUEST Fortran 90/95 package, version 1.44, from https://code.google.com/archive/p/quest-qmc/.

References

  1. 1.

    Greif, D. et al. Site-resolved imaging of a fermionic Mott insulator. Science 351, 953–957 (2016).

    ADS  CAS  Article  Google Scholar 

  2. 2.

    Cheuk, L. W. et al. Observation of 2D fermionic Mott insulators of 40K with single-site resolution. Phys. Rev. Lett. 116, 235301 (2016).

    ADS  Article  Google Scholar 

  3. 3.

    Cocchi, E. et al. Equation of state of the two-dimensional Hubbard model. Phys. Rev. Lett. 116, 175301 (2016).

    ADS  Article  Google Scholar 

  4. 4.

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

    ADS  MathSciNet  CAS  Article  Google Scholar 

  5. 5.

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

    ADS  MathSciNet  CAS  Article  Google Scholar 

  6. 6.

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

    ADS  CAS  Article  Google Scholar 

  7. 7.

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

    ADS  CAS  Article  Google Scholar 

  8. 8.

    Mitra, D. et al. Quantum gas microscopy of an attractive Fermi–Hubbard system. Nat. Phys. 14, 173–177 (2018).

    CAS  Article  Google Scholar 

  9. 9.

    Scalettar, R. T., Cannon, J. W., Scalapino, D. J. & Sugar, R. L. Magnetic and pairing correlations in coupled Hubbard planes. Phys. Rev. B 50, 13419–13427 (1994).

    ADS  CAS  Article  Google Scholar 

  10. 10.

    Maier, T. A. & Scalapino, D. Pair structure and the pairing interaction in a bilayer Hubbard model for unconventional superconductivity. Phys. Rev. B 84, 180513 (2011).

    ADS  Article  Google Scholar 

  11. 11.

    Kancharla, S. S. & Okamoto, S. Band insulator to Mott insulator transition in a bilayer Hubbard model. Phys. Rev. B 75, 193103 (2007).

    ADS  Article  Google Scholar 

  12. 12.

    Golor, M., Reckling, T., Classen, L., Scherer, M. M. & Wessel, S. Ground-state phase diagram of the half-filled bilayer Hubbard model. Phys. Rev. B 90, 195131 (2014).

    ADS  Article  Google Scholar 

  13. 13.

    dos Santos, R. R. Magnetism and pairing in Hubbard bilayers. Phys. Rev. B 51, 15540–15546 (1995).

    ADS  Article  Google Scholar 

  14. 14.

    Rüger, R., Tocchio, L. F., Valentí, R. & Gros, C. The phase diagram of the square lattice bilayer Hubbard model: a variational Monte Carlo study. New J. Phys. 16, 033010 (2014).

    ADS  Article  Google Scholar 

  15. 15.

    Sandvik, A. & Scalapino, D. Order–disorder transition in a two-layer quantum antiferromagnet. Phys. Rev. Lett. 72, 2777–2780 (1994).

    ADS  CAS  Article  Google Scholar 

  16. 16.

    Hafermann, H., Katsnelson, M. & Lichtenstein, A. Metal–insulator transition by suppression of spin fluctuations. Europhys. Lett. 85, 37006 (2009).

    ADS  Article  Google Scholar 

  17. 17.

    Koepsell, J. et al. Robust bilayer charge-pumping for spin-and density-resolved quantum gas microscopy. Phys. Rev. Lett. 125, 010403 (2020).

    ADS  CAS  Article  Google Scholar 

  18. 18.

    Hartke, T., Oreg, B., Jia, N. & Zwierlein, M. Doublon–hole correlations and fluctuation thermometry in a Fermi–Hubbard gas. Phys. Rev. Lett. 125, 113601 (2020).

    ADS  CAS  Article  Google Scholar 

  19. 19.

    Wurz, N. et al. Coherent manipulation of spin correlations in the Hubbard model. Phys. Rev. A 97, 051602 (2018).

    ADS  CAS  Article  Google Scholar 

  20. 20.

    Scalettar, R. T. Magnetism and spin liquid behavior in a two layer Hubbard model. J. Low Temp. Phys. 99, 499–504 (1995).

    ADS  CAS  Article  Google Scholar 

  21. 21.

    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).

    ADS  CAS  Article  Google Scholar 

  22. 22.

    Bouadim, K., Batrouni, G. G., Hébert, F. & Scalettar, R. Magnetic and transport properties of a coupled hubbard bilayer with electron and hole doping. Phys. Rev. B 77, 144527 (2008).

    ADS  Article  Google Scholar 

  23. 23.

    Varney, C. N. et al. Quantum Monte Carlo study of the two-dimensional fermion Hubbard model. Phys. Rev. B 80, 075116 (2009).

    ADS  Article  Google Scholar 

Download references

Acknowledgements

This work has been supported by BCGS, the Alexander-von-Humboldt Stiftung, DFG (SFB/TR 185 project B4), Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 - 390534769 and Stiftung der deutschen Wirtschaft.

Author information

Affiliations

Authors

Contributions

The idea for the experiment was conceived by M.G., N.W., C.F.C. and M.K. Data taking was performed by M.G., N.W. and C.F.C. with contributions by J.S. Data analysis was primarily performed by M.G. and N.W. Numerical simulations were performed by C.F.C. and N.W. The results were discussed and interpreted by all coauthors, and the manuscript was written by M.K. with contributions from all coauthors.

Corresponding author

Correspondence to Michael Köhl.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks the 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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gall, M., Wurz, N., Samland, J. et al. Competing magnetic orders in a bilayer Hubbard model with ultracold atoms. Nature 589, 40–43 (2021). https://doi.org/10.1038/s41586-020-03058-x

Download citation

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