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# Observation of antiferromagnetic correlations in an ultracold SU(N) Hubbard model

## Abstract

Mott insulators are paradigmatic examples of strongly correlated physics from which many phases of quantum matter with hard-to-explain properties emerge. Extending the typical SU(2) spin symmetry of Mott insulators to SU(N) is predicted to produce exotic quantum magnetism at low temperatures. In this work, we experimentally observe nearest-neighbour spin correlations in a SU(6) Hubbard model realized by ytterbium atoms in optical lattices. We study one-dimensional, two-dimensional square and three-dimensional cubic lattice geometries. The measured SU(6) spin correlations are enhanced compared with the SU(2) correlations, owing to strong Pomeranchuk cooling. The experimental data for a one-dimensional lattice agree qualitatively with our theoretical calculations, with an error of at most 30%, without any fitting parameters. Detailed comparison between theory and experiment allows us to infer the temperature to be the lowest achieved for a cold-atom Fermi–Hubbard model. For three-dimensional lattices, the experiments reach entropies below the regime where our calculations converge, highlighting the importance of these experiments as quantum simulations.

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## Data availability

All the data presented in this paper are available from the corresponding author upon reasonable request.

## Code availability

The mathematical codes that support the findings of the study are available from the corresponding author upon reasonable request.

## References

1. Read, N. & Newns, D. M. On the solution of the Coqblin–Schreiffer Hamiltonian by the large-N expansion technique. J. Phys. C 16, 3273–3295 (1983).

2. Affleck, I. Large-N limit of SU(N) quantum “spin” chains. Phys. Rev. Lett. 54, 966–969 (1985).

3. Bickers, N. E. Review of techniques in the large-N expansion for dilute magnetic alloys. Rev. Mod. Phys. 59, 845–939 (1987).

4. Auerbach, A. Interacting Electrons and Quantum Magnetism (Springer-Verlag, 1994).

5. Tóth, T. A., Läuchli, A. M., Mila, F. & Penc, K. Three-sublattice ordering of the SU(3) Heisenberg model of three-flavor fermions on the square and cubic lattices. Phys. Rev. Lett. 105, 265301 (2010).

6. Bauer, B. et al. Three-sublattice order in the SU(3) Heisenberg model on the square and triangular lattice. Phys. Rev. B 85, 125116 (2012).

7. Nataf, P. & Mila, F. Exact diagonalization of Heisenberg SU(N) models. Phys. Rev. Lett. 113, 127204 (2014).

8. Corboz, P., Läuchli, A. M., Penc, K., Troyer, M. & Mila, F. Simultaneous dimerization and SU(4) symmetry breaking of 4-color fermions on the square lattice. Phys. Rev. Lett. 107, 215301 (2011).

9. Hermele, M. & Gurarie, V. Topological liquids and valence cluster states in two-dimensional SU(N) magnets. Phys. Rev. B 84, 174441 (2011).

10. Romen, C. & Läuchli, A. M. Structure of spin correlations in high-temperature SU(N) quantum magnets. Phys. Rev. Res. 2, 043009 (2020).

11. Yamamoto, D., Suzuki, C., Marmorini, G., Okazaki, S. & Furukawa, N. Quantum and thermal phase transitions of the triangular SU(3) Heisenberg model under magnetic fields. Phys. Rev. Lett. 125, 057204 (2020).

12. Wu, C. Hidden symmetry and quantum phases in spin-3/2 cold atomic systems. Mod. Phys. Lett. B 20, 1707–1738 (2006).

13. Li, Y. Q., Ma, M., Shi, D. N. & Zhang, F. C. SU(4) theory for spin systems with orbital degeneracy. Phys. Rev. Lett. 81, 3527–3530 (1998).

14. Tokura, Y. Orbital physics in transition-metal oxides. Science 288, 462–468 (2000).

15. Goerbig, M. O. Electronic properties of graphene in a strong magnetic field. Rev. Mod. Phys. 83, 1193–1243 (2011).

16. Cazalilla, M. A., Ho, A. F. & Ueda, M. Ultracold gases of ytterbium: ferromagnetism and Mott states in an SU(6) Fermi system. N. J. Phys. 11, 103033 (2009).

17. Gorshkov, A. V. et al. Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms. Nat. Phys. 6, 289–295 (2010).

18. Cazalilla, M. A. & Rey, A. M. Ultracold Fermi gases with emergent SU(N) symmetry. Rep. Prog. Phys. 77, 124401 (2014).

19. Affleck, I. & Marston, J. B. Large-n limit of the Heisenberg–Hubbard model: implications for high-Tc superconductors. Phys. Rev. B 37, 3774–3777 (1988).

20. Honerkamp, C. & Hofstetter, W. Ultracold fermions and the SU(N) Hubbard model. Phys. Rev. Lett. 92, 170403 (2004).

21. Assaad, F. F. Phase diagram of the half-filled two-dimensional SU(N) Hubbard–Heisenberg model: a quantum Monte Carlo study. Phys. Rev. B 71, 075103 (2005).

22. Hermele, M., Gurarie, V. & Rey, A. M. Mott insulators of ultracold fermionic alkaline earth atoms: underconstrained magnetism and chiral spin liquid. Phys. Rev. Lett. 103, 135301 (2009).

23. Del Re, L. & Capone, M. Selective insulators and anomalous responses in three-component fermionic gases with broken SU(3) symmetry. Phys. Rev. A 98, 063628 (2018).

24. D., Tusi et al. Flavour-selective localization in interacting lattice fermions via SU(N) symmetry breaking. Preprint at https://arxiv.org/abs/2104.13338 (2021).

25. 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).

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

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

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

29. Scalapino, D. J. Superconductivity and spin fluctuations. J. Low Temp. Phys. 117, 179–188 (1999).

30. Paglione, J. & Greene, R. L. High-temperature superconductivity in iron-based materials. Nat. Phys. 6, 645–658 (2010).

31. Ozawa, H., Taie, S., Takasu, Y. & Takahashi, Y. Antiferromagnetic spin correlation of SU(N) Fermi gas in an optical superlattice. Phys. Rev. Lett. 121, 225303 (2018).

32. Trotzky, S., Chen, Y.-A., Schnorrberger, U., Cheinet, P. & Bloch, I. Controlling and detecting spin correlations of ultracold atoms in optical lattices. Phys. Rev. Lett. 105, 265303 (2010).

33. Manmana, S. R., Hazzard, K. R. A., Chen, G., Feiguin, A. E. & Rey, A. M. SU(N) magnetism in chains of ultracold alkaline-earth-metal atoms: Mott transitions and quantum correlations. Phys. Rev. A 84, 043601 (2011).

34. Taie, S., Yamazaki, R., Sugawa, S. & Takahashi, Y. An SU(6) Mott insulator of an atomic Fermi gas realized by large-spin Pomeranchuk cooling. Nat. Phys. 8, 825–830 (2012).

35. Bonnes, L., Hazzard, K. R. A., Manmana, S. R., Rey, A. M. & Wessel, S. Adiabatic loading of one-dimensional SU(N) alkaline-earth-atom fermions in optical lattices. Phys. Rev. Lett. 109, 205306 (2012).

36. Messio, L. & Mila, F. Entropy dependence of correlations in one-dimensional SU(N) antiferromagnets. Phys. Rev. Lett. 109, 205306 (2012).

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

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

39. Hofrichter, C. et al. Direct probing of the Mott crossover in the SU(N) Fermi–Hubbard model. Phys. Rev. X 6, 021030 (2016).

40. Imriška, J. et al. Thermodynamics and magnetic properties of the anisotropic 3D Hubbard model. Phys. Rev. Lett. 112, 115301 (2014).

41. Greif, D., Jotzu, G., Messer, M., Desbuquois, R. & Esslinger, T. Formation and dynamics of antiferromagnetic correlations in tunable optical lattices. Phys. Rev. Lett. 115, 260401 (2015).

42. Ibarra-García-Padilla, E. et al. Thermodynamics and magnetism in the two-dimensional to three-dimensional crossover of the Hubbard model. Phys. Rev. A 102, 033340 (2020).

43. Ibarra-García-Padilla, E. et al. Universal thermodynamics of an SU(N) Fermi–Hubbard model. Phys. Rev. A 104, 043316 (2021).

44. Wang, D. et al. Competing orders in the 2D half-filled SU(2N) Hubbard model through the pinning-field quantum Monte Carlo simulations. Phys. Rev. Lett. 112, 156403 (2014).

45. Zhou, Z., Cai, Z., Wu, C. & Wang, Y. Quantum Monte Carlo simulations of thermodynamic properties of SU(2N) ultracold fermions in optical lattices. Phys. Rev. B 90, 235139 (2014).

## Acknowledgements

We thank H. Ozawa for contributions in the early stage of the experiment. The experimental work was supported by JSPS grants-in-aid for scientific research nos. JP17H06138, JP18H05405 and JP18H05228, the Impulsing Paradigm Change through Disruptive Technologies (ImPACT) programme, JST CREST (no. JPMJCR1673) and MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) grant no. JPMXS0118069021. Y. K. acknowledges the support of the Grant-in-Aid for JSPS Fellows (no. 17J00486). The work of K.R.A.H., E.I.-G.-P. and H.-T.W. was supported in part by the Welch Foundation through grant no. C1872 and the National Science Foundation through grant no. PHY1848304. The work of R.T.S. was supported by grant DE-SC0014671 funded by the U.S. Department of Energy Office of Science.

## Author information

Authors

### Contributions

S.T., N.N. and Y. Takasu carried out experiments. S.T. analysed the data. Y.K. gave advice on experiments from the theoretical viewpoint. Y. Takahashi conducted the whole experiment. E.I.-G.-P. developed the SU(N) DQMC code extended from R.T.S.’s SU(2) code, performed the DQMC calculations, many ED calculations and the LDA calculations with ED and DQMC results, and carried out the theoretical data analysis. H.-T.W. developed the ED code and performed many of the ED calculations. R.T.S. and K.R.A.H. conceived and supervised the theoretical calculations. E.I.-G.-P., H.-T.W., R.T.S. and K.R.A.H. all contributed to interpreting the theory and theory–experiment comparisons. All the authors contributed to preparing the manuscript. S.T. and E.I.-G.-P. contributed equally to this work.

### Corresponding author

Correspondence to Shintaro Taie.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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Nature Physics thanks Peter Schauß and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Extended data

### Extended Data Fig. 1 Temperature of a 1D SU(6) Fermi gas at U/t = 15.3.

The vertical dashed lines indicate the range of the largest experimentally measured STO imbalance in 1D that is consistent with error bars. The temperature of this datapoint is inferred from the finite-size scaling curves and the results are summarized in panel (d). The fss error is a conservative estimate of the finite-size error, the difference between the finite-size scaled results and the L = 8 site chain. The exp error comes from the experimental uncertainty on the correlations.

### Extended Data Fig. 2 Interaction dependence of the nearest neighbor correlations.

Behavior of (a) STO amplitude and (b) singlet-triplet imbalance in 1D and 3D lattices are shown. Experimental data is shown for SU(6) systems with initial entropy S/NptclkB = 1.4 ± 0.1. The error bars are extracted from the error of fit in the analysis of the STO signal. Solid lines are the result of exact diagonalization calculations for S/NptclkB = 1.4, and the error bars correspond to the sum in quadrature of the finite-size error and the basis-state truncation error. The inset presents the entropy per particle extracted by fitting it to reproduce the experimentally measured spin correlations. Results saturate at S/NptclkB = 1.818 ± 0.005. Error bars in the inset come from the experimental uncertainty on the correlations.

### Extended Data Fig. 3 Normalized STO amplitude for an SU(6) Fermi gas in an L = 5 site chain with U/t = 15.3 for different truncations of the Hilbert space.

Basis states with an on-site energy larger than the energy cutoff Ecut, as well as those that exceed the maximum particle number $${N}_{\max }$$ are disregarded. There is no visible difference between calculations. The inset shows the absolute value of the difference between the $${E}_{{{{\rm{cut}}}}}=2U,{N}_{\max }=7$$ and $${E}_{{{{\rm{cut}}}}}=U,{N}_{\max }=5$$ curves.

### Extended Data Fig. 4 Trap anharmonicity.

(a) Full external potential (optical + gravity) profile along the direction of gravity z. The shaded region is excluded from the calculation of DOS. (b) Density of states calculated from the external potential for the 3D cubic geometry. The corresponding harmonic approximation is also shown. The atoms are sensitive only to the density of states for E/t 10.

### Extended Data Fig. 5 Finite-size scaling in 1D.

Normalized STO amplitude and imbalance for SU(2) and SU(6) Fermi gases in L-site chains and the results after finite-size scaling (fss) for U/t = 15.3. The inset in panel (b) illustrates the finite-size scaling procedure.

### Extended Data Table 1 Error estimates for the DQMC calculation at U/t = 15.3.

Errors are presented at kBT/t = 1. Most errors decrease with increasing temperature. The top two rows estimate errors by considering homogeneous systems as follows: “Finite Tcut” is the error by a non-infinite value of Tcut﻿ in entropy integrations estimated by the difference of kBTcut = 1000 and 500. “Statistical” estimates is the standard error of the mean, taken at its largest value over all μ and T considered. The bottom 5 rows come from results after adiabatic loading in the trap, as follows. “Finite size (2D)” is the finite-size error estimated by the difference between A(S) in 44 and 6﻿6 geometries, “Trotter-step” is the Trotter error estimated by the difference of A(S) for Trotter step size Δμ = 0.05 and 0.025, “μ-grid coarseness” and “T-grid coarseness” are the estimates of the errors associated with discretizing μ and T by taking the difference of the result after doubling the grid spacing, and “Statistical (adiabatic loading)” is the standard error of the mean of the DQMC.

### Extended Data Table 2 Error estimates for the different error sources involved in the DQMC calculation for U/t = 8 and N = 3.

Row labels are the same as in Extended Data Table 1. The first three rows are for homogeneous systems while the last is for the A(S) after adiabatic loading. For homogeneous system results, we report the largest error in the whole range of temperatures/entropies considered. The T grid coarseness error for S vs T monotonically increases from 2.0 × 10−2 at kBT/t = 0.64 to the value reported in the table, 1.3 × 10−1, at kBT/t = 4.

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Taie, S., Ibarra-García-Padilla, E., Nishizawa, N. et al. Observation of antiferromagnetic correlations in an ultracold SU(N) Hubbard model. Nat. Phys. (2022). https://doi.org/10.1038/s41567-022-01725-6

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• DOI: https://doi.org/10.1038/s41567-022-01725-6