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Antiferromagnetic phase transition in a 3D fermionic Hubbard model

Abstract

The fermionic Hubbard model (FHM)1 describes a wide range of physical phenomena resulting from strong electron–electron correlations, including conjectured mechanisms for unconventional superconductivity. Resolving its low-temperature physics is, however, challenging theoretically or numerically. Ultracold fermions in optical lattices2,3 provide a clean and well-controlled platform offering a path to simulate the FHM. Doping the antiferromagnetic ground state of a FHM simulator at half-filling is expected to yield various exotic phases, including stripe order4, pseudogap5, and d-wave superfluid6, offering valuable insights into high-temperature superconductivity7,8,9. Although the observation of antiferromagnetic correlations over short10 and extended distances11 has been obtained, the antiferromagnetic phase has yet to be realized as it requires sufficiently low temperatures in a large and uniform quantum simulator. Here we report the observation of the antiferromagnetic phase transition in a three-dimensional fermionic Hubbard system comprising lithium-6 atoms in a uniform optical lattice with approximately 800,000 sites. When the interaction strength, temperature and doping concentration are finely tuned to approach their respective critical values, a sharp increase in the spin structure factor is observed. These observations can be well described by a power-law divergence, with a critical exponent of 1.396 from the Heisenberg universality class12. At half-filling and with optimal interaction strength, the measured spin structure factor reaches 123(8), signifying the establishment of an antiferromagnetic phase. Our results provide opportunities for exploring the low-temperature phase diagram of the FHM.

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Fig. 1: Experimental scheme and setup.
Fig. 2: Spin structure factor Sπ as a function of U/t.
Fig. 3: Spin structure factor Sπ as a function of initial entropy per particle s.
Fig. 4: Spin structure factor Sπ as a function of average density per site n.

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Data availability

The data that support the findings of this study are available at Zenodo (https://doi.org/10.5281/zenodo.11195759) (ref. 72).

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Acknowledgements

We thank Y.-F. Song for the help with DQMC simulations and Y.-Y. He, J.-P. Hu, J. Schmiedmayer, Y. Qi and Q.-J. Chen for their discussions. This work is supported by the Innovation Program for Quantum Science and Technology (grant no. 2021ZD0301900), the NSFC of China (grant nos. 11874340 and 12275263), the Chinese Academy of Sciences (CAS), the Anhui Initiative in Quantum Information Technologies, the Shanghai Municipal Science and Technology Major Project (grant no. 2019SHZDZX01) and the New Cornerstone Science Foundation. Y. Deng is also supported by the NSF of Fujian province of China (grant no. 2023J02032).

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Y.-A.C. and J.-W.P. conceived the research. X.-C.Y., Y.-A.C. and J.-W.P. designed the experiment. H.-J.S., Y.-X.W., D.-Z.Z., Y.-S.Z. and X.-C.Y. developed the homogeneous optical lattice. H.-J.S., Y.-X.W., D.-Z.Z., Y.-S.Z., H.-N.S., S.-Y.C., C.Z. and X.-C.Y. performed the experiment and collected the data. H.-J.S., Y.-X.W., D.-Z.Z., Y.-S.Z., Z.-J.F., Y.D., X.-C.Y., Y.-A.C. and J.-W.P. contributed to the data analysis. Y.D., X.-C.Y., Y.-A.C. and J.-W.P. wrote the paper with input from all authors.

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Correspondence to Xing-Can Yao, Yu-Ao Chen or Jian-Wei Pan.

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Extended data figures and tables

Extended Data Fig. 1 Characterization of the box trap.

a and c show the intensity profiles of the ring and line beams, respectively. b and d show the intensity cut lines of the ring and line beams, respectively, with solid lines representing fitting curves characterized by a double-peak Gaussian function. The insets in b and d display the residual intensity in the central “dark region”, with the dashed lines indicating the boundary lines in a and c, respectively.

Extended Data Fig. 2 Temperature determination of the homogeneous Fermi gas.

a, The measured n2D(x, z, t) (i.e., optical density) at t = 4 ms. The image is an average of 50 results at a density of n 1. b, Numerical result of n2D(x, z, t) at t = 4 ms, using Eq. (5) with T = 56.6(6) nK, µ/kB = 1383.9(3) nK, and T/TF = 0.041(1). a and b share the same color bar. c, Difference image between a and b. The uniformly distributed, negligible residual noise in the optical density, confirms the accuracy of our thermometry.

Extended Data Fig. 3 Initial entropy per particle in the box trap as a function of heating time.

Each data point (cyan dot) with a standard error bar is a calculated result based on approximately 50 independent measurements. The cyan solid line is obtained by fitting the data to a double exponential function. The inset shows the measured atom number as a function of heating time. The achieved relative uncertainty in atom number is approximately 0.3%, depicted as a standard error bar.

Extended Data Fig. 4 Calibrating the potential depth of the box trap.

The measured \({\mathcal{A}}\)(κ) for a series of κ is shown. The inset displays the remaining atom number as a function of ξr, with ξl being fixed at 1 V. The data points and error bars represent the mean and standard error of the mean, respectively, calculated from approximately 30 independent measurements.

Extended Data Fig. 5 Characterizing the flat-top light field.

The statistical distribution of pixel intensities within the central region of the flat-top beam (i.e., the disk in the inset with a diameter of 49.5 µm). The horizontal axis denotes the relative pixel intensity, which is normalized to the average pixel intensity, while the vertical axis shows the frequency of occurrence of each pixel intensity. The blue solid line represents a Gaussian fit with a 1/e2 full width of 0.03. The inset displays the beam profile of the lattice laser x, measured by a 16× magnification imaging system with a pixel resolution of 0.34 µm. It features a uniform intensity region with a diameter exceeding 60 µm.

Extended Data Fig. 6 Density distribution of atoms in the homogeneous optical lattice.

Numerical results for the statistical distribution of the number of atoms at each lattice site (refer to the main text for details). The occupancy fraction, where one atom occupies a lattice site (i.e., atom counts ranging from 0.995 to 1.005), is approximately 68%. The inset illustrates the numerical results for the 1D density distribution along the x-axis.

Extended Data Fig. 7 Calibrating the depth of the optical lattice.

The population ratio between the s and d bands, Nd/Ns, as a function of the intensity modulation frequency, ωmod. Each data point (green circle) is the average of approximately 10 independent measurements, with the standard statistical error being calculated. The solid line represents the Gaussian fit of the data. The left inset shows the calculated band structure of 41K atoms at a lattice depth of 100Er. The right inset displays an exemplary image measured in the vicinity of resonance.

Extended Data Fig. 8 Calibrating the on-site interaction U.

The atom number in level |2 is plotted as a function of the frequency offset ωrf − ω23 at B = 588.47 G. Each data point includes a standard error bar, which is obtained by averaging over 10 measurements. The solid line results from fitting the data to a double Gaussian function. The left and right peaks correspond to the spectra of sites occupied by one and two atoms, respectively. The data points in the inset are the measured ∆ω(B) values at four different magnetic fields. The error bars for ∆ω(B) (if visible) indicate the standard error, as determined from the curve fitting. The yellow solid line is the theoretical prediction as described in the text.

Extended Data Fig. 9 Measurement of the nearest-neighbor correlations.

a, Linear doublon growth rate as a function of the lattice modulation frequency. Error bars represent the fitting errors. The blue solid line is a Gaussian fit to the data, and the blue shaded area is proportional to \({\mathcal{P}}\). The upper inset shows the measured D(τ) as a function of τ, obtained at closely spaced time intervals. The lower inset displays measured D(τ) at τ = τ0 + l/ωmod, where τ0 is the initial modulation time measured and l denotes non-negative integers. Red solid lines are fits to the measured D(τ), employing a combination of linear and sinusoidal functions for the upper inset, and linear fitting for the lower one. Error bars denote one standard error. b, Nearest-neighbor correlations \({\mathcal{P}}\) (upper panel) and nearest-neighbor spin correlations Css (lower panel) as a function of initial entropy per particle s. Error bars result from the Gaussian fit of ΓD(ωmod).

Extended Data Fig. 10 Quantum Monte Carlo (QMC) results for disordered or hole-doped 3D FHM.

a, The spin structure factor Sπ for L = 6 and T/t = 0.30. The blue and orange data points represent results for disorder strength σ/t = 1 and the standard case without disorder, respectively. The solid curves are from cubic-spline interpolation and their maxima (marked by the shadowed stripes) are from the bootstrap method. The inset displays the estimated \(\left\langle {\rm{sign}}\right\rangle \). b, Results for L = 8, with T and σ the same as for a. For U/t 9, the fermion sign problem becomes so severe that no reliable estimate of Sπ can be obtained. c, Results of \(\left\langle {\rm{sign}}\right\rangle \) and Sπ versus average density per site n for U/t = 11.75, clearly demonstrating that a small doping can lead to sharp decreasing of both \(\left\langle {\rm{sign}}\right\rangle \) and Sπ. The insets present results of Sπ 0 for 0.4 n 0.9, which are unphysical and meaningless.

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Shao, HJ., Wang, YX., Zhu, DZ. et al. Antiferromagnetic phase transition in a 3D fermionic Hubbard model. Nature (2024). https://doi.org/10.1038/s41586-024-07689-2

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