A cold-atom Fermi–Hubbard antiferromagnet

Journal name:
Nature
Volume:
545,
Pages:
462–466
Date published:
DOI:
doi:10.1038/nature22362
Received
Accepted
Published online

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.

At a glance

Figures

  1. Probing antiferromagnetism in the Hubbard model with a quantum gas microscope.
    Figure 1: Probing antiferromagnetism in the Hubbard model with a quantum gas microscope.

    a, Schematic of the two-dimensional Hubbard phase diagram, including predicted phases. We explore the trajectories traced by the red arrows for a Hubbard model with U/t = 7.2(2). The strongest antiferromagnetic order is observed at the starred point. b, Experimental set-up. We trap 6Li atoms in a two-dimensional square optical lattice. We use the combined potential of the optical lattice and the anticonfinement that is generated by the digital micromirror device (DMD) to trap the atoms in a central sample Ω of homogeneous density, surrounded by a dilute reservoir, as shown in the plot. The system is imaged with 671-nm light along the same beam path as the projected 650-nm potential, and separated from it by a dichroic mirror. c, Exemplary raw (left) and processed (right) images of the atomic distribution of single experimental realizations, with both spin components present (upper; corresponding to the starred point in a) and with one spin component removed (lower). The observed chequerboard pattern in the spin-removed images indicates the presence of an antiferromagnet.

  2. Observing antiferromagnetic long-range order.
    Figure 2: Observing antiferromagnetic long-range order.

    a, The spin correlator Cd is plotted for different displacements d = (dx, dy) ranging across the entire sample for five temperatures T/t. We record more than 200 images for each temperature (Methods). Correlations extend across the entire sample for the coldest temperatures, whereas for the hottest temperature only nearest-neighbour correlations remain. b, The sign-corrected correlation function (−1)iCd is obtained through an azimuthal average. The exponential fits to the data (|d| = d > 2 sites) are shown in blue, from which we determine the correlation length ξ; the fit of the coldest sample is plotted in grey in the other panels for comparison. c, The measured spin structure factor Sz(q) − Sz(0) obtained from Fourier transformations of single images. A peak at momentum qAFM = (π/a, π/a) signals the presence of an antiferromagnet. d, The measured correlation length ξ (data), fitted to equation (2) (curve), diverges exponentially as a function of temperature T/t and is comparable to the system size for the lowest temperature. The temperature is varied by holding the atoms in the trap for a variable time. The inset is a semi-logarithmic plot of the same quantity versus inverse temperature. e, The measured corrected staggered magnetization (large blue circles) increases markedly below temperatures T/t ≈ 0.4. We find good agreement with quantum Monte Carlo calculations of the Hubbard model (small grey circles). The trajectory followed in this figure is shown schematically in the phase diagram in the inset. Error bars in d and e are standard deviations of the sampled mean; error bars in b (smaller than the markers) are computed as in Methods. The figure is based on 2,282 experimental realizations.

  3. Full counting statistics of the staggered magnetization operator.
    Figure 3: Full counting statistics of the staggered magnetization operator.

    a, Selected images with one spin component removed (chequerboard overlaid to guide the eye) show a large variation in ordering strength at the coldest temperature. This variation is a consequence of the SU(2) symmetry of the underlying Hamiltonian, which leads to different orientations of the staggered spin-ordering vector relative to the measurement axis z, as shown schematically by the spin vectors (red and blue arrows) relative to the axis defined by (black arrows). b, Measured distributions of the staggered magnetization operator, , are plotted at different temperatures T/t (histograms). We find excellent agreement with quantum Monte Carlo simulations of the Heisenberg model with no free fitting parameters (black lines). The figure is based on 2,282 experimental realizations.

  4. Doping the antiferromagnet.
    Figure 4: Doping the antiferromagnet.

    a, We dope the system with holes and reduce the density from half-filling, with (corresponding to 0.95 ≥ ns ≥ 0.73). The corrected staggered magnetization settles at the critical hole doping δc ≈ 0.15. The trajectory followed in this figure is shown schematically in the phase diagram in the inset. b, The relative strength of the sign-corrected spin correlations (−1)iCd decreases less rapidly with hole doping at smaller distances (d = 1.0) than at larger distances (d = 3.6). For large doping, only the nearest-neighbour correlator is appreciable, so this correlation is predominantly responsible for the non-zero staggered magnetization away from the antiferromagnetic phase. c, We show the spin structure factor Sz(q) − Sz(0), as in Fig. 2c, for each doping value. Error bars in a are standard deviations of the sampled mean; those in b are computed as in Methods. The figure is based on 1,470 experimental realizations.

  5. Amplitude of light fields applied to atoms.
    Extended Data Fig. 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.

  6. Average density profile in the system.
    Extended Data Fig. 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.

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

  8. Alternative basis measurement.
    Extended Data Fig. 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.

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

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Author information

Affiliations

  1. Department of Physics, Harvard University, Cambridge, Massachusetts, USA

    • Anton Mazurenko,
    • Christie S. Chiu,
    • Geoffrey Ji,
    • Maxwell F. Parsons,
    • Márton Kanász-Nagy,
    • Richard Schmidt,
    • Fabian Grusdt,
    • Eugene Demler,
    • Daniel Greif &
    • Markus Greiner

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.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

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

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Extended data figures and tables

Extended Data Figures

  1. Extended Data Figure 1: Amplitude of light fields applied to atoms. (162 KB)

    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.

  2. Extended Data Figure 2: Average density profile in the system. (151 KB)

    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.

  3. Extended Data Figure 3: Comparison of staggered magnetizations obtained directly through single-spin images and from spin correlations. (90 KB)

    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.

  4. Extended Data Figure 4: Alternative basis measurement. (73 KB)

    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.

  5. Extended Data Figure 5: Staggered magnetization obtained from spin correlations, with and without the nearest-neighbour contribution included. (104 KB)

    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.

Additional data