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.
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Extended data figures and tables
Extended Data Figures
- 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.
- 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.
- 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.
- 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.
- 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.