Site-resolved measurement of the spin-correlation function in the Fermi-Hubbard model

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Science  16 Sep 2016:
Vol. 353, Issue 6305, pp. 1253-1256
DOI: 10.1126/science.aag1430


Exotic phases of matter can emerge from strong correlations in quantum many-body systems. Quantum gas microscopy affords the opportunity to study these correlations with unprecedented detail. Here, we report site-resolved observations of antiferromagnetic correlations in a two-dimensional, Hubbard-regime optical lattice and demonstrate the ability to measure the spin-correlation function over any distance. We measure the in situ distributions of the particle density and magnetic correlations, extract thermodynamic quantities from comparisons to theory, and observe statistically significant correlations over three lattice sites. The temperatures that we reach approach the limits of available numerical simulations. The direct access to many-body physics at the single-particle level demonstrated by our results will further our understanding of how the interplay of motion and magnetism gives rise to new states of matter.

Quantum many-body systems exhibiting magnetic correlations underlie a wide variety of phenomena. High-temperature superconductivity, for example, can arise from the correlated motion of holes in an antiferromagnetic (AFM) Mott insulator (1, 2). It is possible to probe physical analogs of such systems using ultracold atoms in lattices, which introduce a degree of control that is not available in conventional solid-state systems (3, 4). Recent experiments exploring the Hubbard model with cold atoms are accessing temperatures where AFM correlations form but have only observed these correlations via measurements that were averages over inhomogeneous systems (5, 6). The recent development of site-resolved imaging for fermionic quantum gases (713) provides the ability to directly detect the correlations and fluctuations present in a fermionic quantum many-body state. As demonstrated in experiments with both bosons (14, 15) and fermions (12, 13, 16), microscopy gives access to the spatial variation in observables that occurs in an inhomogeneous system, yielding precise comparisons with theory. The low energy scales in cold-atom systems also allow for time-resolved observations of many-body dynamics, which typically occur on millisecond time scales. In bosonic systems, temporal and spatial resolution have been combined to observe strongly correlated quantum walks (17), to measure entanglement entropy (18), and to study the dynamics of magnetic correlations (19, 20).

Here, we report trap-resolved observations of magnetic correlations in a Fermi-lattice system. Fermionic atoms in a two-dimensional (2D) optical lattice can be well described by the Hubbard Hamiltonian, a simple model in which there is a competition between the nearest-neighbor tunneling energy t and the on-site interaction energy U. Despite the apparent simplicity of the Hubbard model, it has a rich phase diagram, containing, for example, the transition from a metal to a Mott insulator. AFM spin correlations begin to appear near half-filling when the temperature scale becomes comparable to the exchange energy, which in the strongly interacting regime is Embedded Image. In the thermodynamic limit, what happens as the temperature is lowered further depends on the dimensionality of the system: In three dimensions, there is a finite-temperature phase transition to a state with long-range AFM order, whereas in two dimensions, such an order is prohibited by the Mermin-Wagner-Hohenberg theorem (21). Nonetheless, AFM correlations do arise, decaying exponentially with a correlation length Embedded Image that diverges as the temperature goes to zero as Embedded Image, where Embedded Image is of order unity (22). We use quantum gas microscopy to reveal precisely these correlations, which for our finite-size 2D system are expected to lead to long-range order at a finite temperature, where Embedded Image becomes comparable to the system size.

Our experiments begin with a low-temperature, 2D gas of fermionic Embedded Image atoms in a mixture of the two lowest hyperfine states (Embedded Image and Embedded Image), as described in (12). By adjusting a magnetic bias field in the vicinity of the Feshbach resonance located at 832G, we set the s-wave scattering length to Embedded Image, where Embedded Image is the Bohr radius (23). Using a 30-ms linear ramp, the atoms are adiabatically loaded into an isotropic, square optical lattice with a depth of Embedded Image, where the recoil energy is Embedded Image with Planck constant Embedded Image. We detect magnetic correlations by removing atoms in either spin state and measuring the resulting charge correlations with site-resolved imaging (24), as shown in Fig. 1. Because our imaging technique removes doubly occupied sites, both doubly occupied and unoccupied sites show up as empty sites after imaging. However, proper linear combinations of the different particle and hole correlators (measured both with and without spin-dependent removal) will account for the contribution to the signal from imperfect unity filling (24). From this, we determine the spin correlator (24) Embedded Image (1)Here, Embedded Image, with Embedded Image denoting the number of particles of spin Embedded Image on the site at Embedded Image. We take an average of Embedded Image over all Embedded Image where Embedded Image to obtain Embedded Image. The nearest-neighbor, diagonal next-nearest-neighbor, straight next-nearest-neighbor, etc., correlators are given by Embedded Image, Embedded Image, and Embedded Image, etc. From images where neither spin was removed, we directly obtain a spatial map of the single-occupation probabilityEmbedded Image, which also corresponds to the local moment Embedded Image.

Fig. 1 Experimental technique for measuring spin correlations.

(A) After loading the atoms into an optical lattice, we use a spin-removal technique to map the spin correlations onto charge correlations, which can then be detected using site-resolved imaging. The two spin states are denoted by green and orange balls. By driving cycling optical transitions for either spin state with the spin-removal beam, we can eject one spin state from the trap. We can then combine charge correlations measured in images where we remove each spin state and where no removal is performed to compute the local spin correlation (24). (B) A typical image where no atoms are removed. (C) A typical image with one of the spins removed. Atoms in doubly occupied sites are removed in both the spin-removal and imaging procedures as a result of light-assisted collisions.

After loading atoms into the lattice, we observe AFM correlations for nearest neighbors and diagonal next-nearest neighbors. These correlations are strongest in the cloud center, where the local chemical potential is set to approximately half-filling. The spatial mapsEmbedded Image, Embedded Image, and Embedded Image for colder (top) and hotter (bottom) temperatures are shown in panels A, B, and C, respectively, of Fig. 2. For these data, the interaction is tuned to Embedded Image, with Embedded Image (Embedded Image). The chemical potential is tuned to approximately Embedded Image in the center of the cloud for the colder data by adjusting the atom number to maximize Embedded Image in the center. In Fig. 2A, the suppression of Embedded Image in the center of the cloud is caused by the formation of doubly occupied sites and indicates that the chemical potential in the center of the cloud slightly exceeds Embedded Image. To heat the cloud, we hold the atoms in the optical dipole trap for 4 s before loading the lattice. After heating, the maximum detected occupation decreases from 0.89(1) to 0.84(1), with a slight broadening of the density profile, whereas the largest magnitude of the nearest-neighbor correlator decreases from 0.154(3) to 0.052(6). In this regime, where the exchange energy is much smaller than both U and the bandwidth, an increase in temperature quickly saturates the entropy available in the spin degree of freedom while creating little entropy in the charge degree of freedom, making the nearest-neighbor correlator much more sensitive than the density to temperature changes. For the colder data, we observe significant negative correlations in Embedded Image away from half-filling, which requires further theoretical investigation.

Fig. 2 Local observation of density and spin correlations.

(A to C) Spatial maps and azimuthally averaged profiles (mirrored about Embedded Image and corrected for ellipticity) of the detected density, nearest-neighbor correlator and diagonal next-nearest neighbor correlator for a cold (top) and hot (bottom) cloud. A combined fit determines the temperature T and chemical potential Embedded Image (solid lines). (D) Green symbols show the nearest-neighbor correlator in the center of the cloud for samples prepared at different temperatures. Listed are the values of Embedded Image from fits of a numerical linked-cluster expansion to the radial profile and Embedded Image obtained by comparing the central correlator value to a quantum Monte Carlo calculation at half-filling (solid line) (22). For the coldest data in (D) and (E) (squares), the NLCE theory error is too large for a fit, and we report only the QMC result. (E) An exponential fit to the correlator in the center of the cloud versus d allows us to extract the correlation length for data sets at three different temperatures, giving 0.24(9), 0.39(2), and 0.51(4) sites for decreasing temperature. The asterisk denotes the nearest-neighbor correlator value from the QMC calculation in (D) as Embedded Image. Error bars on Embedded Image and Embedded Image are standard errors after averaging at least 20 sets of combined correlation maps and averaging azimuthally (24). All data shown are at Embedded Image. Horizontal errors in (D) are fit errors.

We take azimuthal averages along the equipotentials of the underlying harmonic trap to obtain Embedded Image and Embedded Image. The resulting profiles are simultaneously fit to the results of a numerical linked-cluster expansion (NLCE) of the 2D Hubbard model under a local density approximation (LDA) (2426). From these fits, we obtain a temperature Embedded Image and chemical potential Embedded Image for the cooler data and Embedded Image and Embedded Image for the hotter data. The excellent agreement with theory provides a strong indication that the local density approximation and the assumption of thermal equilibrium are valid.

By evaporatively cooling further before lattice loading, we are able to prepare samples with even larger nearest-neighbor correlations. However, for this data, the NLCE theory error is too large away from half-filling for the fit to converge, owing to the low temperature. Because the averaged correlator in the center may not be at exactly half-filling, by comparing this value for the coldest data set to a quantum Monte Carlo (QMC) calculation at half-filling (22), we can determine an upper bound on the temperature. The correlator value of Embedded Image gives Embedded Image, the lowest temperature reported for a Hubbard-regime cold-atom system. The QMC calculation predicts that the nearest-neighbor correlator settles as Embedded Image to a value of Embedded Image; our largest measured nearest-neighbor correlation is therefore 53% of the largest predicted value. In Fig. 2D, we plot our largest measured value of the correlator for samples prepared at different temperatures, the temperature derived from the NLCE fits where they converge (x axis), and the QMC upper bound. We find very good agreement between our data and theoretical prediction, which is consistent with half-filling at the cloud center.

We see statistically significant antiferromagnetic correlations to distances of three sites, and the sign of every measured correlator value is consistent with antiferromagnetic ordering. Our ability to measure correlations at all length scales allows us to directly extract the correlation length (Fig. 2E). Samples are prepared at three different temperatures, with the atom number optimized to achieve half-filling in the center of the cloud. Values for the correlator are taken by averaging the spatial maps over a region in the center of the cloud with a six-site radius. To determine the correlation length, we perform an exponential fit of Embedded Image in the center of the cloud versus d, where i = 0 (1) if d is such that the two sites are on the same (different) sublattice. The correlation lengths are 0.24(9), 0.39(2), and 0.51(4) sites for temperatures of Embedded Image 1.53(18), 0.54(7), and 0.45(2), respectively. The asterisk in Fig. 2, D and 2 shows the QMC prediction of -0.36 for the nearest-neighbor correlator at half-filling as Embedded Image.

Quantum gas microscopy also allows for a detailed study of the thermalization of the atomic cloud when loading into the lattice. We investigate the formation of spin correlations and the thermalization of the density distribution for different lattice loading times in Fig. 3. For these data the lattice is ramped on linearly with a varying duration Embedded Image. We determine the radius Embedded Image, where Embedded Image is maximized. For a cloud in thermal equilibrium with Embedded Image not in the center of the cloud, Embedded Image corresponds to half-filling (Embedded Image). Figure 3A shows Embedded Image and Embedded Image as a function of loading time. The detected density grows from 0.6 at very short loading times and settles at about 0.9 for Embedded Image ms. The loading time required for the density to settle also corresponds to the maximum absolute values for both the nearest-neighbor and diagonal next-nearest-neighbor spin correlators. The matching time scales suggest that the suppression of magnetic correlations at Embedded Image ms is caused by the low initial density and not by exchange dynamics. The density at short loading times is determined by the confinement of the optical dipole trap preceding the lattice loading (12). For loading times larger than 10 ms, both Embedded Image and Embedded Image decay, consistent with heating. The faster decay of Embedded Image is further indication that the spin correlators are much more sensitive than the density to temperature in this regime of parameters.

Fig. 3 Thermalization dynamics during lattice loading.

(A) (Upper) Detected density. (Lower) The nearest- and diagonal next-nearest-neighbor spin correlator. Both are measured at Embedded Image as a function of lattice loading time Embedded Image, where Embedded Image is the radius where Embedded Image is maximized. (B) Computed Embedded Image of simultaneous fits of the density and nearest-neighbor correlator profiles to NLCE data. A value Embedded Image indicates a good fit, consistent with our model, which assumes thermal equilibrium. Embedded Image settles to approximately 1 at a lattice loading time of 20 ms, indicated by the shaded region. (C) Sample profile fits for three different loading times.

We also study thermalization by fitting the data for different loading times to the NLCE theory and performing a reduced chi-squared (Embedded Image) analysis. Figure 3B shows Embedded Image versus loading time, and Fig. 3C shows individual NLCE fits at Embedded Image of 0.4 ms, 20 ms, and 150 ms from top to bottom. The value of Embedded Image settles to approximately 1 on a 20-ms time scale, which is slightly longer than the settling times for the density and spin correlator. The value of Embedded Image remains near unity up to our largest loading times, showing that the density and spin-correlator distributions remain consistent with thermal equilibrium.

Whereas in Bose-Hubbard systems AFM correlations appear only in the Heisenberg limit Embedded Image, Fermi-Hubbard systems exhibit AFM correlations at all Embedded Image, with a maximum in the correlations occurring near Embedded Image. For large Embedded Image, AFM correlations are suppressed because the exchange energy becomes small compared with the temperature. For Embedded Image, where the interaction energy is smaller than the bandwidth, charge fluctuations destroy the magnetic correlations. We study these effects by varying the scattering length for fixed Embedded ImageHz. In Fig. 4, we plot Embedded Image versus the scattering length, along with the predictions of the NLCE theory for three different temperatures. We show the calculated Embedded Image from Wannier functions in the lowest band; for our parameters, corrections to this single-band approximation could play a role (27). The data show the expected dependence on Embedded Image from the simple picture mentioned above. We also compare our data with theoretical isothermal curves at half-filling. In this comparison, additional factors should be considered. First, the atom number is fixed, so the chemical potential in the center of the cloud varies with Embedded Image. Second, we anticipate the loading entropy to be approximately fixed, as opposed to the temperature, so the data are not expected to strictly follow a single isotherm. The comparison of data with the theory reflects differences between the thermodynamic preparation of atomic and conventional solid-state systems.

Fig. 4 Varying the interaction strength.

Nearest-neighbor correlator at half-filling for varying scattering length. The top y axis gives computed values of Embedded Image for each scattering length (24). The solid lines are isothermal theory curves from the NLCE theory.

Our ability to observe the in situ, site-resolved distribution of spin correlations at all distances has enabled high-precision comparison with numerical studies and detailed verification that the atomic sample behaves in a manner consistent with thermal equilibrium. These experimental benchmarks on thermal equilibrium affirm our understanding of the entropy distribution, paving the way for the implementation of entropy redistribution techniques to achieve finite-system-size long-range order (28, 29). Implementation of such techniques would require precise trap-shaping protocols, which have been fruitfully demonstrated in bosonic quantum gas microscopes (30). Numerical simulations provide evidence that a pseudogap phase in the hole-doped 2D Hubbard model arises in conjunction with long-range AFM correlations (31) and should therefore be accessible in our experiment in the near future. At lower temperatures of Embedded Image a d-wave superconductor is expected (32). Further thought is required to understand how the real-space observables that we can measure might shed light on these low-temperature phases. Beyond equilibrium physics, we could also exploit our ability to take temporally resolved snapshots of the correlations in a many-body wave function, allowing for in-depth studies of nonequilibrium physics beyond the capability of existing theoretical tools (33).

Supplementary Materials

Supplementary Text

Figs. S1 to S6

Table S1

Reference (36)

References and Notes

  1. See the supplementary materials on Science Online.
Acknowledgments: We thank E. Khatami and M. Rigol for providing the NLCE calculations, as well as T. Paiva and N. Trivedi for the Quantum Monto Carlo calculations at half-filling. We also thank J. P. F. LeBlanc and E. Gull for providing additional data based on a dynamical cluster expansion, used for theory verification. We thank E. Demler, A. Eberlein, F. Grusdt, J. Hoffman, A. Kaufman, M. Kanász-Nagy, M. Lemeshko, L. Tarruell, L. Cheuk, M. Nichols, K. Lawrence, M. Okan, H. Zhang, and M. Zwierlein for insightful discussions. Recently, antiferromagnetic correlations have been observed in the Munich lithium quantum gas microscope and the \MIT potassium quantum gas microscope (34, 35). We acknowledge support from the Air Force Office of Scientific Research, the Multi-University Research Initiative, and NSF. D.G. acknowledges support from the Harvard Quantum Optics Center and the Swiss National Foundation. M.F.P, A.M., and C.S.C. acknowledge support from the NSF. The authors declare no competing financial interests.
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