Antiferromagnetic Correlations in Two-Dimensional Fermionic Mott-Insulating and Metallic Phases

We experimentally study the emergence of antiferromagnetic correlations between ultracold fermionic atoms in a two-dimensional optical lattice with decreasing temperature. We determine the uniform magnetic susceptibility of the two-dimensional Hubbard model from simultaneous measurements of the in situ density distribution of both spin components. At half filling and strong interactions our data approach the Heisenberg model of localized spins with antiferromagnetic correlations. Moreover, we observe a fast decay of magnetic correlations when doping the system away from half filling.

Figure 1

Spin-resolved density distributions and deduced spin correlations. (a) In situ density distributions of spin-up and spin-down atoms residing on singly occupied lattice sites averaged over 36 experimental realisations for $U/t=8.2(5)$ and $k_{B}T/t=0.63_{-0.02}^{+0.09}$. (b) Spatial map of the spin structure factor $S_{\mathit{q}=0}$ extracted from the data set shown in (a). The data have been spatially averaged over a region of $3\times 3$ pixels for better visibility. (c) Spin structure factor $S_{\mathit{q}=0}$ (circles) and local moment $C_{0,0}$ (diamonds) averaged over regions of constant chemical potential $\mu$. Half filling is realized for $\mu =U/2$. The solid line is a fit of the local moment to NLCE data [27], serving as an independent thermometer.

Figure 2

Spin correlations at half filling. (a) Measured spin structure factor $S_{\mathit{q}=0}$ (circles) for different interaction strengths and temperatures. For comparison, data from a numerical linked cluster expansion of the two-dimensional Hubbard model [27] for $S_{\mathit{q}=0}$ (solid lines) and $|C_{0,0}|-4|C_{0,1}|$ (dashed lines) are shown for $U/t=2$ (blue), 8 (green), and 12 (yellow). Crosses show results from quantum Monte Carlo (QMC) calculations of the two-dimensional Hubbard model [28] of $S_{\mathit{q}=0}$ for $U/t=2$ (blue) and 8 (green). (b) Magnetic susceptibility $\chi$ extracted from the data shown in (a). Solid lines and crosses show corresponding results from NLCE and QMC calculations. The solid black line shows the magnetic susceptibility $\chi$ for vanishing interactions. The inset shows the spin correlation length $\xi$ extracted from fits to the correlation data versus distance [24]. (c) Magnetic susceptibility scaled in units of the superexchange interaction $J$ and comparison to the antiferromagnetic Heisenberg model [29] (dashed-dotted line). Vertical error bars display statistical errors. Horizontal error bars show the propagated systematic uncertainty of the on-site interaction $U$, which dominates over the statistical error of the temperature.

Figure 3

Effect of doping a Mott insulator. We compare $S_{\mathit{q}=0}$ (full symbols) and local moment (open symbols) vs filling $n$ at $U/t=8.2(5)$ for temperatures (a) $k_{B}T/t=0.63_{-0.02}^{+0.09}$, (b) $0.72_{-0.07}^{+0.06}$, and (c) $0.85_{-0.12}^{+0.13}$. The dashed lines show the theoretical local moment and the solid lines show NLCE calculations of local moment plus nearest-neighbor correlations $|C_{0,0}|-4|C_{0,1}|$. For all data a gradual disappearance of magnetic correlations with increasing doping $0.5-n$ is observed. The cross in (a) shows $S_{q=0}$ from QMC calculations at half filling for $k_{B}T/t=0.67$.