Strong electron correlations lie at the origin of high-temperature superconductivity. Its essence is believed to be captured by the Fermi-Hubbard model of repulsively interacting fermions on a lattice. Here we report on the site-resolved observation of charge and spin correlations in the two-dimensional (2D) Fermi-Hubbard model realized with ultracold atoms. Antiferromagnetic spin correlations are maximal at half-filling and weaken monotonically upon doping. At large doping, nearest-neighbor correlations between singly charged sites are negative, revealing the formation of a correlation hole, the suppressed probability of finding two fermions near each other. As the doping is reduced, the correlations become positive, signaling strong bunching of doublons and holes, in agreement with numerical calculations. The dynamics of the doublon-hole correlations should play an important role for transport in the Fermi-Hubbard model.

Imaging a microscopic power struggle: Strongly interacting fermions in a two-dimensional lattice form a checkerboard pattern, with spins of opposite directions occupying neighboring sites of the lattice. When an external magnetic field is applied, the situation becomes more complicated—should the spins align with the field, or try to preserve the checkerboard order? Brown et al. studied this problem using ⁶ Li atoms in an optical lattice with unequal numbers of two spin components; the imbalance between the two played the role of an effective magnetic field. With the field applied, the checkerboard pattern correlations of the spin component perpendicular to the field became stronger than those of the spin component parallel to the field, indicating that the system was approaching the so-called canted antiferromagnetic state. Science , this issue p. 1385

We investigate ground state and finite temperature properties of the half-filled Hubbard model on a honeycomb lattice using quantum Monte Carlo and series expansion techniques. Unlike the square lattice, for which magnetic order exists at $T=0$ for any nonzero $U$, the honeycomb lattice is known to have a semimetal phase at small $U$ and an antiferromagnetic one at large $U$. We investigate the phase transition at $T=0$ by studying the magnetic structure factor and compressibility using quantum Monte Carlo simulations and by calculating the sublattice magnetization, uniform susceptibility, spinwave, and single hole dispersion using series expansions around the ordered phase. Our results are consistent with a single continuous transition at $U_c∕t$ in the range 4–5. Finite-temperature signatures of this phase transition are seen in the behavior of the specific heat, $C\left(T\right)$, which changes from a two-peaked structure for $U>U_c$ to a one-peaked structure for $U<U_c$. Furthermore, the $U$ dependence of the low temperature coefficient of $C\left(T\right)$ exhibits an anomaly at $U\approx U_c$.

One of the major challenges in realizing antiferromagnetic and superfluid phases in optical lattices is the ability to cool fermions. We determine constraints on the entropy for observing these phases in two-dimensional Hubbard models using determinantal quantum Monte Carlo simulations. We find that an entropy per particle $\simeq \ln 2$ is sufficient to observe the insulating gap in the repulsive Hubbard model at half-filling, or the pairing pseudogap in the attractive case. Observing antiferromagnetic correlations or superfluidity in 2D systems requires a further reduction in entropy by a factor of 3 or more. In contrast with higher dimensions, we find that adiabatic cooling is not useful to achieve the required low temperatures. We also show that double-occupancy measurements are useful for thermometry for temperatures greater than the nearest-neighbor hopping energy.

The critical temperature for the attractive Hubbard model on a square lattice is determined from the analysis of two independent quantities, the helicity modulus ${\rho }_s$ and the pairing correlation function $P_s.$ These quantities have been calculated through quantum Monte Carlo simulations for lattices up to $18\times 18,$ and for several densities, in the intermediate-coupling regime. Imposing the universal-jump condition for an accurately calculated ${\rho }_s,$ together with thorough finite-size scaling analyses (in the spirit of the phenomenological renormalization group) of $P_s,$ suggests that $T_c$ is considerably higher than hitherto assumed.