Ultracold atoms in optical lattices have great potential to contribute to a better understanding of some of the most important issues in many-body physics, such as high-temperature (high-Tc) superconductivity [1]. Thirty years ago, Anderson suggested that the Hubbard model, a simplified representation of fermions moving on a periodic lattice, may contain the essence of copper oxide superconductivity [2]. The Hubbard model describes many of the features shared by the copper oxides, including an interaction-driven Mott insulating state and an antiferromagnetic (AFM) state. Optical lattices filled with a two-spin-component Fermi gas of ultracold atoms can faithfully realise the Hubbard model with readily tunable parameters, and thus provide a platform for the systematic exploration of its phase diagram [3,4]. Realisation of strongly correlated phases, however, has been hindered by the need to cool the atoms to temperatures as low as the magnetic exchange energy, and also by the lack of reliable thermometry [5]. Here we demonstrate spin-sensitive Bragg scattering of light to measure AFM spin correlations in a realisation of the three-dimensional (3D) Hubbard model at temperatures down to 1.4 times that of the AFM phase transition. This temperature regime is beyond the range of validity of a simple high-temperature series expansion, which brings our experiment close to the limit of the capabilities of current numerical techniques. We reach these low temperatures using a unique compensated optical lattice technique [6], in which the confinement of each lattice beam is compensated by a blue-detuned laser beam. The temperature of the atoms in the lattice is deduced by comparing the light scattering to determinantal quantum Monte Carlo [7] (DQMC) and numerical linked-cluster expansion [8] (NLCE) calculations. Further refinement of the compensated lattice may produce even lower temperatures which, along with light scattering thermometry, would open avenues for achieving and characterising other novel quantum states of matter, such as the pseudogap regime of the 2D Hubbard model.A two-spin-component Fermi gas in a simple cubic optical lattice may be described by a single-band Hubbard model with nearest-neighbour tunnelling t and on-site interaction U > 0. At a density n of one atom per site, and for sufficiently large U/t there is a crossover from a 'metallic' state to a Mott insulating regime [9] as the temperature T is reduced below U . The Mott regime has been demonstrated with ultracold atoms in an optical lattice by observing the reduction of doubly occupied sites [10] and the related reduction of the global compressibility [11]. For T below the Néel ordering temperature T N , which for U t is approximately equal to the exchange energy J = 4t 2 /U , the system undergoes a phase transition to an AFM state [12]. In the context of quantum simulations, AFM phases of Ising spins have been previously engineered with bosonic atoms in an optical lattice [13] and with spin-1 2 ions [14,15]. Also, nearest-neighbour AFM correlat...

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.

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. We investigate antiferromagnetic correlations in the repulsive model at half filling and superfluidity of s-wave pairs in the attractive case away from half filling using determinantal quantum Monte Carlo simulations that are free of the fermion sign problem. We find that an entropy per particle ≃ ln 2 is sufficient to observe the charge gap in the repulsive Hubbard model 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 three or more. In contrast to 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.PACS numbers: 71.10. Fd, 37.10.Jk, 71.27.+a An exciting new development in cold atoms is the ability to realize in the laboratory simple models of strongly correlated fermions in optical lattices [1,2,3,4,5,6]. These studies are motivated by their relevance to spectacular phenomena in condensed matter physics, like high T c superconductivity, that are not fully understood. The best known model is the fermion Hubbard Hamiltonian [7,8] that captures the physics of antiferromagnetism and, at least qualitatively, d-wave superconductivity in two dimensions (2D). The Hubbard model is well understood in one dimension (1D), using exact solutions and bosonization [9], and also in the limit of large dimensions, using dynamical mean field theory (DMFT) [10]. The two dimensional problem, of direct relevance to layered high T c superconductors, is the least well understood theoretically. New insights into the 2D Hubbard model can come from cold atom emulators, given their high degree of tunability (interaction strength, chemical potential) and absence of disorder and material complications.The principal challenges for the optical lattice emulators are to cool down to sufficiently low temperature to see interesting phases and to observe the characteristic order. In this paper we present detailed quantitative results from quantum Monte Carlo (QMC) simulations [7] of (i) the 2D repulsive (U > 0) Hubbard model at half filling and (ii) the 2D attractive (U < 0) Hubbard model at any filling, that are both of direct relevance to ongoing experiments. The reasons for focusing on these systems are threefold. Determinantal QMC simulations are free of the fermion "sign problem" [11] in both cases and we can obtain non-perturbative results on finite systems without any approximations. These two systems exhibit phenomena of great interest: strong antiferromagnetic correlations and Mott physics for U > 0, and a Berezinskii-Kosterlitz-Thouless (BKT) transition to an swave superfluid [12] with a pairin...

Understanding the magnetic response of the normal state of the cuprates is considered a key piece in solving the puzzle of their high-temperature superconductivity [1]. The essential physics of these materials is believed to be captured by the Fermi-Hubbard model [2], a minimal model that has been realized with cold atoms in optical lattices [3, 4]. Here we report on site-resolved measurements of the Fermi-Hubbard model in a spin-imbalanced atomic gas, allowing us to explore the response of the system to large effective magnetic fields. We observe short-range canted antiferromagnetism at half-filling with stronger spin correlations in the direction orthogonal to the magnetization, in contrast with the spin-balanced case where identical correlations are measured for any projection of the pseudospin. The rotational anisotropy of the spin correlators is found to increase with polarization and with distance between the spins. Away from half-filling, the polarization of the gas exhibits non-monotonic behavior with doping for strong interactions, resembling the behavior of the magnetic susceptibility in the cuprates [5]. We compare our measurements to predictions from Determinantal Quantum Monte Carlo (DQMC) [6] and Numerical Linked Cluster Expansion (NLCE) [7] algorithms and find good agreement. Calculations on the doped system are near the limits of these techniques, illustrating the value of cold atom quantum simulations for studying strongly-correlated materials.Ultracold quantum gases have emerged as a powerful tool to study strongly correlated many-body physics. A two-component Fermi gas in an optical lattice can realize the repulsive Hubbard model, which describes fermions in a periodic potential with onsite interaction U and tunneling matrix element t between neighboring sites [8]. The recent introduction of quantum gas microscopes for fermionic atoms [9][10][11][12][13][14][15] has led to rapid development in the experimental study of the 2D Hubbard model. The number-squeezed nature of the Mott insulating phasepreviously inferred from bulk measurements [3, 4]-has been explicitly revealed. Furthermore, site-resolved measurements probe antiferromagnetic correlations beyond the nearest neighbor [16][17][18], which was not possible in previous studies [19][20][21].In this work, we investigate the Fermi-Hubbard model with imbalanced spin populations described by the HamiltonianHere c † i,σ is the creation operator for a fermion with spin σ on site i and n i,σ = c † i,σ c i,σ . Theoretical studies of spin-imbalance in the Hubbard model have predicted an interesting magnetic structure in trapped gases arising from the interplay of spin-imbalance and antiferromagnetic and Stoner instabilities [22][23][24]. Experimentally, * wbakr@princeton.edu the polarization of our two-component atomic Fermi gas is a controllable quantity that is conserved due to the absence of spin-relaxation mechanisms. Thermodynamically, a non-zero polarization corresponds to the introduction of an effective Zeeman field h = (µ ↑ − µ ↓ )/2, wh...

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 non-zero U , the honeycomb lattice is known to have a semi-metal 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, spin-wave and single hole dispersion using series expansions around the ordered phase. Our results are consistent with a single continuous transition at Uc/t in the range 4 − 5. Finite temperature signatures of this phase transition are seen in the behavior of the specific heat, C(T ), which changes from a two-peaked structure for U > Uc to a one-peaked structure for U < Uc. Furthermore, the U dependence of the low temperature coefficient of C(T ) exhibits an anomaly at U ≈ Uc.

The critical temperature for the attractive Hubbard model on a square lattice is determined from the analysis of two independent quantities, the helicity modulus s and the pairing correlation function P s . These quantities have been calculated through quantum Monte Carlo simulations for lattices up to 18ϫ18, and for several densities, in the intermediate-coupling regime. Imposing the universal-jump condition for an accurately calculated s , together with thorough finite-size scaling analyses ͑in the spirit of the phenomenological renormalization group͒ of P s , suggests that T c is considerably higher than hitherto assumed.

A major challenge in realizing antiferromagnetic (AF) and superfluid phases in optical lattices is the ability to cool fermions. We determine the equation of state for the 3D repulsive FermiHubbard model as a function of the chemical potential, temperature and repulsion using unbiased determinantal quantum Monte Carlo methods, and we then use the local density approximation to model a harmonic trap. We show that increasing repulsion leads to cooling, but only in a trap, due to the redistribution of entropy from the center to the metallic wings. Thus, even when the average entropy per particle is larger than that required for antiferromagnetism in the homogeneous system, the trap enables the formation of an AF Mott phase.PACS numbers: 71.10. Fd, 37.10.Jk, 71.27.+a Introduction: One of the most exciting themes in condensed matter physics is how complex states of matter emerge from simple Hamiltonians. In particular, the repulsive Fermi-Hubbard model gives rise to a rich variety of behavior, including a Mott insulating regime, an antiferromagnetically ordered Néel state, and possibly a "high-temperature" d-wave superfluid.Cold atomic gases are unique in being clean and tunable systems that offer tremendous promise for exploring such Hamiltonians. The Fermi-Hubbard model can be emulated using an optical lattice with two hyperfine species of fermions [2]. Several experimental feats have already been accomplished: the observation of sharp Fermi surfaces for free fermions in an optical lattice [3], and of the Mott insulating regime for repulsively interacting fermions [4, 5]. The next step in this quest is to go to even lower temperatures, where the local moments order to form a Néel antiferromagnet.In this Letter we present an adiabatic cooling protocol for trapped systems, which we expect to play an important role in the race for finding antiferromagnetism in the repulsive Hubbard model and for opening the door toward the search for the d-wave superfluid state. We first calculate the thermodynamics of a homogeneous system using unbiased determinantal quantum Monte Carlo (DQMC) as a function of filling and temperature, accessing both paramagnetic and AF phases. At half-filling, this allows us to obtain the entropy down to T = 0.1t (see Fig. 1(b)), well below the maximum Néel temperature T N ≈ 0.36t [6], and also well below the temperatures accessed by recent cluster studies [1].We next use the local density approximation to treat the effect of a harmonic trap. We demonstrate that increasing the repulsion U adiabatically leads to substantial cooling, but only in the presence of the trap (see Fig. 2). During this process, the cloud expands and entropy gets redistributed from the center to the metallic wings. Even though the average entropy per particle S/N ≈ 0.65k B is higher than the critical entropy of the

Local moment formation driven by the on-site repulsion U is one of the most fundamental features in the Hubbard model. At the simplest level, the temperature dependence of the local moment is expected to have a single structure at T ∼ U , reflecting the suppression of the double occupancy. In this paper we show new low temperature Quantum Monte Carlo data which emphasize that the local moment also has a signature at a lower energy scale which previously had been thought to characterize only the temperatures below which moments on different sites begin to correlate locally. We discuss implications of these results for the structure of the specific heat, and connections to quasiparticle resonance and pseudogap formation in the density of states.