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.

Machine learning offers an unprecedented perspective for the problem of classifying phases in condensed matter physics. We employ neural-network machine learning techniques to distinguish finite-temperature phases of the strongly correlated fermions on cubic lattices. We show that a three dimensional convolutional network trained on auxiliary field configurations produced by quantum Monte Carlo simulations of the Hubbard model can correctly predict the magnetic phase diagram of the model at the average density of one (half filling). We then use the network, trained at half filling, to explore the trend in the transition temperature as the system is doped away from half filling. This transfer learning approach predicts that the instability to the magnetic phase extends to at least 5% doping in this region. Our results pave the way for other machine learning applications in correlated quantum many-body systems.Comment: 9 pages, 8 figure

We examine the validity of fluctuation-dissipation relations in isolated quantum systems taken out of equilibrium by a sudden quench. We focus on the dynamics of trapped hard-core bosons in one-dimensional lattices with dipolar interactions whose strength is changed during the quench. We find indications that fluctuationdissipation relations hold if the system is nonintegrable after the quench, as well as if it is integrable after the quench if the initial state is an equilibrium state of a nonintegrable Hamiltonian. On the other hand, we find indications that they fail if the system is integrable both before and after quenching. PACS numbers: 05.30.Jp, 03.75.Kk, The fluctuation-dissipation theorem (FDT) [1-3] is a fundamental relation in statistical mechanics which states that typical deviations from the equilibrium state caused by an external perturbation (within the linear response regime) dissipate in time in the same way as random fluctuations. The theorem applies to both classical and quantum systems as long as they are in thermal equilibrium. Fluctuation-dissipation relations are not, in general, satisfied for out-of-equilibrium systems. In particular, if a system is isolated, it is not clear whether once taken far from equilibrium fluctuation-dissipation relations apply at any later time. Studies of integrable models such as a Luttinger liquid [4] and the transverse field Ising chain [5] have shown that the use of fluctuation-dissipation relations to define temperature leads to values of the temperature that depend on the momentum mode and/or the frequency being considered. More recently, Essler et al. [6] have shown that for a subsystem of an isolated infinite system, the basic form of the FDT holds, and that the same ensemble that describes the static properties also describes the dynamics.The question of the applicability of the FDT to isolated quantum systems is particularly relevant to experiments with cold atomic gases [7,8], whose dynamics is considered to be, to a good approximation, unitary [9]. In that context, the description of observables after relaxation (whenever relaxation to a time-independent value occurs) has been intensively explored in the recent literature [10]. This is because, for isolated quantum systems out of equilibrium, it is not apparent that thermalization can take place. For example, if the system is prepared in an initial pure state |φ ini that is not an eigenstate of the HamiltonianĤ (Ĥ|ψ α = E α |ψ α ) (as in Ref. [9]), then the infinite-time average of the evolution of the observableÔ can be written as Ô (t) = ∑ α |c α | 2 O αα ≡ O diag , where c α = ψ α |φ ini , O αα = ψ α |Ô|ψ α , and we have assumed that the spectrum is nondegenerate. The outcome of the infinite-time average can be thought of as the prediction of a "diagonal" ensemble [11]. O diag depends on the initial state through the c α 's (there is an exponentially large number of them), while the thermal predictions depend only on the total energy φ ini |Ĥ|φ ini ; i.e., they need not agree.The lack of thermal...

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...