Many exotic phenomena in strongly correlated electron systems emerge from the interplay between spin and motional degrees of freedom [1, 2]. For example, doping an antiferromagnet gives rise to interesting phases including pseudogap states and high-temperature superconductors [3]. A promising route towards achieving a complete understanding of these materials begins with analytic and computational analysis of simplified models. Quantum simulation has recently emerged as a complementary approach towards understanding these models [4][5][6][7][8]. Ultracold fermions in optical lattices offer the potential to answer open questions on the lowtemperature regime of the doped Hubbard model [9][10][11], which is thought to capture essential aspects of the cuprate superconductor phase diagram but is numerically intractable in that parameter regime. Already, Mott-insulating phases and short-range antiferromagnetic correlations have been observed, but temperatures were too high to create an antiferromagnet [12][13][14][15]. A new perspective is afforded by quantum gas microscopy [16][17][18][19][20][21][22][23][24][25][26][27][28], which allows readout of magnetic correlations at the site-resolved level [25][26][27][28]. Here we report the realization of an antiferromagnet in a repulsively interacting Fermi gas on a 2D square lattice of approximately 80 sites. Using site-resolved imaging, we detect (finite-size) antiferromagnetic long-range order (LRO) through the development of a peak in the spin structure factor and the divergence of the correlation length that reaches the size of the system. At our lowest temperature of T/t = 0.25(2) we find strong order across the entire sample, where the staggered magnetization approaches the ground-state value. Our experimental platform enables doping away from half filling, where pseudogap states and stripe ordering are expected, but theoretical methods become numerically intractable. In this regime we find that the antiferromagnetic LRO persists to hole dopings of about 15%, providing a guideline for computational methods. Our results demonstrate that quantum gas microscopy of ultracold fermions in optical lattices can now address open questions on the low-temperature Hubbard model.The Hubbard Hamiltonian is a fundamental model for spinful lattice electrons describing a competition between kinetic energy t and interaction energy U [29]. In the limiting case of half-filling (average one particle per site) and dominant interactions (U/t 1) the Hubbard model maps to the Heisenberg model [1]. There, the exchange energy J = 4t 2 /U can give rise to antiferromagnetically ordered states at low temperatures [30]. This order persists for all finite U/t, where charge fluctuations reduce the ordering strength [31]. Away from half-filling, the coupling between motional and spin degrees of freedom is expected to give rise to a rich many-body phase diagram (see Fig. 1a), which is challenging to understand theoretically due to the fermion sign problem [32]. Even so, in the thermodynamic limit com...

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.PACS numbers: 37.10. Jk, 67.85.Lm, 71.10.Fd, 75.10.Jm, Quantum many-body systems exhibiting magnetic correlations underlie a wide variety of phenomena. Hightemperature superconductivity, for example, can arise from the correlated motion of holes on 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 [7][8][9][10][11][12][13] 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 timescales. 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 the first trap-resolved observations of magnetic correlations in a Fermi-lattice system. Fermionic atoms in a two-dimensional 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 comp...

The complexity of quantum many-body systems originates from the interplay of strong interactions, quantum statistics, and the large number of quantum-mechanical degrees of freedom. Probing these systems on a microscopic level with single-site resolution offers important insights. Here we report site-resolved imaging of two-component fermionic Mott insulators, metals, and band insulators, using ultracold atoms in a square lattice. For strong repulsive interactions, we observed two-dimensional Mott insulators containing over 400 atoms. For intermediate interactions, we observed a coexistence of phases. From comparison to theory, we find trap-averaged entropies per particle of 1.0 times the Boltzmann constant (k(B)). In the band insulator, we find local entropies as low as 0.5 k(B). Access to local observables will aid the understanding of fermionic many-body systems in regimes inaccessible by modern theoretical methods.

Understanding strongly correlated quantum many-body states is one of the most difficult challenges in modern physics. For example, there remain fundamental open questions on the phase diagram of the Hubbard model, which describes strongly correlated electrons in solids. In this work we realize the Hubbard Hamiltonian and search for specific patterns within the individual images of many realizations of strongly correlated ultracold fermions in an optical lattice. Upon doping a cold-atom antiferromagnet we find consistency with geometric strings, entities that may explain the relationship between hole motion and spin order, in both pattern-based and conventional observables. Our results demonstrate the potential for pattern recognition to provide key insights into cold-atom quantum many-body systems.Quantum superposition describes quantum systems as simultaneously realizing different configurations. Such behavior is believed to be at the heart of phenomena in strongly correlated quantum many-body systems, which cannot be described by single-particle or mean-field theories. An intriguing consequence of the superposition principle is the existence of hidden order in correlated quantum systems: although every individual configuration is characterized by a particular pattern, the average over these configurations leads to an apparent loss of order. By contrast, instantaneous projective measurements have the potential to reveal these underlying patterns.One notable example of a system with hidden order is the one-dimensional (1D) Fermi-Hubbard model at strong coupling [1, 2]. Although 1D chains with additional holes or particles beyond an average of one particle per site (doped) yield average two-point spin correlations which decay more rapidly with distance than chains with an average of one particle per site (half-filled), this magnetic ordering can be revealed by accounting for the fluctuating positions across individual configurations of the additional dopants within each chain. The apparent loss of magnetic order is in fact hidden order, hidden by the dopants and their varying positions [3,4]. Although direct detection of this hidden string order remains inaccessible in solids, experiments with ultracold atoms enable projective measurements, or "snapshots", and generally can provide access to such structures [5]. In particular, quantum gas microscopy [6] enables siteresolved imaging and access to correlators which have been constructed to reveal the hidden order [7].The hidden order in 1D is well understood, but the physics of the 2D Hubbard model is fundamentally more * Corresponding author. Email: greiner@physics.harvard.edu. complex due to an intricate interplay between spin and charge degrees of freedom; as a result, formulating an appropriate correlation function to search for hidden order becomes considerably more challenging. The 2D Hubbard model is believed to capture the rich physics of high-temperature superconductivity and other phases [8-10] such as the strange metal, stripe, antiferromagnet (AFM), or p...

When a mobile hole is moving in an anti-ferromagnet it distorts the surrounding Néel order and forms a magnetic polaron. Such interplay between hole motion and anti-ferromagnetism is believed to be at the heart of high-temperature superconductivity in cuprates. In this article we study a single hole described by the t − Jz model with Ising interactions between the spins in two dimensions. This situation can be experimentally realized in quantum gas microscopes with Mott insulators of Rydberg-dressed bosons or fermions, or using polar molecules. We work at strong couplings, where hole hopping is much larger than couplings between the spins. In this regime we find strong theoretical evidence that magnetic polarons can be understood as bound states of two partons, a spinon and a holon carrying spin and charge quantum numbers respectively. Starting from first principles, we introduce a microscopic parton description which is benchmarked by comparison with results from advanced numerical simulations. Using this parton theory, we predict a series of excited states that are invisible in the spectral function and correspond to rotational excitations of the spinon-holon pair. This is reminiscent of mesonic resonances observed in high-energy physics, which can be understood as rotating quark antiquark pairs carrying orbital angular momentum. Moreover, we apply the strong coupling parton theory to study far-from equilibrium dynamics of magnetic polarons observable in current experiments with ultracold atoms. Our work supports earlier ideas that partons in a confining phase of matter represent a useful paradigm in condensedmatter physics and in the context of high-temperature superconductivity in particular. While direct observations of spinons and holons in real space are impossible in traditional solid-state experiments, quantum gas microscopes provide a new experimental toolbox. We show that, using this platform, direct observations of partons in and out-of equilibrium are now possible. Extensions of our approach to the t−J model are also discussed. Our predictions in this case are relevant to current experiments with quantum gas microscopes for ultracold atoms.Here we study the problem of a single hole moving in an anti-ferromagnet from a different perspective, focusing on the t − J z model for simplicity. In contrast to most earlier works, we consider the strong coupling regime, t J z . Starting from first principles, we derive a microscopic parton theory of magnetic polarons. This approach not only provides new conceptual insights to the physics of magnetic polarons, but it also enables semianalytical derivations of their properties. We benchmark arXiv:1712.01874v1 [cond-mat.quant-gas] 5 Dec 2017 1. Implementation of the t − Jz model H Ising = 1 2 i,j