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...

Quantum gas microscopes for ultracold atoms can provide high-resolution real-space snapshots of complex many-body systems. We implement machine learning to analyze and classify such snapshots of ultracold atoms. Specifically, we compare the data from an experimental realization of the twodimensional Fermi-Hubbard model to two theoretical approaches: a doped quantum spin liquid state of resonating valence bond type, and the geometric string theory, describing a state with hidden spin order. This approach considers all available information without a potential bias towards one particular theory by the choice of an observable and can therefore select the theory which is more predictive in general. Up to intermediate doping values, our algorithm tends to classify experimental snapshots as geometric-string-like, as compared to the doped spin liquid. Our results demonstrate the potential for machine learning in processing the wealth of data obtained through quantum gas microscopy for new physical insights. arXiv:1811.12425v1 [cond-mat.quant-gas]

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Image-like data from quantum systems promises to offer greater insight into the physics of correlated quantum matter. However, the traditional framework of condensed matter physics lacks principled approaches for analyzing such data. Machine learning models are a powerful theoretical tool for analyzing image-like data including many-body snapshots from quantum simulators. Recently, they have successfully distinguished between simulated snapshots that are indistinguishable from one and two point correlation functions. Thus far, the complexity of these models has inhibited new physical insights from such approaches. Here, we develop a set of nonlinearities for use in a neural network architecture that discovers features in the data which are directly interpretable in terms of physical observables. Applied to simulated snapshots produced by two candidate theories approximating the doped Fermi-Hubbard model, we uncover that the key distinguishing features are fourth-order spin-charge correlators. Our approach lends itself well to the construction of simple, versatile, end-to-end interpretable architectures, thus paving the way for new physical insights from machine learning studies of experimental and numerical data.

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