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The repulsive Hubbard model has been immensely useful in understanding strongly correlated electron systems and serves as the paradigmatic model of the field. Despite its simplicity, it exhibits a strikingly rich phenomenology reminiscent of that observed in quantum materials. Nevertheless, much of its phase diagram remains controversial. Here, we review a subset of what is known about the Hubbard model based on exact results or controlled approximate solutions in various limits, for which there is a suitable small parameter. Our primary focus is on the ground state properties of the system on various lattices in two spatial dimensions, although both lower and higher dimensions are discussed as well. Finally, we highlight some of the important outstanding open questions. 

Quantum ground states that arise at atomically controlled oxide interfaces provide an opportunity to address key questions in condensed matter physics, including the nature of two-dimensional metallic behaviour often observed adjacent to superconductivity. At the superconducting LaAlO₃/SrTiO₃interface, a metallic ground state emerges upon the collapse of superconductivity with field-effect gating and is accompanied with a pseudogap. Here we utilize independent control of carrier density and disorder of the interfacial superconductor using dual electrostatic gates, which enables the comprehensive examination of the electronic phase diagram approaching zero temperature. We find that the pseudogap corresponds to precursor pairing, and the onset of long-range phase coherence forms a two-dimensional superconducting dome as a function of the dual-gate voltages. The gate-tuned superconductor–metal transitions are driven by macroscopic phase fluctuations of Josephson coupled superconducting puddles.