The Hubbard Model

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