In quantum field theory, models are often defined with one or more continuously adjustable parameters. Different regions of parameter-space in which the vacuum state has qualitatively different properties are called different phases, and the thresholds between those regions are called phase transitions. For many of the models that we know how to construct nonperturbatively, the only known nonperturbative constructions involve pretending that spacetime is discrete. Understanding the phase structure of models defined in discrete spacetime can inform the search for interesting continuum limits. This article reviews the phase structure of some simple models of scalar quantum fields, emphasizing how it depends both on the type of internal symmetry and on the number of dimensions of spacetime.
