Quasiperiodic modulation can prevent isolated quantum systems from equilibrating by localizing their degrees of freedom. Such localization has been well-known since the work of Aubry and André and, in modern quantum optical systems, quasiperiodic `disorder' has become a standard tool for inducing Anderson localization.
In this talk, I will describe how the interplay of symmetry breaking and localization in the presence of quasiperiodic modulation leads to a surprisingly rich range of phenomena in transverse field Ising chains. We will start at `infinite temperature', where we will find a 'quasiperiodic Ising glass' phase which is dynamically stable at all energy densities due to localization. This glass melts in several ways, which we will touch on briefly. We will then turn to zero temperature, which exhibits a bewildering array of symmetry breaking phases and phase transitions with extended, localized, and critically delocalized low-energy excitations. We will focus on the most striking feature: a new quantum Ising transition whose properties are intermediate between those of the clean and infinite randomness Ising transitions.