The ideas of quantum criticality have proven a powerful unifying framework for a broad range of strongly correlated quantum systems. Experimental probes with ever greater sensitivity have resolved subtle new states of electronic matter in the vicinity of quantum critical points. Attempts to understand these have led to a remarkable cross-fertilisation of ideas. This has involved insights from numerics, the novel applications of some old ideas from condensed matter field theory, and the import of completely new ideas from quantum information theory and string theory. par I will review the basic notions underlying quantum criticality and the different types of systems to which they apply. Following this, I will discuss three different approaches to understanding quantum critical systems and indicate how it is hoped to combine them into a future, unified view of quantum critical instabilities.

University College London