Overview

This chapter begins by describing the *D* > 1-dimensional classical statistical mechanical models which are `equivalent' (in a sense to be made precise) to the *d*-dimensional quantum Ising and rotor models introduced in Chapter 1. The universal properties of these transitions are discussed, and we argue that these are described by certain continuum field theories. At the level of the these continuum theories it is argued quite generally that, at least in a formal sense, there is a classical statistical mechanical model associated with every second order quantum phase transition. The nature of this general quantum-classical mapping is discussed and its limitations and utility are highlighted. In particular, we stress that a direct treatment of the *d*-dimensional quantum model is crucial for many experimentally important properties: these are effectively inextricable from the observables of the corresponding *D*-dimensional classical theory.

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