Chapter 2
The mapping to classical statistical mechanics: single site models

This chapter discusses the reason for the central importance of the quantum Ising and rotor models in the theory of quantum phase transitions, quite apart from any experimental motivations: quantum transitions in these models in d dimensions are intimately connected to certain well-studied finite temperature phase transitions in classical statistical mechanics models in D = d+1 dimensions. This mapping is described in some detail in its simplest context of d = 0, D = 1: we will consider single site quantum Ising and rotor models, and explicitly discuss their mapping to classical statistical mechanics models in D = 1. These very simple classical models in D = 1, actually do not have any phase transitions. Nevertheless, it is quite useful to examine them thoroughly as they do have regions in which the correlation `length' x becomes very large: the properties of these regions are very similar to those in the vicinity of the phase transition points in higher dimensions. In particular, we will introduce the central ideas of the scaling limit and universality in this very simple context. We will then go on to map the classical models to equivalent zero-dimensional quantum models and demonstrate that this mapping becomes exact in the scaling limit.