This chapter will discuss 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. It turns out that the 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. We will then be able to transfer much of the sophisticated technology developed to analyze these classical models to the quantum models of interest here. (It is important not to confuse this very general and formal mapping with the fact that some d-dimensional quantum systems in the vicinity of finite temperature phase transitions are described by effective d-dimensional classical models, as in the shaded region of Fig. 1.2).

We will discuss this mapping here in the 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 (the cases d > 0 will then be discussed in Chapter 3). 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” ξ becomes very large; the properties of these regions are very similar to those in the vicinity of the phase transition points in higher dimensions.