As we noted in the preface, this and the following chapter are at a more advanced level, and some readers may wish to skip ahead to Chapter 8.

In Chapter 5 we studied the O(N) quantum rotor model in the large-N limit for a number of values of the spatial dimensionality, including d = 1. We noted that the results provided an adequate description of the static properties in d = 1 for N ≥ 3. This will be justified in the present chapter where we will obtain a number of exact results for the same static observables. We also noted that the large-N limit did a very poor job of describing dynamical properties at nonzero temperatures. This will be repaired in this chapter by simple physical arguments that lead to a fairly complete (and believed exact) description of the long-time behavior. Some of the discussion in this chapter will be specialized to the O(N = 3) model, which is also the case of greatest physical importance; the properties of the O(N > 3) models are very similar, and many of our results will be quoted for general N. Of the remaining cases, the d = 1, N = 1 model has been already considered in Chapter 4, and study of the d = 1, N = 2 model is postponed to Section 14.3.

The physical picture of the T = 0, N = 3 state that emerged in Chapter 5 was very simple.