Chapter 7
The d = 2, O(N 3) rotor models

The large N limit of quantum rotor models in d = 2 was examined in Chapter 5. There we claimed that the large N results provided a satisfactory description of the crossovers in the static and thermodynamic observables for N 3. This chapter establishes this claim, and also treats the dynamic correlations of n at nonzero temperatures. The low T region on the quantum paramagnetic side of the quantum critical point is described in an effective model of quasi-classical particles closely related to those developed in Chapters 4 and 6. On the other low T region on the magnetically ordered side, we obtain a `dual' model of quasi-classical waves , which is connected to that developed in Chapter 6. Finally, in the intermediate `quantum critical' or continuum high T region, neither of these descriptions is adequate: quantum and thermal, particle- and wave-like behavior, all play important roles, and we use a menage of these concepts to obtain a complete picture in this, and the following two chapters.

The results for the quasi-classical wave regime described in this chapter are obtained by a combination of analytical and numerical techniques, which become exact in the low T limit. For the other two regions, we shall use the large N expansion. This approximate approach is satisfactory for most purposes, but fails in the very low frequency regime, w << T. A proper description of the low frequency dynamical correlators of n must await alternative techniques which will be developed in Chapter 8.

The cases N = 1,2, d = 2 are special because they permit phase transitions at non-zero temperatures; we defer their discussion to Chapter 8.

We do not consider time-dependent correlations of the angular momentum L in this chapter. The conservation of the total L implies that its low frequency dynamics obeys the diffusion equation. So the problem reduces to the determination of a `transport coefficient' (the spin diffusion constant Ds), and we defer discussion of the transport problem to Chapter 9.