Chapter 6
The d = 1, O(N ³ 3) rotor models

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 3: this is justified in the present chapter where we obtain 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 is repaired in this chapter by simple physical arguments which lead to a fairly complete (and believed exact) description of the long-time behavior.

The physical picture of the T = 0, N = 3 state which emerged in Chapter 5 was very simple. The ground state was a quantum paramagnet which did not break any symmetries. There was an energy gap, D, above the ground state, and the low-lying excitations were a triplet of particles with dispersion ek = (c2 k2 + D2)1/2; this picture is verified here by a more complete renormalization group analysis. At non-zero temperatures, the dynamical crossovers between a low T and a high T regime is described. The dynamics of the low T region is described by an effective model of quasi-classical particles , closely related to the particle model developed in Chapter 4 for the Ising chain. For the high T region, we develop a new, `dual', description in a model of quasi-classical waves .

As indicated in Chapter 5, and discussed more extensively in Chapter 13, the d = 1, O(3) rotor model describes a large class of quasi one-dimensional spin gap compounds. The low T regime is applicable to all such systems, while the high T, quasi-classical wave regime applies only if the continuum quantum field theory description for the lattice model holds at these elevated temperature. We make contact with recent experiments.