The

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 e_{k} = (*c*^{2} *k*^{2} + D^{2})^{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.

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