The

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 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 *D*_{s}),
and we defer discussion of the transport problem to
Chapter 9.

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