Transport in

We considered time-dependent correlations of the conserved
angular momentum, **L**(*x*,*t*), of the O(3) quantum rotor model
in *d* = 1 in Chapter 6. We found, using
effective semiclassical models, that the dynamic fluctuations
of **L**(*x*,*t*) were characterized by a *diffusive* form
at long times and distances, and were able to obtain values for
the spin diffusion constant *D*_{s} at low *T* and high *T*.
The purpose of this chapter is to study the analogous correlations
in *d* = 2 for *N* ³ 2; the case *N* = 1 has no conserved angular-momentum,
and so no possibility of diffusive spin correlations. Rather than
thinking about fluctuations of the conserved angular momentum in equilibrium,
we shall find it more convenient here to consider instead the response
to an external space and time dependent
`magnetic' field **H** (*x*,*t*), and to examine how the system
transports the conserved angular momentum under its influence.

In principle, it is possible to address these issues in the high *T*
region using the
non-linear classical wave problem developed in
Chapter 8 in the context of the e = 3-*d*
expansion. However, an attempt to do this quickly shows that the
correlators of **L** contain ultraviolet divergences when
evaluated in the effective classical theory. Physically, this is a
signal that, for small e, transport properties are *not* dominated by
excitations with energy << *T* (while the order parameter
fluctuations, considered in Chapter 8, were),
and it is necessary to include
fluctuations with higher energy, which must then be treated
quantum mechanically. It is this quantum transport problem we
address here. We show that it is necessary to solve a quantum
transport equation for the quantized particle excitations to
describe diffusion in the high *T* and the low *T* quantum
paramagnetic regions: in the former regime we find a conductivity
which is simply a universal number times fundamental constants
of nature.

We note, in passing, a new recent proposal [58]
applying classical wave models
to transport directly in *d* = 2.

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