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 Ds 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  applying classical wave models to transport directly in d = 2.