This chapter turns to a systematic analysis of transport of conserved charges in the quantum rotor model. We introduced some general concepts in Section 8.3, and these are illustrated here by explicit computations at higher orders.

For d = 1, we considered time-dependent correlations of the conserved angular momentum, L(x, t), of the O(3) quantum rotor model in Chapter 12. We found, using effective semiclassical models, that the dynamic fluctuations of L(x, t) were characterized by a diffusive form (see (12.26)) at long times and distances, and we were able to obtain values for the spin diffusion constant Ds at low T and high T (see Table 12.1). The purpose of this chapter is to study the analogous correlations in d = 2 for N ≥ 2; the case N = 1 has no conserved angular momentum, and so there is no possibility of diffusive spin correlations. Rather than thinking about fluctuations of the conserved angular momentum in equilibrium, we 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 nonlinear classical wave problem developed in Section 14.3 in the context of the ∈ = 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.