Low temperature spin diffusion in the one-dimensional quantum $O(3)$ nonlinear $\sigma$-model

An effective, low temperature, classical model for spin transport in the one-dimensional, gapped, quantum $O(3)$ non-linear $\sigma$-model is developed. Its correlators are obtained by a mapping to a model solved earlier by Jepsen. We obtain universal functions for the ballistic-to-diffusive crossover and the value of the spin diffusion constant, and these are claimed to be exact at low temperatures. Implications for experiments on one-dimensional insulators with a spin gap are noted.