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

TL;DR

This paper develops a classical model for low-temperature spin transport in a one-dimensional quantum $O(3)$ non-linear sigma model, providing universal functions for the crossover and diffusion constant, with implications for experimental systems.

Contribution

It introduces a low-temperature classical model for spin transport in the 1D quantum $O(3)$ sigma model, deriving exact universal functions for the crossover and diffusion constant.

Findings

Universal functions for ballistic-to-diffusive crossover.

Exact low-temperature spin diffusion constant.

Implications for experiments on 1D insulators with a spin gap.

Abstract

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.