Spin dynamics and transport in gapped one-dimensional Heisenberg antiferromagnets at nonzero temperatures

We discuss the theory of the nonzero temperature $\mathrm{}\left(T\right)\mathrm{}$ spin dynamics and transport in one-dimensional Heisenberg antiferromagnets with a gap $\Delta$. For $T\ll \Delta$, we develop a semiclassical picture of thermally excited particles. Multiple inelastic collisions between the particles are crucial, and are described by a two-particle $\mathcal{S}$ matrix which is shown to have a superuniversal form at low momenta. This is established by computations on the $O\left(3\right)\mathrm{}$ $\sigma$ model, and strong- and weak-coupling expansions (the latter using a Majorana fermion representation) for the two-leg $S=1/2\mathrm{}$ Heisenberg antiferromagnetic ladder. As an aside, we note that the strong-coupling calculation reveals an $S=1\mathrm{}$, two-particle bound state which leads to the presence of a second peak in the $T=0\mathrm{}$ inelastic neutron-scattering (INS) cross section for a range of values of momentum transfer. We obtain exact, or numerically exact, universal expressions for the thermal broadening of the quasiparticle peak in the INS cross section, the spin diffusivity, and for the field dependence of the NMR relaxation rate ${1/T}_1$ of the effective semiclassical model; these are expected to be asymptotically exact for the quantum antiferromagnets in the limit $T\ll \Delta$. The results for ${1/T}_1$ are compared with the experimental findings of Takigawa et al. [Phys. Rev. Lett. 76, 2173 (1996)] and the agreement is quite good. In the regime $\Delta <T<\mathrm{}$(a typical microscopic exchange) and we argue that a complementary description in terms of semiclassical waves applies, and give some exact results for the thermodynamics and dynamics.