Spin conductivity in almost integrable spin chains

The spin conductivity in the integrable spin-1∕2 XXZ chain is known to be infinite at finite temperatures T for anisotropies −1<Δ<1. Perturbations, which break integrability, e.g., a next-nearest neighbor coupling J^′, render the conductivity finite. We construct numerically a nonlocal conserved operator J_∥ which is responsible for the finite spin Drude weight of the integrable model and calculate its decay rate for small J^′. This allows us to obtain a lower bound for the spin conductivity σ_s⩾c(T)∕J^′2, where c(T) is finite for J^′→0. We discuss the implication of our result for the general question how nonlocal conservation laws affect transport properties.