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 σₛc(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.