Ballistic-to-diffusive transition in spin chains with broken integrability

We study the ballistic-to-diffusive transition induced by the weak breaking of integrability in a boundary-driven XXZ spin-chain. Studying the evolution of the spin current density J^s as a function of the system size L, we show that, accounting for boundary effects, the transition has a non-trivial universal behavior close to the XX limit. It is controlled by the scattering length L^*~ V^-2, where V is the strength of the integrability breaking term.
In the XXZ model, the interplay of interactions controls the emergence of a transient ``quasi-ballistic'' regime at length scales much shorter than L^*.
This parametrically large regime is characterized by a strong renormalization of the current which forbids a universal scaling, unlike the XX model.
Our results are based on Matrix Product Operator numerical simulations and agree with perturbative analytical calculations.