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 ${\mathcal{J}}^{s}$ as a function of the system size $L$, we show that, accounting for boundary effects, the transition has a nontrivial universal behavior close to the XX limit. It is controlled by the scattering length ${L}^{*}\ensuremath{\propto}{V}^{\ensuremath{-}2}$, where $V$ is the strength of the integrability-breaking term. In the XXZ model, the interplay of interactions controls the emergence of a transient ``quasiballistic'' 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.