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We study how perturbations affect dynamics of integrable many-body quantumsystems, causing transition from integrability to chaos. Looking at spintransport in the Heisenberg chain with impurities we find that in thethermodynamic limit transport gets diffusive already at an infinitesimalperturbation. Small extensive perturbations therefore cause an immediatetransition from integrability to chaos. Nevertheless, there is a remnant ofintegrability encoded in the dependence of the diffusion constant on theimpurity density, namely, at small densities it is proportional to the squareroot of the inverse density, instead of to the inverse density as would followfrom Matthiessen's rule. We show that Matthiessen's rule has to be modified innon-ballistic systems. Results also highlight a nontrivial role of interactingscattering on a single impurity, and that there is a regime where adding moreimpurities can actually increase transport.

Weak integrability breaking: chaos with integrability signature in coherent diffusion