  Weak perturbations can drive an interacting many-particle system far from its
initial equilibrium state if one is able to pump into degrees of freedom
approximately protected by conservation laws. This concept has for example been
used to realize Bose-Einstein condensates of photons, magnons, and excitons.
Integrable quantum systems, like the one-dimensional Heisenberg model, are
characterized by an infinite set of conservation laws. Here we develop a theory
of weakly driven integrable systems and show that pumping can induce large spin
or heat currents even in the presence of integrability breaking perturbations,
since it activates local and quasi-local approximate conserved quantities. The
resulting steady state is qualitatively captured by a truncated generalized
Gibbs ensemble with Lagrange parameters that depend on the structure but not on
the overall amplitude of perturbations nor the initial state. We suggest to use
spin-chain materials driven by terahertz radiation to realize
integrability-based spin and heat pumps.
