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On the thermalization of a Luttinger liquid after a sequence of sudden interaction quenches

We present a comprehensive analysis of the relaxation dynamics of a Luttinger liquid subject to a sequence of sudden interaction quenches. We express the critical exponent $β$ governing the decay of the steady-state propagator as an explicit functional of the switching protocol. At long distances $β$ depends only on the initial state while at short distances it is also history dependent. Continuous protocols of arbitrary complexity can be realized with infinitely long sequences. For quenches of finite duration we prove that there exist no protocol to bring the initial non-interacting system in the ground state of the Luttinger liquid. Nevertheless memory effects are washed out at short-distances. The adiabatic theorem is then investigated with ramp-switchings of increasing duration, and several analytic results for both the propagator and the excitation energy are derived.