Does instability imply thermalization in many-body interacting systems?

We study the relaxation process of a 1D model of interacting spins quenched far from equilibrium. Our approach is based on the direct correspondence between the classical and quantum spread of energy, initially concentrated in a particular spin-particle. The classical dynamics are quantified by the largest Lyapunov exponent, in close correspondence with the spread rate of the quantum wave packet in the many-body representation. Our results show that the initial (classical or quantum) exponential spread of energy does not necessarily imply thermalization.