Splitting a critical spin chain

We study a quench protocol that conserves the entanglement spectrum of a bipartition of a quantum system. As an example we consider the splitting of a critical Ising chain in two chains, and compare it with the well known case of joining of two chains. We show that both the out of equilibrium time evolution of global properties and the equilibrium regime after the quench of local properties are different in the two scenarios. Since the two quenches only differ in the presence/absence of the conservation of the entanglement spectrum, our results suggest that this conservation plays a fundamental role in both the out-of-equilibrium dynamics and the subsequent equilibration mechanism. We discuss the relevance of our results in the context of quantum simulators.