Conserved Quantities in Generalized Gibbs Ensemble from Entanglement Hamiltonian

The description and understanding of relaxation and thermalization of isolated quantum many-body systems have been elusive for decades. It has been shown that a subset of conserved quantities may lead an integrable system to relax into a nonthermal state described by a generalized Gibbs ensemble (GGE), however, which sets of conserved quantities constrain the relaxation and which do not is still unclear. Recently, a method of getting subregionally (quasi)local conserved quantities from the entanglement Hamiltonian of a bipartite system has been proposed. We find that the conserved quantities which constrain the relaxation of a 1+1d free-fermion system can be got by the same method while making the system a "coarse-grained subsystem" of a larger system. Generically, these conserved quantities may not commute to each other but should all be included in GGE.