  Relaxed quantum systems with conservation laws are believed to be
approximated by the Generalized Gibbs Ensemble (GGE), which incorporates the
constraints of certain conserved quantities serving as integrals of motion. By
drawing analogy between reduced density matrix and GGE, we demonstrate for free
fermions a generic entanglement Hamiltonian superdensity matrix (EHSM)
framework for determining the set of conserved quantities in GGE. The framework
proposes that such conserved quantities are linear superposition of eigenstate
entanglement Hamiltonians of a larger auxiliary system, where the eigenstates
are Fock states occupying the common eigenmodes. For 1D homogeneous free
fermions with periodic boundary condition, which maps to 1D hardcore bosons,
these conserved quantities lead to an non-Abelian GGE, which predicts the
relaxation of both fermion and boson bilinears more accurately than the
conventional Abelian GGE. Generalization of the framework to interacting models
may provide novel numerical insights for quantum integrability.
