Conserved quantities from entanglement Hamiltonian

We show that the subregion entanglement Hamiltonians of excited eigenstates of a quantum many-body system are approximately linear combinations of subregionally (quasi)local approximate conserved quantities, with relative commutation errors O(subregionboundaryareasubregionvolume). By diagonalizing an entanglement Hamiltonian superdensity matrix (EHSM) for an ensemble of eigenstates, we can obtain these conserved quantities as the EHSM eigenoperators with nonzero eigenvalues. For free fermions, we find the number of nonzero EHSM eigenvalues is cut off around the order of subregion volume, and some of their EHSM eigenoperators can be rather nonlocal, although subregionally quasilocal. In the interacting XYZ model, we numerically find the nonzero EHSM eigenvalues decay roughly as a power law if the system is integrable, with the exponent s≈1 (s≈1.5–2) if the eigenstates are extended (many-body localized). For fully chaotic systems, only two EHSM eigenvalues are significantly nonzero, the eigenoperators of which correspond to the identity and the subregion Hamiltonian.