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We investigate of the relationship between the entanglement and subsystemHamiltonians in the perturbative regime of strong coupling between subsystems.One of the two conditions that guarantees the proportionality between theseHamiltonians obtained by using the nondegenerate perturbation theory within thefirst order is that the unperturbed ground state has a trivial entanglementHamiltonian. Furthermore, we study the entanglement Hamiltonian of theHeisenberg ladders in a time-dependent magnetic field using the degenerateperturbation theory, where couplings between legs are considered as aperturbation. In this case, when the ground state is two-fold degenerate, andthe entanglement Hamiltonian is proportional to the Hamiltonian of a chainwithin first-order perturbation theory, even then also the unperturbed groundstate has a nontrivial entanglement spectrum.

Entanglement spectrum of the degenerative ground state of Heisenberg ladders in a time-dependent magnetic field