Reconstructing Entanglement Hamiltonian via Entanglement Eigenstates

The entanglement Hamiltonian $H_E$, defined through the reduced density matrix of a subsystem $\rho_A=\exp(-H_E)$, is an important concept in understanding the nature of quantum entanglement in many-body systems and quantum field theories. In this work, we explore a numerical scheme which explicitly reconstructs the entanglement Hamiltonian using one entangled mode (i.e., an eigenstate) of $\rho_A$. We demonstrate and benchmark this scheme on quantum spin lattice models. The resulting $H_E$ bears a form similar to a physical Hamiltonian with spatially varying couplings, which allows us to make quantitative comparison with perturbation theory and conformal field theory.