From Eigenstate To Hamiltonian: Prospects For Ergodicity And Localization

This paper addresses the so-called inverse problem which consists in searching for (possibly multiple) parent target Hamiltonian(s), given a single quantum state as input. Starting from Psi(0), an eigenstate of a given local Hamiltonian H-0, we ask whether or not there exists another parent Hamiltonian H-p for Psi(0), with the same local form as H-0. Focusing on one-dimensional quantum disordered systems, we extend the recent results obtained for Bose-glass ground states [M. Dupont and N. Laflorencie, Phys. Rev. B 99, 020202(R) (2019)] to Anderson localization, and the many-body localization (MBL) physics occurring at high energy. We generically find that any localized eigenstate is a very good approximation for an eigenstate of a distinct parent Hamiltonian, with an energy variance sigma(2)(P)(L) = < H-P(2)>(Psi 0) - < H-P >(2)(Psi 0) vanishing as a power law of system size L. This decay is microscopically related to a chain-breaking mechanism, also signaled by bottlenecks of vanishing entanglement entropy. A similar phenomenology is observed for both Anderson and MBL. In contrast, delocalized ergodic many-body eigenstates uniquely encode the Hamiltonian in the sense that sigma(2)(P)(L) remains finite at the thermodynamic limit, i.e., L -> +infinity. As a direct consequence, the ergodic-MBL transition can be very well captured from the scaling of sigma(2)(P) (L).