From eigenstate to Hamiltonian: Prospects for ergodicity and localization

Abstract

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 ${\mathrm{\ensuremath{\Psi}}}_{0}$, an eigenstate of a given local Hamiltonian ${\mathcal{H}}_{0}$, we ask whether or not there exists another parent Hamiltonian ${\mathcal{H}}_{\mathrm{P}}$ for ${\mathrm{\ensuremath{\Psi}}}_{0}$, with the same local form as ${\mathcal{H}}_{0}$. 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 ${\ensuremath{\sigma}}_{\mathrm{P}}^{2}(L)={\ensuremath{\langle}{\mathcal{H}}_{\mathrm{P}}^{2}\ensuremath{\rangle}}_{{\mathrm{\ensuremath{\Psi}}}_{0}}\ensuremath{-}{\ensuremath{\langle}{\mathcal{H}}_{\mathrm{P}}\ensuremath{\rangle}}_{{\mathrm{\ensuremath{\Psi}}}_{0}}^{2}$ 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 ${\ensuremath{\sigma}}_{\mathrm{P}}^{2}(L)$ remains finite at the thermodynamic limit, i.e., $L\ensuremath{\rightarrow}+\ensuremath{\infty}$. As a direct consequence, the ergodic-MBL transition can be very well captured from the scaling of ${\ensuremath{\sigma}}_{\mathrm{P}}^{2}(L)$.