# From eigenstate to Hamiltonian: Prospects for ergodicity and localization

## TL;DR

This paper investigates whether localized eigenstates in disordered quantum systems can serve as approximate eigenstates for different Hamiltonians, revealing distinct scaling behaviors that differentiate localized from ergodic states and capturing the ergodic-MBL transition.

## Contribution

It extends inverse Hamiltonian problems to Anderson and many-body localization, showing localized states approximate eigenstates of different Hamiltonians with size-dependent variance decay.

## Key findings

- Localized eigenstates approximate other Hamiltonians with variance vanishing as a power law of system size.
- Delocalized ergodic states maintain finite variance, encoding the Hamiltonian uniquely.
- Scaling of variance effectively captures the ergodic-MBL transition.

## 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 $\Psi_0$, 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 $\Psi_0$, 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 $\sigma_\mathrm{P}^2(L)=\langle\mathcal{H}_\mathrm{P}^2\rangle_{\Psi_0}-\langle\mathcal{H}_\mathrm{P}\rangle_{\Psi_0}^2$ vanishing as a power law of system size $L$. This decay is microscopically related to a chain-breaking mechanism, also signalled 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 $\sigma_\mathrm{P}^2(L)$ remains finite at the thermodynamic limit, i.e., $L\to+\infty$. As a direct consequence, the ergodic-MBL transition can be very well captured from the scaling of $\sigma_\mathrm{P}^2(L)$.