From eigenstate to Hamiltonian: Prospects for ergodicity and localization

Maxime Dupont; Nicolas Macé; Nicolas Laflorencie

doi:10.1103/PhysRevB.100.134201

paper

From eigenstate to Hamiltonian: Prospects for ergodicity and localization

arXiv:1907.12124 · doi:10.1103/PhysRevB.100.134201

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 $Ψ_{0}$ , an eigenstate of a given local Hamiltonian $H_{0}$ , we ask whether or not there exists another parent Hamiltonian $H_{P}$ for $Ψ_{0}$ , 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 $σ_{P}^{2} (L) = ⟨ H_{P}^{2} ⟩_{Ψ_{0}} - ⟨ H_{P} ⟩_{Ψ_{0}}^{2}$ 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 $σ_{P}^{2} (L)$ 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 $σ_{P}^{2} (L)$ .

arXiv comment14 pages, 15 figures

From eigenstate to Hamiltonian: Prospects for ergodicity and localization

Abstract

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