From eigenstate to Hamiltonian: Prospects for ergodicity and localization

Abstract

This work 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 Ψ0, an eigenstate of a given local Hamiltonian H0, we ask whether or not there exists another parent Hamiltonian HP for Ψ0, with the same local form as H0. 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 manybody 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 2 P (L) = H 2 P Ψ 0 − HP 2 Ψ 0 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 2 P (L) remains finite at the thermodynamic limit, i.e., L → +∞. As a direct consequence, the ergodic-MBL transition can be very well captured from the scaling of σ 2 P (L).

Chain breaking and Kosterlitz-Thouless scaling at the many-body localization transition in the random field Heisenberg spin chain

“…Building on previous ideas [89], we are now ready to understand the spin freezing mechanism. If we approximate the Anderson orbitals by simple exponential functions…”

When can a local Hamiltonian be recovered from a steady state?

“…Many algorithms have been proposed to recover the Hamiltonian by making quantum measurements on its eigenstate [1,[14][15][16][17][18], dynamics [19][20][21][22] and quantum quench process [23]. Several algorithms have been employed to successfully recover some local Hamiltonians with a specific pattern [24][25][26].…”

Optimal parent Hamiltonians for time-dependent states

“…In the time-independent case, the inverse problem consists in reconstructing a stationary local Hamiltonian starting from a stationary state, for example an eigenstate. Following the pioneering work by Chertkov and Clark 5 , several works have been devoted to this question [6][7][8][9][10] . A key role in the determi-nation of the parent Hamiltonian is played by the so-called Quantum Covariance Matrix (QCM) of the state.…”