Does a single eigenstate encode the full Hamiltonian

By J. Garrison, T. Grover. 2015

265 citations recorded by the source corpus.

The Eigenstate Thermalization Hypothesis (ETH) posits that the reduced density matrix for a subsystem corresponding to an excited eigenstate is "thermal." Here we expound on this hypothesis by asking: for which class of operators, local or non-local, is ETH satisfied? We show that this question is directly related to a seemingly unrelated question: is the Hamiltonian of a system encoded within a single eigenstate? We formulate a strong form of ETH where in the thermodynamic limit, the reduced density matrix of a subsystem corresponding to a pure, finite energy density eigenstate asymptotically becomes equal to the thermal reduced density matrix, as long as the subsystem size is much less than the total system size, irrespective of how large the subsystem is compared to any intrinsic length scale of the system. This allows one to access the properties of the underlying Hamiltonian at arbitrary energy densities/temperatures using just a $\textit{single}$ eigenstate. We provide support for our conjecture by performing an exact diagonalization study of a non-integrable 1D lattice quantum model with only energy conservation. In addition, we examine the case in which the subsystem size is a finite fraction of the total system size, and find that even in this case, a large class of operators continue to match their canonical expectation values. Specifically, the von Neumann entanglement entropy equals the thermal entropy as long as the subsystem is less than half the total system. We also study, both analytically and numerically, a particle number conserving model at infinite temperature which substantiates our conjectures.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4e486c7b0ef2e733b15ec255698d4494d6dd72c1

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In Atomic and Molecular Physics, and Optics

In Statistical and Nonlinear Physics

Citations in the corpus

Quench dynamics and relaxation in isolated integrable quantum spin chains (2016, background): “An example would be expectation values of the momentum distribution function, which is the Fourier transform of the single particle Green’s function [24, 158].”
Quantum thermalization through entanglement in an isolated many-body system (2016, methodology): “Our studies, and beautiful recent theoretical work [11, 12, 35], hint towards a microscopic origin for entropy maximization in a single quantum state, namely that induced by the entanglement we measure.”
The randomized measurement toolbox (2022, background): “Perhaps surprisingly, when H is sufficiently generic, low-weight Pauli observables with respect to ψ 0 contain enough information to reconstruct H up to an overall scale factor (and an unobservable energy shift) 90–92 .”
Quantum Entanglement in Neural Network States (2017, background): “Yet, as has been realized in a number of diﬀerent physical contexts, the entanglement entropy cannot capture the full entanglement structure of the system [96–101].”