Abstract
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.
Highlights
Given a local Hamiltonian, what information about the system is encoded in a single eigenstate? If the eigenstate happens to be a ground state of the Hamiltonian, a tremendous amount of progress can be made on this question for Lorentz invariant systems [1,2], especially conformal field theories (CFTs) [3,4,5,6], and for topological phases [7,8,9]
The same intuition is tied to the fact that the ground state entanglement satisfies a “boundary law” of entanglement entropy [11,12]; that is, the von Neumann entanglement entropy S1 1⁄4 −trAðρA logðρAÞÞ of the ground state corresponding to a subsystem A scales with the size of the boundary of subsystem A
While we find in this case that Eq (1) fails to hold for some operators, our results are consistent with the possibility that eigenstate thermalization hypothesis (ETH) holds for all operators, as long as a microcanonical ensemble is used for comparison. [We will return to this case shortly, after first considering the implications of Eq (1).] The satisfaction of Eq (1) for all operators in a subsystem A is equivalent to the statement that the reduced density matrix ρAðjψiβÞ 1⁄4 trAjψiββhψj corresponding to an eigenstate jψiβ is given by [23]
Summary
Given a local Hamiltonian, what information about the system is encoded in a single eigenstate? If the eigenstate happens to be a ground state of the Hamiltonian, a tremendous amount of progress can be made on this question for Lorentz invariant systems [1,2], especially conformal field theories (CFTs) [3,4,5,6], and for topological phases [7,8,9]. We will argue that ETH allows one to calculate thermodynamical quantities, as well as correlators, at all energy densities (or temperatures) using only a single eigenstate. We will demonstrate this explicitly by studying a quantum 1D model numerically. We introduce a third version of ETH that we explore in this paper, given by EhψjOjψiE 1⁄4 tr 1⁄2OρmcðEÞ: ð4Þ If this equation holds for all operators O within subregion A, it implies the equivalence ρAðjψ iEÞ 1⁄4 ρA;mcðEÞ; ð2cÞ where ρA;mcðEÞ ≡ trA 1⁄2ρmcðEÞ is the microcanonical ensemble restricted to subregion A.
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