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arXiv:1712.01999 · Physical Review Letters

Shiraishi and Mori Reply

Naoto Shiraishi, Takashi Mori

Published Jul 17, 2018• 19 citations
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Abstract

This paper is a Reply paper to the Comment paper by Mondaini et.al. [arXiv:1711.06279]. We first distinguish the diagonal and the off-diagonal eigenstate thermalization hypothesis (ETH) in each sector and in the whole Hilbert space, and then clarify their properties and their roles in thermalization. We argue that our formulation of the ETH in our letter is a standard and natural one in the context of thermalization, and that our results are qualitatively new and unexpected.

DOI
10.1103/physrevlett.121.038902

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This paper is a Reply paper to the Comment paper by Mondaini et.al. [arXiv:1711.06279]. We first distinguish the diagonal and the off-diagonal eigenstate thermalization hypothesis (ETH) in each sector and in the whole Hilbert space, and then clarify their properties and their roles in thermalization. We argue that our formulation of the ETH in our letter is a standard and natural one in the context of thermalization, and that our results are qualitatively new and unexpected.
In a recent Letter [PhysRevLett.119.030601 (2017), arXiv:1702.08227], Shiraishi and Mori claim to provide a general method for constructing local Hamiltonians that do not exhibit eigenstate thermalization. We argue that the claim is based on a misunderstanding of the eigenstate thermalization hypothesis (ETH). More specifically, on the assumption that ETH is valid for the entire Hamiltonian matrix instead of each symmetry sector independently. We discuss what happens if one mixes symmetry sectors in the two-dimensional transverse field Ising model.
The so-called eigenstate thermalization hypothesis (ETH), which has been tested in various many-body models by numerical simulations, supplies a way of understanding eventual thermalization and is believed to be important for understanding processes of thermalization. Two functions play important roles in the application of ETH, one for averaged diagonal elements and the other for the variance of off-diagonal elements of an observable addressed by ETH on the energy basis. For the former function, a semiclassical expression is known of the zeroth order of ℏ, while, little is known analytically for the latter. In this paper, a semiclassical expression is derived for the former diagonal function, which includes higher-order contributions of ℏ. And, another semiclassical expression is derived for the scale of the height of a central platform of the latter off-diagonal function. These analytical results are checked numerically in the Lipkin-Meshkov-Glick model. Possible relevance of the latter result for future investigation is discussed.
Quantum thermalization is well understood via the eigenstate thermalization hypothesis (ETH). The general form of ETH, describing all the relevant correlations of matrix elements, may be derived on the basis of a "typicality" argument of invariance with respect to local rotations involving nearby energy levels. In this Letter, we uncover the close relation between this perspective on ETH and free probability theory, as applied to a thermal ensemble or an energy shell. This mathematical framework allows one to reduce in a straightforward way higher-order correlation functions to a decomposition given by minimal blocks, identified as free cumulants, for which we give an explicit formula. This perspective naturally incorporates the consistency property that local functions of ETH operators also satisfy ETH. The present results uncover a direct connection between the eigenstate thermalization hypothesis and the structure of free probability, widening considerably the latter's scope and highlighting its relevance to quantum thermalization.