Abstract
The eigenstate thermalization hypothesis (ETH) explains how closed unitary quantum systems can exhibit thermal behavior in pure states. In this work we examine a recently proposed microscopic model of a black hole in AdS2, the so-called Sachdev-Ye-Kitaev (SYK) model. We show that this model satisfies the eigenstate thermalization hypothesis by solving the system in exact diagonalization. Using these results we also study the behavior, in eigenstates, of various measures of thermalization and scrambling of information. We establish that two-point functions in finite-energy eigenstates approximate closely their thermal counterparts and that information is scrambled in individual eigenstates. We study both the eigenstates of a single random realization of the model, as well as the model obtained after averaging of the random disordered couplings. We use our results to comment on the implications for thermal states of a putative dual theory, i.e. the AdS2 black hole.
Highlights
We introduce the model to be examined, discuss its properties, and introduce some pertinent notions of many-body thermalization necessary to follow the remainder of the work
First and foremost we establish, by exactly diagonalizing the complex SYK Hamiltonian for up to 17 sites, that expectation values of non-extensive — that is those involving a few sites only — operators are to a very good approximation thermal. In particular their matrix elements take on the expected form encapsulated in the eigenstate thermalization hypothesis
By studying off-diagonal matrix elements of non-extensive operators we establish that the SYK model behaves like a random-matrix theory (RMT) for a certain range of energies, but more generally deviates from such RMT behavior
Summary
First and foremost we establish, by exactly diagonalizing the complex SYK Hamiltonian for up to 17 sites, that expectation values of non-extensive — that is those involving a few sites only — operators are to a very good approximation thermal In particular their matrix elements take on the expected form encapsulated in the eigenstate thermalization hypothesis. To the extent that we can extrapolate these results to large values of N this suggests that individual eigenstates can in some sense be considered to be dual to the black-hole geometry in the putative dual, we make no statement about its interior (see discussion section) This motivates us to consider measures of scrambling in eigenstates, which we find to behave in accordance with expectations from combining known results in the canonical ensemble with our results on eigenstate thermalization. We expect that there is a large-N eigenstate equivalent of the maximal scrambling exponent satisfied by the SYK model, as detailed in (2.11) below
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