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arXiv:1804.07924 · Physical review. D/Physical review. D.

Probing thermality beyond the diagonal

Enrico M. Brehm, Diptarka Das, Shouvik Datta

Published Dec 28, 2018• 72 citations
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Abstract

We investigate the off-diagonal sector of eigenstate thermalization using both local and nonlocal probes in two-dimensional conformal field theories. A novel analysis of the asymptotics of operator product expansion coefficients via the modular bootstrap is performed to extract the behavior of the off-diagonal matrix elements. We also probe this sector using semiclassical heavy-light Virasoro blocks. The results demonstrate signatures of thermality and confirm the entropic suppression of the off-diagonal elements as necessitated by the eigenstate thermalization hypothesis.

DOI
10.1103/physrevd.98.126015

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The eigenstate thermalization hypothesis (ETH) provides a way to understand how an isolated quantum mechanical system can be approximated by a thermal density matrix. We find a class of operators in (1+1)-dimensional conformal field theories, consisting of quasiprimaries of the identity module, which satisfy the hypothesis only at the leading order in large central charge. In the context of subsystem ETH, this plays a role in the deviation of the reduced density matrices, corresponding to a finite energy density eigenstate from its hypothesized thermal approximation. The universal deviation in terms of the square of the trace-square distance goes as the eighth power of the subsystem fraction and is suppressed by powers of inverse central charge (c). Furthermore, the nonuniversal deviations from subsystem ETH are found to be proportional to the heavy-light-heavy structure constants which are typically exponentially suppressed in sqrt[h/c], where h is the conformal scaling dimension of the finite energy density state. We also examine the effects of the leading finite-size corrections.
We study the matrix elements of few-body observables, focusing on the off-diagonal ones, in the eigenstates of the two-dimensional transverse field Ising model. By resolving all symmetries, we relate the onset of quantum chaos to the structure of the matrix elements. In particular, we show that a general result of the theory of random matrices, namely, the value 2 of the ratio of variances (diagonal to off-diagonal) of the matrix elements of Hermitian operators, occurs in the quantum chaotic regime. Furthermore, we explore the behavior of the off-diagonal matrix elements of observables as a function of the eigenstate energy differences and show that it is in accordance with the eigenstate thermalization hypothesis ansatz.