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A<sc>bstract</sc> The eigenstate thermalization hypothesis (ETH) is the leading conjecture for the emergence of statistical mechanics in generic isolated quantum systems and is formulated in terms of the matrix elements of operators. An analog known as the ergodic bipartition (EB) describes entanglement and locality and is formulated in terms of the components of eigenstates. In this paper, we significantly generalize the EB and unify it with the ETH, extending the EB to study higher correlations and systems out of equilibrium. Our main result is a diagrammatic formalism that computes arbitrary correlations between eigenstates and operators based on a recently uncovered connection between the ETH and free probability theory. We refer to the connected components of our diagrams as generalized free cumulants. We apply our formalism in several ways. First, we focus on chaotic eigenstates and establish the so-called subsystem ETH and the Page curve as consequences of our construction. We also improve known calculations for thermal reduced density matrices and comment on an inherently free probabilistic aspect of the replica approach to entanglement entropy previously noticed in a calculation for the Page curve of an evaporating black hole. Next, we turn to chaotic quantum dynamics and demonstrate the ETH as a sufficient mechanism for thermalization, in general. In particular, we show that reduced density matrices relax to their equilibrium form and that systems obey the Page curve at late times. We also demonstrate that the different phases of entanglement growth are encoded in higher correlations of the EB. Lastly, we examine the chaotic structure of eigenstates and operators together and reveal previously overlooked correlations between them. Crucially, these correlations encode butterfly velocities, a well-known dynamical property of interacting quantum systems.
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Zero modes of local operators in 2d CFT on a cylinder
A bstract Studies of Eigenstate Thermalization Hypothesis (ETH) in two-dimensional CFTs call for calculation of the expectation values of local operators in highly excited energy eigenstates. This can be done efficiently by representing zero modes of these operators in terms of the Virasoro algebra generators. In this paper we present a pedagogical introduction explaining how this calculation can be performed analytically or using computer algebra. We illustrate the computation of zero modes by a number of examples and list explicit expressions for all local operators from the vacuum family with the dimension of less or equal than eight. Finally, we derive an explicit expression for the quantum KdV generator Q 7 in terms of the Virasoro algebra generators. The obtained results can be used for quantitative studies of ETH at finite value of central charge.
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- Award ID(s):
- 1720374
- PAR ID:
- 10282605
- Date Published:
- Journal Name:
- Journal of High Energy Physics
- Volume:
- 2020
- Issue:
- 7
- ISSN:
- 1029-8479
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/JHEP07(2020)172
