Abstract
We study the behavior of Rényi entropies for pure states from standard assumptions about chaos in the high-energy spectrum of the Hamiltonian of a many-body quantum system. We compute the exact long-time averages of Rényi entropies and show that the quantum noise around these values is exponentially suppressed in the microcanonical entropy. For delocalized states over the microcanonical band, the long-time average approximately reproduces the equilibration proposal of H. Liu and S. Vardhan, with extra structure arising at the order of non-planar permutations. We analyze the equilibrium approximation for AdS/CFT systems describing black holes in equilibrium in a box. We extend our analysis to the situation of an evaporating black hole, and comment on the possible gravitational description of the new terms in our approximation.
Highlights
JHEP08(2021)014 have some additional structure that enters at the level of non-planar diagrams of the previous approximation
We study the behavior of Rényi entropies for pure states from standard assumptions about chaos in the high-energy spectrum of the Hamiltonian of a many-body quantum system
Since the replicas are already connected through the boundary conditions, in bulk terms this path integral will be dominated by an Euclidean black hole at inverse temperature 2β which will connect the two replicas.5. This connected term matches the contribution of the ‘replica wormhole’ in previous replica calculations in gravity. These observations lead to the hypothesis introduced in [5] that replica computations of the purity of the radiation using the gravitational path integral (TrR ρ2R)grav are effectively reproducing each term of the equilibrium approximation (5.7)
Summary
We will study the behavior of Rényi entropies for pure states under mild assumptions about chaos in a high-energy microcanonical band of the Hamiltonian of a many-body quantum system. The first sum exactly reproduces the terms in the equilibrium ansatz of [5], but again one has to consider the microscopic equilibration density matrix ρ = |ci|2 |Ei Ei| which has information about the initial state |Ψ of the system. In particular we can see that |π| > 1 implies that the new terms enter the long-time value at least at the order of permutations with non-planar diagrams. Note, that this hierarchy is less pronounced when |Ψ involves a small number of energy eigenstates, and in this case the new terms can become comparable to certain ‘planar’ permutations. For a single eigenstate, all the terms in (2.13) are of the same order of magnitude
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