Abstract
Recent work found an enhanced correction to the entanglement entropy of a subsystem in a chaotic energy eigenstate. The enhanced correction appears near a phase transition in the entanglement entropy that happens when the subsystem size is half of the entire system size. Here we study the appearance of such enhanced corrections holo-graphically. We show explicitly how to find these corrections in the example of chaotic eigenstates by summing over contributions of all bulk saddle point solutions, including those that break the replica symmetry. With the help of an emergent rotational symmetry, the sum over all saddle points is written in terms of an effective action for cosmic branes. The resulting Renyi and entanglement entropies are then naturally organized in a basis of fixed-area states and can be evaluated directly, showing an enhanced correction near holographic entanglement transitions. We comment on several intriguing features of our tractable example and discuss the implications for finding a convincing derivation of the enhanced corrections in other, more general holographic examples.
Highlights
The primary goal of this paper is to study the appearance of such enhanced corrections in holographic theories near entanglement transitions where two competing HRT surfaces have about the same area
We show explicitly how to find these corrections in the example of chaotic eigenstates by summing over contributions of all bulk saddle point solutions, including those that break the replica symmetry
We studied the details of entanglement transition in subsystem Renyi entropy SnA in the context of high energy eigenstates
Summary
We begin by studying the enhanced correction to the entanglement of subsystem A in chaotic high energy eigenstates, as in [10]. The difference between microstate and micro-canonical ensemble Renyi entropy can be computed via the saddle point approximation by: This shows that for smaller-than-half subsystems, the microstate and micro-canonical ensemble agree in terms of Renyi entropy up to corrections suppressed exponentially. We shall instead focus on Sndom, it is a good proxy for the microstate Renyi entropy Sn. For generic n away from n = 1, the saddle point E2 for F2 in (2.16) is smaller than E/2:. The two symmetric saddles (E2, E − E2) of Fdom in the full range 0 < E < E collide as n → 1 When this happens, (2.36) is no longer a good approximation, i.e. dominant contributions to Sndom do not consist of two independent full Gaussian integrals.
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