Enhanced corrections near holographic entanglement transitions: a chaotic case study
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Springer
Received: July 23, 2020 Accepted: September 22, 2020 Published: November 4, 2020
Xi Donga and Huajia Wangb,c a
Department of Physics, University of California, Santa Barbara, CA 93106, U.S.A. b Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, U.S.A. c Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
E-mail: [email protected], [email protected] 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 holographically. 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. Keywords: AdS-CFT Correspondence, Black Holes, Nonperturbative Effects, Random Systems ArXiv ePrint: 2006.10051
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)007
JHEP11(2020)007
Enhanced corrections near holographic entanglement transitions: a chaotic case study
Contents 1 Introduction
1
2 Chaotic high energy eigenstates 2.1 Away from f = 1/2 2.2 Transition point f = 1/2
3 6 7 11 11 13 16 20
4 Discussion
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A Disorder Wick contraction
28
1
Introduction
Entanglement and Renyi entropies of subsystems are important quantities that encode key properties of a quantum system as a whole. In suitable limits where the number of relevant degrees of freedom becomes large, entanglement and Renyi entropies become analytically tractable and have been studied in (at least) two classes of examples. The first involves energy eigenstates in the thermodynamic limit of a chaotic system [1–3]. The second involves holographic states whose entanglement entropy in the large-N limit is given by the Ryu-Takayanagi (RT) or Hubeny-Rangamani-Takayanagi (HRT) formula in terms of the area of a bulk extremal surface [4–6], and whose Renyi entropies are determined from similar areas of appropriate cosmic branes [7].1 As the size of the subsystem is varied, entanglement and Renyi entropies can experience phase transitions in the limit of a large number of degrees of freedom, signifying major rearrangements of the entanglement structure.
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