TY - JOUR

T1 - Fun with replicas

T2 - tripartitions in tensor networks and gravity

AU - Penington, Geoff

AU - Walter, Michael

AU - Witteveen, Freek

N1 - Publisher Copyright: © 2023, The Author(s).

PY - 2023

Y1 - 2023

N2 - We analyse a simple correlation measure for tripartite pure states that we call G(A : B : C). The quantity is symmetric with respect to the subsystems A, B, C, invariant under local unitaries, and is bounded from above by log dA dB. For random tensor network states, we prove that G(A : B : C) is equal to the size of the minimal tripartition of the tensor network, i.e., the logarithmic bond dimension of the smallest cut that partitions the network into three components with A, B, and C. We argue that for holographic states with a fixed spatial geometry, G(A : B : C) is similarly computed by the minimal area tripartition. For general holographic states, G(A : B : C) is determined by the minimal area tripartition in a backreacted geometry, but a smoothed version is equal to the minimal tripartition in an unbackreacted geometry at leading order. We briefly discuss a natural family of quantities Gn(A : B : C) for integer n ≥ 2 that generalize G = G 2. In holography, the computation of Gn(A : B : C) for n > 2 spontaneously breaks part of a ℤ n × ℤ n replica symmetry. This prevents any naive application of the Lewkowycz-Maldacena trick in a hypothetical analytic continuation to n = 1.

AB - We analyse a simple correlation measure for tripartite pure states that we call G(A : B : C). The quantity is symmetric with respect to the subsystems A, B, C, invariant under local unitaries, and is bounded from above by log dA dB. For random tensor network states, we prove that G(A : B : C) is equal to the size of the minimal tripartition of the tensor network, i.e., the logarithmic bond dimension of the smallest cut that partitions the network into three components with A, B, and C. We argue that for holographic states with a fixed spatial geometry, G(A : B : C) is similarly computed by the minimal area tripartition. For general holographic states, G(A : B : C) is determined by the minimal area tripartition in a backreacted geometry, but a smoothed version is equal to the minimal tripartition in an unbackreacted geometry at leading order. We briefly discuss a natural family of quantities Gn(A : B : C) for integer n ≥ 2 that generalize G = G 2. In holography, the computation of Gn(A : B : C) for n > 2 spontaneously breaks part of a ℤ n × ℤ n replica symmetry. This prevents any naive application of the Lewkowycz-Maldacena trick in a hypothetical analytic continuation to n = 1.

KW - AdS-CFT Correspondence

KW - Black Holes in String Theory

UR - http://www.scopus.com/inward/record.url?scp=85158032006&partnerID=8YFLogxK

U2 - 10.1007/JHEP05(2023)008

DO - 10.1007/JHEP05(2023)008

M3 - Journal article

AN - SCOPUS:85158032006

VL - 2023

JO - Journal of High Energy Physics (Online)

JF - Journal of High Energy Physics (Online)

SN - 1126-6708

IS - 5

M1 - 8

ER -