Holographic duality from random tensor networks

By Patrick M. Hayden, Sepehr Nezami, Xiao-Liang Qi, Nathaniel Thomas, Michael Walter, Zhao Yang. 2016

597 citations recorded by the source corpus.

A bstractTensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0b1040c7776bd91d50026bbde2486aad85ba0aa5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. A. Nahum
2. Matthew Fisher
3. Brian Swingle
4. V. Khemani
5. D. Stanford
6. D. Harlow
7. S. Vijay
8. R. Myers

In Nuclear and High Energy Physics

1. D. Stanford
2. R. Myers
3. D. Harlow
4. J. Maldacena
5. S. Shenker
6. S. Raju
7. A. Nahum
8. B. Yoshida

In Statistical and Nonlinear Physics

1. D. Stanford
2. R. Myers
3. D. Harlow
4. A. Nahum
5. S. Shenker
6. S. Raju
7. T. Takayanagi
8. J. Maldacena

Citations in the corpus

• Measurement and Entanglement Phase Transitions in All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory (2020, methodology): “costs for non-uniform boundary conditions [33–35] (see Sec.”
• Random Quantum Circuits (2022, background): “In the general case with N layers each of U(t) and U(t)∗, the “spin” σ labelling different patterns of pairing becomes an element of the permutation group, σ ∈ SN (132, 134, 113).”
• Statistical mechanics of quantum error correcting codes (2020, background): “[25–28] for random Haar unitary circuits without measurements, where it was first pointed out that the entanglement entropy can be viewed as free energies of “entanglement domain walls”.”
• Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, methodology): “More recently, tensor networks have been used to construct toy models of holographic principles in hyperbolic geometries (Hayden et al., 2016; Pastawski et al., 2015; Swingle, 2012).”
• TASI Lectures on the Emergence of Bulk Physics in AdS/CFT (2017, background): “There are many important references for this material, including [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].”
• Quantum extremal islands made easy. Part II. Black holes on the brane (2020, background): “Yet another direction to take would be to consider our setup from the perspective of tensor networks and error correction codes [135, 140–142].”
• Entanglement Membrane in Chaotic Many-Body Systems (2019, methodology): “The approach could also be adapted to study tensor networks that do not involve a time direction, for example in the context of the AdS-CFT correspondence where random tensor networks have been used [42, 43].”
• Emergent Gravity and the Dark Universe (2016, methodology): “For this one makes use of the fact that the bulk states are maximally entangled with the boundary states [20, 21].”
• Entanglement of purification through holographic duality (2017, background): “Let us regard a time slice of AdS as a tensor network which describes a quantum state |ψ〉CFT by following [4, 7, 8, 30].”
• Replica wormholes and the black hole interior (2019, background): “how that the non-perturbative corrections to the Ryu-Takayanagi formula from additional extremal surfaces are identical in a) \xed-area" states in AdS/CFT [60, 61] and b) random tensor networks [89]. This is consistent with the intuition from [60, 61] that xed-area states are analogous to tenso”