Nonarchimedean Holographic Entropy from Networks of Perfect Tensors

Heydeman, Matthew; Marcolli, Matilde; Parikh, Sarthak; Saberi, Ingmar

doi:10.48550/arxiv.1812.04057

Cited by 10 publications

(15 citation statements)

References 86 publications

(119 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In order to understand entanglement wedge reconstruction, many toy models have been considered. A particularly successful class of toy models are the ones based on tensor networks [10,13,19], including the HaPPY code. The HaPPY code is without a doubt the most celebrated holographic quantum error correcting code based on tensor networks.…”

Section: Introductionmentioning

confidence: 99%

The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics

Gesteau¹,

Kang²

2020

Preprint

View full text Add to dashboard Cite

We construct an infinite-dimensional analog of the HaPPY code as a growing series of stabilizer codes defined respective to their Hilbert spaces. The Hilbert spaces are related by isometric maps, which we define explicitly. We construct a Hamiltonian that is compatible with the infinite-dimensional HaPPY code and further study the stabilizer of our code, which has an inherent fractal structure. We use this result to study the dynamics of the code and map a nontrivial bulk Hamiltonian to the boundary. We find that the image of the mapping is scale invariant, but does not create any long-range entanglement in the boundary, therefore failing to reproduce the features of a CFT. This result shows the limits of the HaPPY code as a model of the AdS/CFT correspondence, but also hints that the relevance of quantum error correction in quantum gravity may not be limited to the CFT context.

show abstract

Section: Introductionmentioning

confidence: 99%

The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics

Gesteau¹,

Kang²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Tensor networks were also interpreted in[20] as the wavefunction of a p-adic CFT, and used to study entanglement entropy. The behavior of this entropy remains puzzling, and it is not clear whether the tree structure can recover the expected entanglement entropy of the dual theory.3 It was already noted in[1] that the Witten diagrams naturally contain bulk-boundary propagators that treat the Bruhat-Tits tree as space-time, rather than as a time slice.…”

mentioning

confidence: 99%

p-adic CFT is a holographic tensor network

Hung

Melby-Thompson

2019

J. High Energ. Phys.

View full text Add to dashboard Cite

The p-adic AdS/CFT correspondence relates a CFT living on the p-adic numbers to a system living on the Bruhat-Tits tree. Modifying our earlier proposal [1] for a tensor network realization of p-adic AdS/CFT, we prove that the path integral of a p-adic CFT is equivalent to a tensor network on the Bruhat-Tits tree, in the sense that the tensor network reproduces all correlation functions of the p-adic CFT. Our rules give an explicit tensor network for any p-adic CFT (as axiomatized by Melzer), and can be applied not only to the p-adic plane, but also to compute any correlation functions on higher genus p-adic curves. Finally, we apply them to define and study RG flows in p-adic CFTs, establishing in particular that any IR fixed point is itself a p-adic CFT.

show abstract

“…It is also interesting to ask if more general objects than hyperbolic buildings are welladapted to the construction of holographic codes. In particular, it would be interesting to understand if quotients of our buildings can be taken in order to describe nontrivial bulk topologies in the spirit of the BTZ-like topologies constructed in [19]. We expect that the setup of latin square designs [7] might be helpful to think about this kind of problem.…”

Section: Discussionmentioning

confidence: 99%

“…Holographic tensor networks have an interesting geometric structure, which ranges from tesselations of the hyperbolic plane [26] or higher-dimensional spaces [24] to p-adic spaces [19] (see [6] for a complimentary perspective). Although it is often briefly mentioned that hyperbolic tesselations have to do with Coxeter systems [24], the interplay between holographic tensor networks and hyperbolic geometry has been left largely unexplored.…”

Section: Introductionmentioning

confidence: 99%

Holographic tensor networks from hyperbolic buildings

Gesteau¹,

Marcolli²,

Parikh³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We introduce a unifying framework for the construction of holographic tensor networks, based on the theory of hyperbolic buildings. The underlying dualities relate a bulk space to a boundary which can be homeomorphic to a sphere, but also to more general spaces like a Menger sponge type fractal. In this general setting, we give a precise construction of a large family of bulk regions that satisfy complementary recovery. For these regions, our networks obey a Ryu-Takayanagi formula. The areas of Ryu-Takayanagi surfaces are controlled by the Hausdorff dimension of the boundary, and consistently generalize the behavior of holographic entanglement entropy in integer dimensions to the non-integer case. Our construction recovers HaPPY-like codes in all dimensions, and generalizes the geometry of Bruhat-Tits trees. It also provides examples of infinite-dimensional nets of holographic conditional expectations, and opens a path towards the study of conformal field theory and holography on fractal spaces.

show abstract

Nonarchimedean Holographic Entropy from Networks of Perfect Tensors

Cited by 10 publications

References 86 publications

The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics

The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics

p-adic CFT is a holographic tensor network

Holographic tensor networks from hyperbolic buildings

Contact Info

Product

Resources

About