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Towards Explicit Discrete Holography: Aperiodic Spin Chains from Hyperbolic Tilings
by Pablo Basteiro, Giuseppe Di Giulio, Johanna Erdmenger, Jonathan Karl, René Meyer, Zhuo-Yu Xian
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Pablo Basteiro · Giuseppe Di Giulio · Johanna Erdmenger · Rene Meyer |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2205.05693v2 (pdf) |
| Date accepted: | 2022-09-06 |
| Date submitted: | 2022-08-29 09:45 |
| Submitted by: | Di Giulio, Giuseppe |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We propose a new example of discrete holography that provides a new step towards establishing the AdS/CFT duality for discrete spaces. A class of boundary Hamiltonians is obtained in a natural way from regular tilings of the hyperbolic Poincar\'e disk, via an inflation rule that allows to construct the tiling using concentric layers of tiles. The models in this class are aperiodic spin chains, whose sequences of couplings are obtained from the bulk inflation rule. We explicitly choose the aperiodic XXZ spin chain with spin 1/2 degrees of freedom as an example. The properties of this model are studied by using strong disorder renormalization group techniques, which provide a tensor network construction for the ground state of this spin chain. This can be regarded as discrete bulk reconstruction. Moreover we compute the entanglement entropy in this setup in two different ways: a discretization of the Ryu-Takayanagi formula and a generalization of the standard computation for the boundary aperiodic Hamiltonian. For both approaches, a logarithmic growth of the entanglement entropy in the subsystem size is identified. The coefficients, i.e. the effective central charges, depend on the bulk discretization parameters in both cases, albeit in a different way.
Author comments upon resubmission
we are grateful for the useful and insightful comments which arose from a thorough and meticulous analysis of our manuscript.
We have revised our manuscript and replied to these observations. A detailed account of the changes is given below, ordered as they appear in the manuscript.
List of changes
- We have corrected a typo in equations (7) and (8).
- On page 11, we have added a clarification about the difference between the classification of the types of vertices in $\{p,q\}$ tilings with $p\neq 3$ and the one in $\{3,q\}$ tilings.
- We have added an example of inflation rule between Eq.(12) and Eq.(13)
- To further motivate the choice of binary inflation rules in our setup, we have added two comments, one in the Introduction on page 4 and the other at the end of Sec.2.2 on page 12.
- We have included two comments below Eq.(37) and Eq.(39).
- We have added a comment clarifying the notation between Eq.(43) and Eq.(44).
- We have corrected a typo in Eq.(55).
- We have expanded the discussion at the end of Sec.5.4, adding a schematic computation.
- We have added comments on page 51.
- References added.
Published as SciPost Phys. 13, 103 (2022)