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Aperiodic spin chains at the boundary of hyperbolic tilings

by Pablo Basteiro, Rathindra Nath Das, Giuseppe Di Giulio, Johanna Erdmenger

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Submission summary

Authors (as registered SciPost users): Pablo Basteiro · Giuseppe Di Giulio · Johanna Erdmenger
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-11-13
Date submitted: 2023-08-23 15:09
Submitted by: Di Giulio, Giuseppe
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


In view of making progress towards establishing a holographic duality for theories defined on a discrete tiling of the hyperbolic plane, we consider a recently proposed boundary spin chain Hamiltonian with aperiodic couplings that are chosen such as to reflect the inflation rule, i.e. the construction principle, of the bulk tiling. As a remnant of conformal symmetry, the spin degrees of freedom are arranged in multiplets of the dihedral group under which the bulk lattice is invariant. For the boundary Hamiltonian, we use strong-disorder RG techniques and evaluate correlation functions, the entanglement entropy and mutual information for the case that the ground state is in an aperiodic singlet phase. We find that two-point functions decay as a power-law with exponent equal to one. Furthermore, we consider the case that the spin variables transform in the fundamental representation of $SO(N)$, leading to a gapless system, and find that the effective central charge obtained from the entanglement entropy scales as $\ln N$, reflecting the number of local degrees of freedom. We also determine the dependence of this central charge on the parameters specifying the bulk tiling. Moreover, we obtain an analytical expression for the mutual information, according to which there is no phase transition at any finite value of the distance between the two intervals involved.

Author comments upon resubmission

Dear editor and dear referees,

we are grateful for the insightful comments based on a meticulous analysis of our manuscript. In order to address any queries about novelty as mentioned under `Weaknesses' in both reports, we would like to summarize and highlight the new results of this present paper in comparison to our previous paper Ref.[25]. These are as follows:

1) In the present paper we consider the large N limit of N -component spins as compared to spin 1/2 in our previous paper. For these we show that the effective central charge depends logarithmically on N , allowing for a regime where this effective central charge can be made large by taking N → ∞ . This is a further step towards usual holographic setups.

2) In the present paper we calculate the mutual information additionally to the entanglement entropy. We schematically show how to perform this analysis for generic aperiodic spin chains with aperiodic singlet phases as their ground states, and explicitly carry out computations in two example models. We find fully analytic results which exhibit an interesting behavior as a function of the distance which can be suitably explained in terms of both the properties of the aperiodic singlet phase ground state and the geometric properties of the corresponding tensor network.

Further replies to specific comments have been communicated to the referees individually.

We have revised our manuscript and replied to the referees' observations. A detailed account of the changes is given below, ordered as they appear in the manuscript. In the light of these clarifications and additions that address the referees' queries, we now consider the manuscript ready for publication in SciPost.

List of changes

List of changes:

- On page 13-14, we have added an explanatory paragraph on the factorization properties of the aperiodic singlet phase and how these allow for the computation of higher-point correlation functions.
- On page 26, we have included a paragraph elaborating on the properties of the homogeneous and disorder-induced fixed points with respect to the disorder parameter. We have also added a referral to Ref.~[25] for more details.
- On page 27, we have added a paragraph on the interpretation of the bulk tensor network as a an explicit geometric description of the aperiodic singlet phase on the boundary, together with an explanation of the behavior found for the mutual information in terms of the tensor network geometry.

Published as SciPost Phys. 15, 218 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-11-7 (Invited Report)


The authors have sufficiently addressed the points raised in my previous report. I recommend publication in the present form.

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Anonymous Report 1 on 2023-10-16 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2212.11292v2, delivered 2023-10-16, doi: 10.21468/SciPost.Report.7954


Reading the changes the authors have made to the manuscript, I agree that several issues, including the factorisation of correlation functions and the spectrum of the theory, have been clarified.

As for the connection between the bulk tensor network geometry and the entanglement structure, I also agree with the authors clarification in the newly added paragraph, that there is some connection, except that when the bulk theory is not really in a semi-classical limit, it would be unlikely that geometrical objects such as geodesics would directly be related to either correlation functions or entanglement entropy. That they scale in the right way is perhaps as best as what one could hope for generally. In this light, I think that the paper has done what is possible in the current context and given the scarcity of exact results in describing quantum ground states using tensor networks, the paper generalising previous results to include more generic spins beyond spin 1/2 while providing many analytic results for these general cases would be of value to future studies of tensor networks in general.
I would therefore recommend the paper for publication in sci-post.

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