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Rainbow chains and numerical renormalisation group for accurate chiral conformal spectra
by Attila Szabó
Submission summary
| Authors (as registered SciPost users): | Attila Szabó |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2412.09685v4 (pdf) |
| Code repository: | https://github.com/attila-i-szabo/rainbow-chain-NRG/tree/v1.0 |
| Date accepted: | Aug. 25, 2025 |
| Date submitted: | Aug. 7, 2025, 1:21 p.m. |
| Submitted by: | Attila Szabó |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Based on the relationship between reduced and thermal density matrices in conformal field theory (CFT), we show that the entanglement spectrum of a conformal critical chain with exponentially decaying terms consists of conformal towers of the associated chiral CFT, with only weak finite-size effects. Through free-fermion and interacting examples, we show that these entanglement spectra present a reliable method to extract detailed CFT spectra from single wave functions without access to the parent Hamiltonian. We complement our method with a Wilsonian numerical renormalisation group algorithm for solving interacting, exponentially decaying chain Hamiltonians.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Please find attached the revised version of my paper "Rainbow chains and numerical renormalisation group for accurate chiral conformal spectra". I wish to thank the referees for their positive assessment of the paper and their constructive comments, which have greatly improved the presentation of this version. I believe that this new version is now well-suited for publication in SciPost.
I have replied in detail to all referee comments directly under the reports.
With best wishes,
Attila Szabó
List of changes
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A new discussion for the relationship between inhomogeneous critical spin chains and non-uniform discretisation of CFTs, based on Ref. [25], is presented in Sec. 2, under the heading "Inhomogeneous spin chains"
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Inhomogeneous spin chains are presented in terms of a scaling function f(x) that can be inserted the same way into the different Hamiltonians considered in the paper.
explicit formulas for f(x) in the conformal chain, rainbow chain, and conformal ring geometries are given as Eqs. (7), (8), (12), respectively
the Hamiltonians in Eqs. (14), (15), and (24) are written explicitly in terms of f(x) such that the uniform critical chain corresponds to f(x) = 1
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Fig. 2 has become unnecessary and has been removed.
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Eq. (22) is changed to explicitly contain the first subleading term of the scaling of entanglement energy and the first subleading term of $\varepsilon_\alpha/\delta_{ent}$ is written out explicitly.
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The description of the "MPS unzipping" algorithm in Sec. 4 is expanded to describe all steps of the algorithm in more detail.
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An explicit construction of the MPO for the rainbow-chain Potts model is added in Sec. 4.1 (heading "Representing the rainbow chain as a matrix-product operator") and Fig. 6.
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The discussion of the excited-state entanglement spectrum in Sec. 4.1 is clarified in terms of boundary CFT operators.
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The discussion of the effective length scaling of the NRG flow (Sec. 4.2, heading "Finite \chi_{NRG} effects on NRG stability") is improved using more of the data in Fig. 8, and expanded with a study of the transverse-field Ising model in Appendix C.
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An explicit guide to setting the unzipping bond dimension \chi is added with Eq. (27).
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The outlook part of Sec. 5 is expanded:
the relationship between the entanglement spectra of excited states and other boundary conditions is added as a direction of future work
the usefulness of the method for e.g. parton wave functions is made more explicit
Published as SciPost Phys. 19, 075 (2025)
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