bibkey: estienne2022annulus authors: Benoit Estienne, Yacine Ikhlef, and Andrei Rotaru year: 2022 title: Second Renyi entropy and annulus partition function for one-dimensional quantum critical systems with boundaries doi: 10.21468/SciPostPhys.12.4.141 url: https://arxiv.org/abs/2112.01929v2 claim: The paper computes a second-Renyi boundary-CFT problem through an annulus and records exact diagonal Ising boundary partition functions. strata_touched: [] license: citation-only triage: anchor

Exact Ising annulus as an entropy-certificate benchmark

Primary input

SciPost Physics 12, 141 (2022). The primary arXiv record specifies v2 dated 17 May 2022. Equation (3.3) gives the free Ising boundary state; equation (3.5) gives Z_f|f=sqrt(theta3/eta). Appendix A, equations (A.4) and (A.6), gives the theta and Dedekind products. These are known formulas, reused rather than reclassified as new theorems.

For an annulus of width ell in the Cardy–Tonni conventions, the closed-channel product is exp(ell/(24n)) times product_(j>=1)[1+exp(-(2j-1)ell/n)]. The modularly equivalent open-channel product has fermion energies (2j-1)pi^2/ell at n=1. It yields a normalized positive density operator. The RC benchmark compares its direct fermion entropy with the closed-channel derivative and with the new positive-spectrum envelope.

The paper’s principal second-Renyi problem, an interval away from a boundary, has its own replica geometry. An annulus representation at n=2 alone does not supply a common annulus for arbitrary n. The RC use imports only the exact Ising annulus partition function and the displayed identities. It does not extend the paper’s second-Renyi result into a general von Neumann solution for that different interval geometry.

Certification boundary

The c=1/2 Ising annulus is a continuum CFT benchmark, not a large-c Einstein-gravity dual or new lattice experiment. Finite product sums require tail bounds; the supplied diagnostic retains explicit geometric tails. The vacuum-character-only benchmark elsewhere in RC is a distinct transfer function and is not called the full Ising spectrum.

The primary parsed equation (3.5) and Appendix A were read. Appendix A printed page 19 was viewed as an image; the separate image request for printed page 13 failed. No figure data are used. The RC extremal theorem and its quantitative interval certificate are written derivations, not results attributed to this source, and have no Lean or independent-review status.