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Exact Diagonalization, Matrix Product States and Conformal Perturbation Theory Study of a 3D Ising Fuzzy Sphere Model
by Andreas M. Läuchli, Loïc Herviou, Patrick H. Wilhelm and Slava Rychkov
Submission summary
| Authors (as registered SciPost users): | Loïc Herviou · Slava Rychkov |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202504_00036v2 (pdf) |
| Date accepted: | Aug. 25, 2025 |
| Date submitted: | Aug. 20, 2025, 10 a.m. |
| Submitted by: | Loïc Herviou |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Numerical studies of phase transitions in statistical and quantum lattice models provide crucial insights into the corresponding Conformal Field Theories (CFTs). In higher dimensions, comparing finite-volume numerical results to infinite-volume CFT data is facilitated by choosing the sphere $S^{d-1}$ as the spatial manifold. Recently, the fuzzy sphere regulator [Zhu et al, Phys. Rev. X 13, 021009 (2023)] has enabled such studies with exact rotational invariance, yielding impressive agreement with known 3D Ising CFT predictions, as well as new results. However, systematic improvements and a deeper understanding of finite-size corrections remain essential. In this work, we revisit the fuzzy sphere regulator, focusing on the original Ising model, with two main goals. First, we assess the robustness of this approach using Conformal Perturbation Theory (CPT), to which we provide a detailed guidebook. We demonstrate how CPT provides a unified framework for determining the critical point, the speed of light, and residual deviations from CFT predictions. Applying this framework, we study finite-size corrections and clarify the role of tuning the model in minimizing these effects. Second, we develop a novel method for extracting Operator Product Expansion (OPE) coefficients from fuzzy sphere data. This method leverages the sensitivity of energy levels to detuning from criticality, providing new insights into level mixing and avoided crossings in finite systems. Our work also includes validation of CPT in a 1+1D Ising model away from the integrable limit.
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
We thank the referees for their very positive assessment of our work.
We took into account their remarks — the different changes are listed below. We also provide a direct answer to each referee as a comment to their report.
We updated the bibliography, including journal information to cited works on the arXiv which got published in the meantime.
Changes in the manuscripts are in blue.
Best regards,
The authors.
List of changes
- A sentence and a footnote after Eq. (17).
- Section 2.3. Clarified the relationship between the electron number $N$ and $R$.
- Added $(J>0)$ below Eq. 32.
- Sec 4: corrected the definition of the momentum, and specified that we consider even-length chains.
- Sec 4: clarified the discussion on the effect of the spin in CPT.
- Sec. 4: expanded the caption of Fig.5 and added a sentence at the end of the section.
- Sec. 5.3 the last three paragraphs rewritten.
- Sec 5.4: corrected a typo in the description of Fig. 10.
- Sec. 5.4: added footnote 19.
- Sec. 5.4: the last two sentences rewritten.
- Sec. 6: paragraph below Eq. (41).
- Sec 7 last paragraph modified.
Published as SciPost Phys. 19, 076 (2025)