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arXiv:2609.35960

Advancing Fuzzy-Sphere CFTs: $\mathrm{SO}(3)$-Rotation-Resolving Exact Diagonalization, 3D Ising Heavy Primaries, and Quasi-Hole-Space Projection

Zheng Zhou, Yin-Chen He

Published Sep 28, 2026• 0 citations
arXivHTMLPDF

Abstract

Fuzzy-sphere regularization has recently emerged as a powerful framework for studying three-dimensional conformal field theories (CFTs). In this work, we exploit symmetry and Hilbert-space structures to substantially advance this framework. We develop an exact-diagonalization algorithm that resolves the full $\mathrm{SO}(3)$ rotational symmetry, reducing computational costs by factors of several hundred. For the three-dimensional Ising CFT, we identify around 100 primary operators, including a large number of previously unknown heavy scalar primaries up to scaling dimension $Δ\approx16.5$. We also resolve the tension between previous fuzzy-sphere and bootstrap results regarding the parity-odd scalar primary. Furthermore, we identify a branch of scalar primaries associated with $ϕ^n$ operators in the $ϕ^4$ Lagrangian description, beginning with the identity, $σ$, $ε$, $ε'$, and $σ'$. Their wave functions exhibit a semi-classical structure closely related to states invariant under area-preserving diffeomorphisms of the sphere. Finally, we show that critical points involving fractional quantum Hall states can be studied within a substantially reduced Hilbert space spanned by quasi-hole states of the corresponding model quantum Hall wave functions like the Laughlin state. This suggests a general strategy for exploring CFTs intertwined with topologically ordered phases.

DOI
10.48550/arxiv.2609.35960

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Conformal field theory (CFT) offers a modern viewpoint for understanding phase transitions. However, directly accessing the conformal algebra and microscopically uncovering the emergent conformal symmetry, especially in higher dimensions, remains a significant challenge. Motivated by recent advances in revealing CFT features via the fuzzy two-sphere, here we generalize this approach to higher dimensions and aim to expose the conformality at the (3+1)-D quantum critical point. We demonstrate this framework by investigating quantum phase transitions belonging to the Ising and Yang-Lee universality classes in a (3+1)-D quantum model (equivalent to a classical four-dimensional system), realized via Landau level projection on the fuzzy three-sphere. By computing the energy spectra at criticality, we explicitly verify the state-operator correspondence, a hallmark of conformal invariance. Together with prior advances, this work establishes a new pathway for the microscopic study of emergent conformality in higher-dimensional phase transitions.
Conformal data for the O(2)O(2)O(2) Wilson-Fisher CFT in (2+1)(2+1)(2+1)-dimensional spacetime from exact diagonalization and matrix product states on the fuzzy sphere
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