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arXiv:2506.14904

Free real scalar CFT on fuzzy sphere: spectrum, algebra and wavefunction ansatz

Yin-Chen He

Published Jun 17, 2025• 0 citations
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Abstract

We introduce a simple model to realize the free real scalar CFT on the fuzzy sphere. The model is structurally similar to the original model that realizes the 3D Ising CFT on the fuzzy sphere. Owing to the shift symmetry of the free scalar, the free scalar CFT fixed point in our model can be accessed with only a single tuning parameter-the conformal coupling. We numerically demonstrate that our model correctly reproduces the operator spectrum, correlation functions, and, crucially, the harmonic oscillator algebra of the real scalar CFT. We also examine the fuzzy sphere algebra, a generalization of the Girvin-MacDonald-Platzman algebra, and discuss its potential implications for defining quantum field theories on non-commutative geometries. Finally, we propose wavefunction ansatz for the ground states of both the free scalar and Ising CFTs, which exhibit remarkable agreement with the CFT ground state wavefunctions of the fuzzy sphere model. For instance, the wavefunction overlap between our Ising ansatz and the Ising CFT ground state exceeds $0.99$ for $N=28$ fermions, suggesting a promising direction for tackling the 3D Ising CFT.

DOI
10.48550/arxiv.2506.14904

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We introduce a simple model to realize the free real scalar CFT on the fuzzy sphere. The model is structurally similar to the original model that realizes the 3D Ising CFT on the fuzzy sphere. Owing to the shift symmetry of the free scalar, the free scalar CFT fixed point in our model can be accessed with only a single tuning parameter-the conformal coupling. We numerically demonstrate that our model correctly reproduces the operator spectrum, correlation functions, and, crucially, the harmonic oscillator algebra of the real scalar CFT. We also examine the fuzzy sphere algebra, a generalization of the Girvin-MacDonald-Platzman algebra, and discuss its potential implications for defining quantum field theories on non-commutative geometries. Finally, we propose wavefunction ansatz for the ground states of both the free scalar and Ising CFTs, which exhibit remarkable agreement with the CFT ground state wavefunctions of the fuzzy sphere model. For instance, the wavefunction overlap between our Ising ansatz and the Ising CFT ground state exceeds 0.990.990.99 for N=28N=28N=28 fermions, suggesting a promising direction for tackling the 3D Ising CFT.
Free theories are landmarks in the landscape of quantum field theories: their exact solvability serves as a pillar for perturbative constructions of interacting theories. Fuzzy sphere regularization, which combines quantum Hall physics with state-operator correspondence, has recently been proposed as a promising framework for simulating three-dimensional conformal field theories (CFTs), but so far it has not provided access to free theories. We overcome this limitation by designing a bilayer quantum Hall system that hosts an Ising tricritical point-a nontrivial fixed point where first-order and second-order transitions meet-which flows to the conformally coupled scalar theory in the infrared. The critical energy spectrum and operator structure match those at the Gaussian fixed point, providing nonperturbative evidence for the emergence of a free scalar CFT. Our results expand the landscape of CFTs realizable on the fuzzy sphere and demonstrate that even free bosonic theories-previously inaccessible-can emerge from interacting electrons in this framework.
The fuzzy-sphere regularization is a powerful tool to study conformal field theories (CFT) in three spacetime dimensions. In this paper, we extend its scope to CFTs with local fermionic operators. We realize the free-Majorana-fermion CFT on a setup with one flavor of bosons and one flavor of fermions on the lowest Landau level with a 1 / 2 angular momentum mismatch and allow conversion between two bosons and two fermions, and use a relative chemical potential as the tuning parameter. On the phase diagram, we observe two continuous transitions described, respectively, by a free Majorana fermion and a gauged Ising CFT. We numerically confirm the emergent conformal symmetry through the operator spectrum and the two-point correlation function of the local Majorana fermion. We further establish a correspondence between the fuzzy-sphere models and the field-theory Lagrangians, and extend it to an interacting fermionic CFT—the super-Ising theory with emergent superconformal symmetry.
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Advancing Fuzzy-Sphere CFTs: SO(3)\mathrm{SO}(3)SO(3)-Rotation-Resolving Exact Diagonalization, 3D Ising Heavy Primaries, and Quasi-Hole-Space Projection
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 SO(3)\mathrm{SO}(3)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≈16.5Δ\approx16.5Δ≈16.5. 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 ϕnϕ^nϕn operators in the ϕ4ϕ^4ϕ4 Lagrangian description, beginning with the identity, σσσ, εεε, ε′ε'ε′, and σ′σ'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.
We locate the phase-transition line for the Ising model on the fuzzy sphere from a finite-size scaling analysis of its ground-state energy. Our strategy is to write the latter as EGS(Nm)/Nm=E0+E1/Nm+E3/2/Nm3/2+...E_{GS}(N_m)/N_m = E_{0} + E_1 /N_m + E_{3/2}/N_m^{3/2}+ ...EGS​(Nm​)/Nm​=E0​+E1​/Nm​+E3/2​/Nm3/2​+..., and to search for a minimum of $ χ:=E_{3/2}/E_0$ as a function of the couplings. Conformal perturbation theory predicts that around a CFT, χ=χmin+∑iλi2Nm−ωi+O(λ3)χ= χ_{\rm min} + \sum_i λ_i^2 N_m^{-ω_i} + O(λ^3)χ=χmin​+∑i​λi2​Nm−ωi​​+O(λ3), where λiλ_iλi​ are the couplings associated to perturbations of operators with dimension ΔiΔ_iΔi​, and ωi=d−Δiω_i = d-Δ_iωi​=d−Δi​. This procedure finds the critical curve of [PRX 13 (2023) 021009] and their sweet spot with good precision. Varying two coupling constants allows us to extract the correction-to-scaling exponent ωωω associated to the two leading scalars εεε, and ε′ε'ε′. We find similar results when normalizing by the gap to the stress tensor TTT or first parity-odd operator σσσ instead of E0E_0E0​.