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arXiv:2504.00842 · SciPost Physics

Exact diagonalization, matrix product states and conformal perturbation theory study of a 3D Ising fuzzy sphere model

Andreas Martin Läuchli, Loïc Herviou, Patrick H. Wilhelm, Slava Rychkov

Published Sep 26, 2025• 14 citations

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} 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.

DOI: 10.21468/scipostphys.19.3.076

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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} 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.

The fuzzy sphere regularization provides a powerful framework for studying three-dimensional (3D) conformal field theories (CFTs) by mapping them onto numerically tractable lattice models on the spherical lowest Landau level. However, the system sizes accessible to this method have been limited by the exact diagonalization (ED). In this work, we transcend this limitation by combining the fuzzy sphere regularization with a sophisticated multi-target density matrix renormalization group (DMRG) algorithm. Focusing on the 3D Ising-type model on the spherical lowest Landau level, we calculate the 24 low-lying energies at a larger system size than previously feasible with ED. At criticality, we extract the scaling dimensions of six primary operators, and the results show significantly improved agreement with bootstrap benchmarks compared to previous ED results at smaller sizes. Our approach allows us to efficiently target multiple excited states in larger systems beyond the reach of exact diagonalization. This study establishes the fuzzy sphere regularization combined with advanced DMRG techniques as a powerful and general framework for precision physics in 3D CFTs.

We consider the transverse field Ising model in (2+1)D, putting 12 spins at the vertices of the regular icosahedron. The model is tiny by the exact diagonalization standards, and breaks rotation invariance. Yet we show that it allows a meaningful comparison to the 3D Ising CFT on \mathbb{R}× S^2 ℝ × S 2 , by including effective perturbations of the CFT Hamiltonian with a handful of local operators. This extreme example shows the power of conformal perturbation theory in understanding finite N effects in models on regularized S^2 S 2 . Its ideal arena of application should be the recently proposed models of fuzzy sphere regularization.

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.

A bstract The fuzzy sphere method has enjoyed great success in the study of (2+1)-dimensional unitary conformal field theories (CFTs) by regularizing them as quantum Hall transitions on the sphere. Here, we extend this approach to the Yang-Lee CFT — the simplest non-unitary CFT. We use an Ising quantum-Hall ferromagnet Hamiltonian with a transverse field and an imaginary longitudinal field, the latter breaks the Hermiticity of the Hamiltonian and thus the unitarity of the associated quantum field theory. Non-unitary conformal field theories — particularly the Yang-Lee CFT — pose significant challenges to conventional fuzzy sphere approaches. To overcome these obstacles, here we utilize a different method for determining critical points that requires no a priori knowledge of CFT scaling dimensions. Our method instead leverages the state-operator correspondence while utilizing two complementary criteria: the conformality of the energy spectrum and its consistency with conformal perturbation theory. We also discuss a new finite-size scaling on the fuzzy sphere that makes the sources of systematic error in extracting conformal data explicit, and compare it with the conventional analysis using the (1+1)-dimensional Yang-Lee problem as an example. Our results show broad agreement with previous Monte-Carlo and conformal bootstrap results. We also uncover one previously unknown primary operator and several operator product expansion coefficients.

Conformal data for the

O(2)

Wilson-Fisher CFT in

(2+1)

-dimensional spacetime from exact diagonalization and matrix product states on the fuzzy sphere

We study at zero temperature a microscopic quantum spin-1 model on the fuzzy sphere that realizes the O(2) O ( 2 ) Wilson-Fisher conformal field theory (CFT) in (2+1) ( 2 + 1 ) -dimensional spacetime at a quantum critical point. Here, we use the fuzzy-sphere regularization as it preserves the full spatial SO(3) S O ( 3 ) rotational symmetry of the CFT, enabling the state-operator correspondence that maps energy eigenstates directly to CFT operators. Using exact diagonalization (ED) and matrix product state (MPS) techniques combined with conformal perturbation theory (CPT), we extract conformal data including scaling dimensions and operator product expansion (OPE) coefficients. We identify 48 primary operators and their descendants, organized by the conserved O(2) O ( 2 ) charge S^{z} S z and spatial angular momentum L L . Our numerical results for the scaling dimensions of the lowest primary operators show good agreement with conformal bootstrap predictions. We verify predictions from the large charge expansion, which provides systematic predictions for operators carrying large U(1) U ( 1 ) charge, connecting the Goldstone mode physics in the ordered phase to phonon primaries at the critical point.

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

E_{GS}(N_m)/N_m = E_{0} + E_1 /N_m + E_{3/2}/N_m^{3/2}+ ...

, 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= χ_{\rm min} + \sum_i λ_i^2 N_m^{-ω_i} + O(λ^3)

, where

λ_i

are the couplings associated to perturbations of operators with dimension

Δ_i

, and

ω_i = d-Δ_i

. 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

ω

associated to the two leading scalars

ε

, and

ε'

. We find similar results when normalizing by the gap to the stress tensor

T

or first parity-odd operator

σ

instead of

E_0

.