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Physical review. B./Physical review. B

Locating the Ising conformal field theory via the ground-state energy on the fuzzy sphere

Kay Jörg Wiese

Published Jan 12, 2026• 2 citations
Publisher

Abstract

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. This is similar to what was used to locate the complex CFT of the 5-state Potts model in dimension $d=2$ [PRL 133 (2024) 077101]. There it was shown that a CFT is characterized by a stationarity condition for the measured effective central charge. Our strategy is to write the ground-state energy as $E_{\rm GS}(N)/N_m = E_{0} + E_1 /N_m + E_{3/2}/N_m^{3/2}+ ...$, and to search for a minimum of $ E_{3/2}/E_0$ as a function of the couplings. This procedure finds the critical curve of [PRX 13 (2023) 021009] with good precision, and their sweet spot as well. We find similar results when normalizing by the gap to the stress tensor tensor $T$ or first parity-odd operator $σ$.

DOI
10.1103/qzzt-86fn

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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. This is similar to what was used to locate the complex CFT of the 5-state Potts model in dimension d=2d=2d=2 [PRL 133 (2024) 077101]. There it was shown that a CFT is characterized by a stationarity condition for the measured effective central charge. Our strategy is to write the ground-state energy as EGS(N)/Nm=E0+E1/Nm+E3/2/Nm3/2+...E_{\rm GS}(N)/N_m = E_{0} + E_1 /N_m + E_{3/2}/N_m^{3/2}+ ...EGS​(N)/Nm​=E0​+E1​/Nm​+E3/2​/Nm3/2​+..., and to search for a minimum of E3/2/E0 E_{3/2}/E_0E3/2​/E0​ as a function of the couplings. This procedure finds the critical curve of [PRX 13 (2023) 021009] with good precision, and their sweet spot as well. We find similar results when normalizing by the gap to the stress tensor tensor TTT or first parity-odd operator σσσ.
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​.
Fuzzy-sphere regularization gives numerical access to three-dimensional conformal field theories (CFTs), but locating one requires testing for criticality across the parameter space of microscopic couplings. We reduce the search cost by about two orders of magnitude, combining an exact diagonalization based on states having definite total angular momentum with a certified reduced-basis emulator that reconstructs energies, observables and their derivatives continuously across the couplings from a handful of exact solutions per system size. The combination reproduces the published ground-state-energy analysis of the fuzzy-sphere Ising model and the published conformal data of a three-flavor fuzzy-sphere realization of the O(2) Wilson--Fisher CFT, including quantities defined by parameter derivatives. Applied to the O(2) model with no CFT input, the ground-state-energy criterion locates the critical coupling to within half a percent of a density matrix renormalization group value obtained on systems more than twice the size, and maps the critical line of the model across the coupling plane. Ground-state criteria locate the critical line robustly, but the point they select along it is weakly determined, and the computed dimensions come closest to the bootstrap elsewhere along the line.
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 3D Ising transition, the most celebrated and unsolved critical phenomenon in nature, has long been conjectured to have emergent conformal symmetry, similar to the case of the 2D Ising transition.Yet, the emergence of conformal invariance in the 3D Ising transition has rarely been explored directly, mainly due to unavoidable mathematical or conceptual obstructions.Here, we design an innovative way to study the quantum version of the 3D Ising phase transition on spherical geometry, using the "fuzzy (noncommutative) sphere" regularization.We accurately calculate and analyze the energy spectra at the transition, and explicitly demonstrate the state-operator correspondence (i.e., radial quantization), a fingerprint of conformal field theory.In particular, we identify13 parity-even primary operators within a high accuracy and two parity-odd operators that were not known before.Our result directly elucidates the emergent conformal symmetry of the 3D Ising transition, a conjecture made by Polyakov half a century ago.More importantly, our approach opens a new avenue for studying 3D conformal field theories by making use of the state-operator correspondence and spherical geometry.
The Potts model describes interacting spins with QQQ different components, which is a direct generalization of the Ising model (Q=2Q=2Q=2). Compared to the existing exact solutions in 2D, the phase transitions and critical phenomena in the 3D Potts model have been less explored. Here, we systematically investigate a quantum (2+1)(2+1)(2+1)-D Potts model with Q=3Q=3Q=3 using a fuzzy sphere regularization scheme. We first construct a microscopic model capable of achieving a magnetic phase transition that separates a spin S3S_3S3​ permutationally symmetric paramagnet and a spontaneous symmetry-breaking ferromagnet. Importantly, the energy spectrum at the phase transition point exhibits an approximately conformal symmetry, implying that an underlying conformal field theory may govern this transition. Moreover, when tuning along the phase transition line in the mapped phase diagram, we find that the dimension of the subleading S3S_3S3​ singlet operator flows and drifts around the critical value ∼3\sim 3∼3, which is believed to be crucial for understanding this phase transition, although determining its precise value remains challenging due to the limitations of our finite-size calculations. These findings suggest a discontinuous transition in the 3D 3-state Potts model, characterized by pseudo-critical behavior, which we argue results from a nearby multicritical or complex fixed point.
We present a numerical quantum Monte Carlo (QMC) method for simulating the 3D phase transition on the recently proposed fuzzy sphere [Phys. Rev. X 13, 021009 (2023)]. By introducing an additional SU(2) layer degree of freedom, we reformulate the model into a form suitable for sign-problem-free QMC simulation. From the finite-size-scaling, we show that this QMC-friendly model undergoes a quantum phase transition belonging to the 3D Ising universality class, and at the critical point we compute the scaling dimensions from the state-operator correspondence, which largely agrees with the prediction from the conformal field theory. These results pave the way to construct sign-problem-free models for QMC simulations on the fuzzy sphere, which could advance the future study on more sophisticated criticalities.