We present a model for strongly interacting fermions with internal O(3) symmetry on the fuzzy sphere that (i) preserves the rotational symmetry of the fuzzy sphere and (ii) undergoes a quantum phase transition in the (2+1)-dimensional O(3) Wilson-Fisher universality class. Using exact diagonalization and density matrix renormalization group, we locate the quantum critical point via conformal perturbation theory and obtain scaling dimensions from finite-size spectra. We identify 24 primary operators and determine some of their operator product expansion coefficients through first-order conformal perturbation theory. The results are benchmarked against conformal bootstrap and large quantum-number expansions and reveal a weakly irrelevant operator that plays a role in dimerized antiferromagnets. Our Letter provides a general framework for quantitatively accessing conformal data for O(N) Wilson-Fisher conformal field theories.
We present a model for strongly interacting fermions with internal O(3) symmetry on the fuzzy sphere that (i) preserves the rotational symmetry of the fuzzy sphere and (ii) undergoes a quantum phase transition in the (2+1)-dimensional O(3) Wilson-Fisher universality class. Using exact diagonalization and density matrix renormalization group, we locate the quantum critical point via conformal perturbation theory and obtain scaling dimensions from finite-size spectra. We identify 24 primary operators and determine some of their operator product expansion coefficients through first-order conformal perturbation theory. The results are benchmarked against conformal bootstrap and large quantum-number expansions and reveal a weakly irrelevant operator that plays a role in dimerized antiferromagnets. Our Letter provides a general framework for quantitatively accessing conformal data for O(N) Wilson-Fisher conformal field theories.
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.
The Wilson-Fisher criticality provides a paradigm for a large class of phase transitions in nature (e.g., helium, ferromagnets). In three dimensions, Wilson-Fisher critical points are not exactly solvable due to the strongly correlated feature, so one has to resort to nonperturbative tools such as numerical simulations. Here, we design a microscopic model of Heisenberg magnet bilayer and study the underlying Wilson-Fisher O(3) transition through the lens of fuzzy sphere regularization. We uncover a wealth of crucial information which directly reveals the emergent conformal symmetry regarding this fixed point. Specifically, we accurately calculate and analyze the energy spectra at the transition, and explicitly identify the existence of a conserved Noether current, a stress tensor, and relevant primary fields. Most importantly, the primaries and their descendants form a fingerprint conformal tower structure, pointing to an almost perfect state-operator correspondence. Furthermore, by examining the leading rank-4 symmetric tensor operator, we demonstrate the cubic perturbation is relevant, implying the critical O(3) model is unstable to cubic anisotropy, in agreement with the renormalization group and bootstrap calculations. The successful dissection of conformal content of the Wilson-Fisher universality class extends the horizon of the fuzzy sphere method and paves the way for exploring higher-dimensional conformal field theories.
The three-dimensional cubic conformal field theory governs the critical behavior of Heisenberg magnets with cubic anisotropy. Studying this theory nonperturbatively is challenging because its most easily accessible observables are numerically very close to those of the more symmetric O ( 3 ) model. In this work, we overcome this difficulty using the fuzzy sphere regularization method. By adding a cubic-invariant two-body interaction to the quantum rotor Hamiltonian used for the O ( 3 ) model, we break the continuous rotational symmetry by construction and unambiguously isolate the cubic critical point. Using exact diagonalization and the density matrix renormalization group, we calculate the scaling dimensions of several key operators, including the leading scalar singlets, and resolve the splitting of the O ( 3 ) rank-two traceless symmetric tensor into the E g and T 2 g representations of the cubic group. Our results are consistent with existing Monte Carlo, conformal perturbation theory, and ɛ expansion benchmarks, demonstrating the power of the fuzzy sphere in resolving closely spaced universality classes.
Boundary conformal field theory (BCFT) provides a universal framework for critical phenomena in the presence of boundaries. We determine BCFT data for the normal and ordinary boundary universality classes of the 1+1-dimensional boundaries of the 2+1-dimensional O(2) and O(3) Wilson-Fisher fixed points, realized microscopically by a bilayer Heisenberg model on the fuzzy sphere. Using the fuzzy-sphere state-operator correspondence, we obtain boundary operator spectra, identify low-lying boundary primary operators, extract operator-product-expansion (OPE) data, and estimate the boundary central charges for both boundary conditions. For the normal boundary condition, the universal amplitudes aσ and bt extracted from one- and two-point functions agree quantitatively with Monte Carlo benchmarks where available. For both N=2 and N=3, we find a positive extraordinary-log exponent α, providing independent microscopic evidence for extraordinary-log boundary criticality. Our results extend fuzzy-sphere BCFT spectroscopy beyond the Ising universality class to continuous O(N) symmetry.
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.
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.