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.
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 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.
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 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.
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.
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 revisit the problem of \textit{real-time} quantum dynamics of the paradigmatic two dimensional transverse-field Ising model using the recently developed fuzzy sphere regularization scheme. By linearly ramping the transverse field from the paramagnetic phase to criticality, we study the finite-time scaling behavior of the squared order parameter ⟨mz2⟩, the excitation energy density Q, and the two-point correlation function of mz. We establish numerically that, at intermediate quench rate, ⟨mz2⟩ follows the conventional Kibble-Zurek prediction set by the critical exponents of the 3D Ising universality class, and the correlation function exhibits the expected exponential decay whose correlation length can be used to estimate the non-universal scaling coefficient in the freeze-out time/length. In contrast, the excitation energy density Q does not reach the same scaling regime at available system sizes due to large effective finite-size gap from symmetry-enforced level sparsity in the energy spectrum. At slow quench rates the universal quasi-adiabatic scaling for both ⟨mz2⟩ and Q is recovered. Since the fuzzy sphere construction can realize not only the Ising conformal field theory (CFT), but a broad family of (2+1)d CFTs, our results establish a route to the real-time critical dynamics of strongly coupled CFTs that are otherwise computationally challenging to study.