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.
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.
Advancing Fuzzy-Sphere CFTs: 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) 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≈16.5. 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 ϕn operators in the ϕ4 Lagrangian description, beginning with the identity, σ, ε, ε′, and 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.
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.
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.
The recently introduced “fuzzy-sphere” method has enabled accurate numerical regularizations of certain three-dimensional (3D) conformal field theories (CFTs). The regularization is provided by the noncommutative geometry of the lowest Landau level filled by electrons, such that the charge is trivially gapped due to the Pauli exclusion principle at filling factor ν = 1 , while the electron spins encode the desired CFT. Successful applications of the fuzzy sphere to paradigmatic CFTs, such as the 3D Ising model, raise an important question: How finely tuned does the underlying electron system need to be? Here, we show that the 3D Ising CFT can also be realized at electron fillings. In such cases, the CFT spectrum is intertwined with the charge-neutral spectrum of the underlying fractional quantum Hall state—a feature that is trivially absent in the previously studied ν = 1 case. Remarkably, we show that the mixing between the CFT spectrum and the fractional quantum Hall spectrum is strongly suppressed within the numerically accessible system sizes. Moreover, we demonstrate that the CFT critical point is unaffected by the exchange statistics of the particles and by the nature of topological order in the charge sector. Our results set the stage for the fuzzy-sphere exploration of conformal critical points between topologically ordered states.
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.
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.