Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2604.21091

Studying 3D O(N) Surface CFT on the Fuzzy Sphere

Jiechao Feng, Taige Wang

Published Apr 22, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $a_σ$ and $b_t$ 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.

DOI
10.48550/arxiv.2604.21091

Related papers

Conformal operator content of the Wilson-Fisher transition on fuzzy sphere bilayers

Chao Han, Liangdong Hu, Wei Kang Zhu

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 $\mathrm{O}(3)$ 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 $\mathrm{O}(3)$ 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.

arXiv:2312.04047•Physical review. B./Physical review. B•Sep 9, 2024•25 citations

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

Arjun Dey, Loïc Herviou, Christopher Mudry +2 more

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.

arXiv:2604.18705•SciPost Physics•Oct 14, 2026•0 citations

Conformal Data for the O(3) Wilson-Fisher Conformal Field Theory from Fuzzy Sphere Realization of the Quantum Rotor Model

Arjun Dey, Loïc Herviou, Christopher Mudry +1 more

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.

arXiv:2510.09755•Physical Review Letters•Apr 24, 2026•5 citations

Monodromy Defects of 3D Wilson-Fisher Criticality on the Fuzzy Sphere

Yin Tang, W. J. Zhu

Monodromy (twist) line defects arise as the boundaries of topological symmetry surfaces, which are among the simplest non-trivial conformal defects with direct realizations as magnetic vortex in quantum magnets, as Wilson and t Hooft lines of gauge theories, and as the replica twist operators in quantum information theory. However, their interacting dynamics remains difficult to characterize non-perturbatively. Here we design a way to realize the monodromy boundary conditions in the fuzzy-sphere regularization by threading a symmetry-selective Aharonov-Bohm flux through the two poles of the sphere. We apply this construction to the $\mathbb{Z}_2$ twist of the 3D Ising conformal field theory (CFT), $π$-twist of the 3D O(2) CFT and three inequivalent monodromies of O(3) (centre twist, z-axis $π$-twist and the reflection). For each defect, we extract the low-lying defect spectrum together with bulk-to-defect and defect-defect operator product expansion coefficients, which characterize the operators living on the defect line, their mutual interactions, and how they couple to the surrounding critical bulk. Among the low-lying defect primaries resolved in our spectra, we find no relevant scalar perturbation that preserves all the symmetries of the corresponding defect. The relevant deformations identified in our calculations break at least one preserved symmetry, providing evidence that these monodromy defects are stable against symmetry-preserving relevant perturbations. These results establish a systematic set of non-perturbative conformal data for monodromy defects beyond the Ising twist. Our construction therefore turns the symmetry flux into a tunable knob of the fuzzy-sphere regularization, opening a route to exploring the response of critical matter to distinct defects.

arXiv:2610.31269•Oct 22, 2026•0 citations

Boundary operator product expansion coefficients of the three-dimensional Ising universality class

Dorian Przetakiewicz, Stefan Weßel, Francesco Parisen Toldin

Recent advances in conformal field theory and critical phenomena have focused on the characterization of boundary or defects in a conformally invariant system. In this Letter, we study the critical behavior of the three-dimensional Ising universality class in the presence of a surface, realizing the ordinary, the special, and the normal universality classes. By combining high-precision Monte Carlo simulations of an improved model, where leading scaling corrections are suppressed, with a finite-size scaling analysis informed by conformal field theory, we determine unbiased, accurate estimates of universal boundary operator product expansion coefficients of experimental relevance. Furthermore, we improve the value of the scaling dimension of the surface field at the special transition by the estimate Δ ̂ σ = 0.3531 ( 3 ) .

arXiv:2502.14965•Physical Review Research•Jul 3, 2025•2 citations

Conformal Nature of Quantum Phase Transitions via Fuzzy Three-Sphere Regularization

Xue Meng, Liangdong Hu, W. Zhu

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.

arXiv:2607.11371•Jul 13, 2026•0 citations
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+11+11+1-dimensional boundaries of the 2+12+12+1-dimensional O(2)O(2)O(2) and O(3)O(3)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 aσa_σaσ​ and btb_tbt​ extracted from one- and two-point functions agree quantitatively with Monte Carlo benchmarks where available. For both N=2N=2N=2 and N=3N=3N=3, we find a positive extraordinary-log exponent ααα, 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)O(N)O(N) symmetry.
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)\mathrm{O}(3)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)\mathrm{O}(3)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.
Conformal data for the O(2)O(2)O(2) Wilson-Fisher CFT in (2+1)(2+1)(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 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.
Monodromy (twist) line defects arise as the boundaries of topological symmetry surfaces, which are among the simplest non-trivial conformal defects with direct realizations as magnetic vortex in quantum magnets, as Wilson and t Hooft lines of gauge theories, and as the replica twist operators in quantum information theory. However, their interacting dynamics remains difficult to characterize non-perturbatively. Here we design a way to realize the monodromy boundary conditions in the fuzzy-sphere regularization by threading a symmetry-selective Aharonov-Bohm flux through the two poles of the sphere. We apply this construction to the Z2\mathbb{Z}_2Z2​ twist of the 3D Ising conformal field theory (CFT), πππ-twist of the 3D O(2) CFT and three inequivalent monodromies of O(3) (centre twist, z-axis πππ-twist and the reflection). For each defect, we extract the low-lying defect spectrum together with bulk-to-defect and defect-defect operator product expansion coefficients, which characterize the operators living on the defect line, their mutual interactions, and how they couple to the surrounding critical bulk. Among the low-lying defect primaries resolved in our spectra, we find no relevant scalar perturbation that preserves all the symmetries of the corresponding defect. The relevant deformations identified in our calculations break at least one preserved symmetry, providing evidence that these monodromy defects are stable against symmetry-preserving relevant perturbations. These results establish a systematic set of non-perturbative conformal data for monodromy defects beyond the Ising twist. Our construction therefore turns the symmetry flux into a tunable knob of the fuzzy-sphere regularization, opening a route to exploring the response of critical matter to distinct defects.
Recent advances in conformal field theory and critical phenomena have focused on the characterization of boundary or defects in a conformally invariant system. In this Letter, we study the critical behavior of the three-dimensional Ising universality class in the presence of a surface, realizing the ordinary, the special, and the normal universality classes. By combining high-precision Monte Carlo simulations of an improved model, where leading scaling corrections are suppressed, with a finite-size scaling analysis informed by conformal field theory, we determine unbiased, accurate estimates of universal boundary operator product expansion coefficients of experimental relevance. Furthermore, we improve the value of the scaling dimension of the surface field at the special transition by the estimate Δ ̂ σ = 0.3531 ( 3 ) .
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.