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:2509.08038 · Physical Review X

Free and Interacting Fermionic Conformal Field Theories on the Fuzzy Sphere

Zheng Zhou, Davide Gaiotto A, Yin-Chen He

Published May 11, 2026• 4 citations
arXivHTMLPDF

Abstract

The fuzzy-sphere regularization is a powerful tool to study conformal field theories (CFT) in three spacetime dimensions. In this paper, we extend its scope to CFTs with local fermionic operators. We realize the free-Majorana-fermion CFT on a setup with one flavor of bosons and one flavor of fermions on the lowest Landau level with a 1 / 2 angular momentum mismatch and allow conversion between two bosons and two fermions, and use a relative chemical potential as the tuning parameter. On the phase diagram, we observe two continuous transitions described, respectively, by a free Majorana fermion and a gauged Ising CFT. We numerically confirm the emergent conformal symmetry through the operator spectrum and the two-point correlation function of the local Majorana fermion. We further establish a correspondence between the fuzzy-sphere models and the field-theory Lagrangians, and extend it to an interacting fermionic CFT—the super-Ising theory with emergent superconformal symmetry.

DOI
10.1103/fkkl-9tw2

Related papers

3D Conformal Field Theories with Sp ( N ) Global Symmetry on a Fuzzy Sphere

Zheng Zhou, Yin-Chen He

The quest to discover new 3D CFTs has been intriguing for physicists. For this purpose, fuzzy sphere regularization that studies interacting quantum systems defined on the lowest Landau level on a sphere has emerged as a powerful tool. In this Letter, we discover a series of new CFTs with global symmetry Sp(N) in the fuzzy sphere models that are closely related to the SO(5) deconfined phase transition, and are related to a Sp(N)/[Sp(M)×Sp(N-M)] nonlinear sigma model with a Wess-Zumino-Witten term. We numerically verify the emergent conformal symmetry by observing the integer-spaced conformal multiplets, the quality of conformal generators, and studying the finite-size scaling of the conformality. We discuss possible candidates for these newly discovered CFTs, the most plausible ones being Chern-Simons-matter theories which have N flavors of gapless bosons or fermions coupled to a non-Abelian [viz. Sp(1), Sp(2), etc.] Chern-Simons gauge field. Our Letter provides new avenues for studying interacting CFTs in 3D, possibly facilitating the nonperturbative study of critical gauge theories and previously undiscovered CFTs.

arXiv:2410.00087•Physical Review Letters•Jun 16, 2025•15 citations

Conformal Scalar Field Theory from Ising Tricriticality on the Fuzzy Sphere

Joseph C. Taylor, Cristian Voinea, Zlatko Papić +1 more

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.

arXiv:2506.22539•Physical Review Letters•Dec 24, 2025•7 citations

Free real scalar CFT on fuzzy sphere: spectrum, algebra and wavefunction ansatz

Yin-Chen He

We introduce a simple model to realize the free real scalar CFT on the fuzzy sphere. The model is structurally similar to the original model that realizes the 3D Ising CFT on the fuzzy sphere. Owing to the shift symmetry of the free scalar, the free scalar CFT fixed point in our model can be accessed with only a single tuning parameter-the conformal coupling. We numerically demonstrate that our model correctly reproduces the operator spectrum, correlation functions, and, crucially, the harmonic oscillator algebra of the real scalar CFT. We also examine the fuzzy sphere algebra, a generalization of the Girvin-MacDonald-Platzman algebra, and discuss its potential implications for defining quantum field theories on non-commutative geometries. Finally, we propose wavefunction ansatz for the ground states of both the free scalar and Ising CFTs, which exhibit remarkable agreement with the CFT ground state wavefunctions of the fuzzy sphere model. For instance, the wavefunction overlap between our Ising ansatz and the Ising CFT ground state exceeds $0.99$ for $N=28$ fermions, suggesting a promising direction for tackling the 3D Ising CFT.

arXiv:2506.14904•Jun 17, 2025•0 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

Certified reduced-basis emulation of conformal field theories on the fuzzy sphere

V. Baran

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.

arXiv:2609.11675•Sep 10, 2026•0 citations

Regularizing 3D Conformal Field Theories via Anyons on the Fuzzy Sphere

Cristian Voinea, Ruihua Fan, Nicolas Regnault +1 more

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.

arXiv:2411.15299•Physical Review X•May 20, 2025•11 citations
The fuzzy-sphere regularization is a powerful tool to study conformal field theories (CFT) in three spacetime dimensions. In this paper, we extend its scope to CFTs with local fermionic operators. We realize the free-Majorana-fermion CFT on a setup with one flavor of bosons and one flavor of fermions on the lowest Landau level with a 1 / 2 angular momentum mismatch and allow conversion between two bosons and two fermions, and use a relative chemical potential as the tuning parameter. On the phase diagram, we observe two continuous transitions described, respectively, by a free Majorana fermion and a gauged Ising CFT. We numerically confirm the emergent conformal symmetry through the operator spectrum and the two-point correlation function of the local Majorana fermion. We further establish a correspondence between the fuzzy-sphere models and the field-theory Lagrangians, and extend it to an interacting fermionic CFT—the super-Ising theory with emergent superconformal symmetry.
The quest to discover new 3D CFTs has been intriguing for physicists. For this purpose, fuzzy sphere regularization that studies interacting quantum systems defined on the lowest Landau level on a sphere has emerged as a powerful tool. In this Letter, we discover a series of new CFTs with global symmetry Sp(N) in the fuzzy sphere models that are closely related to the SO(5) deconfined phase transition, and are related to a Sp(N)/[Sp(M)×Sp(N-M)] nonlinear sigma model with a Wess-Zumino-Witten term. We numerically verify the emergent conformal symmetry by observing the integer-spaced conformal multiplets, the quality of conformal generators, and studying the finite-size scaling of the conformality. We discuss possible candidates for these newly discovered CFTs, the most plausible ones being Chern-Simons-matter theories which have N flavors of gapless bosons or fermions coupled to a non-Abelian [viz. Sp(1), Sp(2), etc.] Chern-Simons gauge field. Our Letter provides new avenues for studying interacting CFTs in 3D, possibly facilitating the nonperturbative study of critical gauge theories and previously undiscovered CFTs.
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 introduce a simple model to realize the free real scalar CFT on the fuzzy sphere. The model is structurally similar to the original model that realizes the 3D Ising CFT on the fuzzy sphere. Owing to the shift symmetry of the free scalar, the free scalar CFT fixed point in our model can be accessed with only a single tuning parameter-the conformal coupling. We numerically demonstrate that our model correctly reproduces the operator spectrum, correlation functions, and, crucially, the harmonic oscillator algebra of the real scalar CFT. We also examine the fuzzy sphere algebra, a generalization of the Girvin-MacDonald-Platzman algebra, and discuss its potential implications for defining quantum field theories on non-commutative geometries. Finally, we propose wavefunction ansatz for the ground states of both the free scalar and Ising CFTs, which exhibit remarkable agreement with the CFT ground state wavefunctions of the fuzzy sphere model. For instance, the wavefunction overlap between our Ising ansatz and the Ising CFT ground state exceeds 0.990.990.99 for N=28N=28N=28 fermions, suggesting a promising direction for tackling the 3D Ising CFT.
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.
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.