A Fuzzy Sphere Journey in Critical Phenomena

Abstract

This review discusses the recently proposed fuzzy sphere regularization for studying $2+1$D critical phenomena, particularly three-dimensional (3D) conformal field theory (CFT). The fuzzy sphere scheme not only offers remarkable efficiency in extracting extensive CFT data at low computational cost but also reveals unexpected connections among 3D CFT (critical phenomena), noncommutative geometry, and the quantum Hall effect. We introduce the fundamental ideas of fuzzy sphere regularization, emphasizing its role in demonstrating the state-operator correspondence of 3D CFTs on the $S^2\times \mathbb{R}$ geometry. Additionally, we review key developments in this approach across various directions and outline potential future applications.

Invited review. Submitted May 2025; revised September 2025; published March 2026

References in corpus (47)

• Statistical physics of social dynamics
• Noncommutative Field Theory
• Critical Phenomena and Renormalization-Group Theory
• "Deconfined" quantum critical points
• Bounding scalar operator dimensions in 4D CFT
• Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm
• On Renormalization Group Flows in Four Dimensions
• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• A Four Dimensional Generalization of the Quantum Hall Effect
• Towards the F-Theorem: N=2 Field Theories on the Three-Sphere
• On the RG running of the entanglement entropy of a circle
• Seeing a c-theorem with holography
• A Finite Size Scaling Study of Lattice Models in the three-dimensional Ising Universality Class
• Conformality Lost
• The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT
• Emergent Spacetime and Holographic CFTs
• Walking, Weak first-order transitions, and Complex CFTs
• F-Theorem without Supersymmetry
• Pushing the Limits of Monte Carlo Simulations for the 3d Ising Model
• Emergent $SO(5)$ Symmetry at the Néel to Valence-Bond-Solid Transition
• A Constraint on Defect and Boundary Renormalization Group Flows
• Scaling Limits of the Fuzzy Sphere at one Loop
• Walking, Weak first-order transitions, and Complex CFTs II. Two-dimensional Potts model at $Q>4$
• The Conformal Bootstrap at Finite Temperature
• Three-Dimensional 3-State Potts Model Revisited With New Techniques
• Representations of the Virasoro algebra from lattice models
• Search for charged Higgs bosons in the $H^{\pm }\to tb$ decay channel in $pp$ collisions at $\sqrt{s}=8$ TeV using the ATLAS detector
• Localized magnetic field in the $O(N)$ model
• Phases of the (2+1) dimensional SO(5) non-linear sigma model with topological term
• Landau Level Quantization on the Sphere
• Half-filled Landau levels: a continuum and sign-free regularization for 3D quantum critical points
• Conformal data and renormalization group flow in critical quantum spin chains using periodic uniform matrix product states
• Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula
• Operator Product Expansion Coefficients of the 3D Ising Criticality via Quantum Fuzzy Sphere
• Bootstrapping the 3d Ising Stress Tensor
• New Developments in the Numerical Conformal Bootstrap
• Phase Transitions of Ferromagnetic Potts Models on the Simple Cubic Lattice
• The entropic $g$-theorem in general spacetime dimension
• The $g$-function and Defect Changing Operators from Wavefunction Overlap on a Fuzzy Sphere
• Studying the 3d Ising surface CFTs on the fuzzy sphere
• Phases of (2+1)D SO(5) non-linear sigma model with a topological term on a sphere: multicritical point and disorder phase
• Universal amplitude-exponent relation for the Ising model on sphere-like lattices
• 3D Ising CFT and Exact Diagonalization on Icosahedron: The Power of Conformal Perturbation Theory
• Conformal symmetry of the critical 3D Ising model inside a sphere
• Constructing the Infrared Conformal Generators on the Fuzzy Sphere
• Reclaiming the Lost Conformality in a non-Hermitian Quantum 5-state Potts Model
• Quantum Monte Carlo Simulation of the 3D Ising Transition on the Fuzzy Sphere

Cited by in corpus (1)

• The $O(N)$ Free-Scalar and Wilson-Fisher Conformal Field Theories on the Fuzzy Sphere