Uncovering conformal symmetry in the $3D$ Ising transition: State-operator correspondence from a fuzzy sphere regularization

Abstract

The $3D$ Ising transition, the most celebrated and unsolved critical phenomenon in nature, has long been conjectured to have emergent conformal symmetry, similar to the case of the $2D$ Ising transition. Yet, the emergence of conformal invariance in the $3D$ Ising transition has rarely been explored directly, mainly due to unavoidable mathematical or conceptual obstructions. Here, we design an innovative way to study the quantum version of the $3D$ Ising phase transition on spherical geometry, using the "fuzzy (non-commutative) sphere" regularization. We accurately calculate and analyze the energy spectra at the transition, and explicitly demonstrate the state-operator correspondence (i.e. radial quantization), a fingerprint of conformal field theory. In particular, we have identified 13 parity-even primary operators within a high accuracy and 2 parity-odd operators that were not known before. Our result directly elucidates the emergent conformal symmetry of the $3D$ Ising transition, a conjecture made by Polyakov half a century ago. More importantly, our approach opens a new avenue for studying $3D$ CFTs by making use of the state-operator correspondence and spherical geometry.

arXiv commentClose to the published version with a few extra modifications: 1) Typos (a missing of 1/2 factor in the Hamiltonian) are corrected; 2) Convention of lowest Landau level projection is modified; 3) A typo of pseudo-potentials in Appendix has been corrected

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• The $\mathrm{SO}(5)$ Deconfined Phase Transition under the Fuzzy Sphere Microscope: Approximate Conformal Symmetry, Pseudo-Criticality, and Operator Spectrum
• Operator Product Expansion Coefficients of the 3D Ising Criticality via Quantum Fuzzy Sphere
• New Developments in the Numerical Conformal Bootstrap
• The $g$-function and Defect Changing Operators from Wavefunction Overlap on a 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
• Studying the 3d Ising surface CFTs on the fuzzy sphere
• 3D Ising CFT and Exact Diagonalization on Icosahedron: The Power of Conformal Perturbation Theory
• Constructing the Infrared Conformal Generators on the Fuzzy Sphere
• Reclaiming the Lost Conformality in a non-Hermitian Quantum 5-state Potts Model
• Exact Diagonalization, Matrix Product States and Conformal Perturbation Theory Study of a 3D Ising Fuzzy Sphere Model
• A new series of 3D CFTs with $\mathrm{Sp}(N)$ global symmetry on fuzzy sphere
• Quantum Monte Carlo Simulation of the 3D Ising Transition on the Fuzzy Sphere
• Three-dimensional Abelian and non-Abelian gauge Higgs theories
• Regularizing 3D conformal field theories via anyons on the fuzzy sphere
• The most uniform distribution of points on the sphere
• Machine Learning the Operator Content of the Critical Self-Dual Ising-Higgs Gauge Model
• Three-dimensional real space renormalization group with well-controlled approximations
• Critical Majorana fermion at a topological quantum Hall bilayer transition
• Numerical extraction of crosscap coefficients in microscopic models for (2+1)D conformal field theory
• Conformal Data for the O(3) Wilson-Fisher Conformal Field Theory from Fuzzy Sphere Realization of the Quantum Rotor Model
• Finite-size scaling on the torus with periodic projected entangled-pair states
• Continuously varying critical exponents in an exactly solvable long-range cluster XY mode
• Free and Interacting Fermionic Conformal Field Theories on the Fuzzy Sphere
• Boundary operator product expansion coefficients of the three-dimensional Ising universality class
• Essential difference between 2D and 3D from the perspective of real-space renormalization group
• The $O(N)$ Free-Scalar and Wilson-Fisher Conformal Field Theories on the Fuzzy Sphere
• Lattice-reflection symmetry in tensor-network renormalization group with entanglement filtering in two and three dimensions
• Oscillator Calculus on Coadjoint Orbits and Index Theorems
• Analytic framework for self-dual criticality in ${\mathbb{Z}}_k$ gauge theory with matter
• Deformation of Matrix Geometry via Landau Level Evolution
• Higher-form anomalies and state-operator correspondence beyond conformal invariance