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.

Close 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

References in corpus (6)

• Towards a derivation of holographic entanglement entropy
• Holographic c-theorems in arbitrary dimensions
• The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT
• Landau Level Quantization on the Sphere
• Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula
• Hopf Maps, Lowest Landau Level, and Fuzzy Spheres

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