Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization

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 (noncommutative) 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 identify13 parity-even primary operators within a high accuracy and two 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 conformal field theories by making use of the state-operator correspondence and spherical geometry.