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

3D Ising CFT and exact diagonalization on icosahedron: The power of conformal perturbation theory

“…Recently, Ref. [10] considered a Hamiltonian on S 2 realizing a quantum phase transition in the (2 + 1)D Ising universality class, which preserves exact rotation invariance. Already for small systems N ≤ 24, where N is the number of electrons on the sphere, results were obtained for CFT operator dimensions [10], operator product expansion (OPE) coefficients [11], and the four-point functions [12] which compare well with the conformal bootstrap [13][14][15][16], up to small deviations associated to finite N effects.…”

“…Why does the model of [10] work so well? One possibility is that the model is somehow very special (beyond the fact that it realizes exact rotation invariance).…”

“…In the 2 even scalar sector, after the relevant ε of dimension ∆ ε ≈ 1.41 and the leading irrelevant ε ′ of dimension ∆ ε ′ ≈ 3.83 the next irrelevant operator has a rather high dimension ∆ ε ′′ ≈ 6.89 [17]. Given that [10] performs a double tuning of the model's parameters, a possible scenario is that the operators ε and ε ′ are both tuned away, while the corrections associated with ε ′′ are expected to scale as 1/N α with a high power α = 1 2 (∆ ε ′′ − 3), and may be small even at moderate N .…”

“…These considerations can be more constructively formulated as the following Conjecture 1. Finite N effects in the spectrum on S 2 of the model of [10] can be understood by an effective theory, perturbing the CFT Hamiltonian by integrals of local CFT operators times small couplings.…”

“…3 Ref. [10] achieves exact rotation invariance via an alternative smart construction of "fuzzy sphere regularization" which we will not use here. Our main point here will be that, even with a broken rotation invariance and in a very small model with only 12 spins on the sphere, we will still be able to understand deviations of the exact diagonalization spectra from CFT via an effective theory.…”