3D Ising CFT and Exact Diagonalization on Icosahedron: The Power of Conformal Perturbation Theory

TL;DR

This paper demonstrates how conformal perturbation theory can effectively analyze finite-size effects in the 3D Ising model on a regularized sphere, using a small 12-spin system on an icosahedron.

Contribution

It shows that conformal perturbation theory can be applied to small, symmetry-breaking systems to extract meaningful CFT data, especially relevant for fuzzy sphere models.

Findings

Successful comparison between 12-spin icosahedron model and 3D Ising CFT

Effective perturbations capture finite N effects

Highlights potential for fuzzy sphere regularization studies

Abstract

We consider the transverse field Ising model in $(2+1)$D, putting 12 spins at the vertices of the regular icosahedron. The model is tiny by the exact diagonalization standards, and breaks rotation invariance. Yet we show that it allows a meaningful comparison to the 3D Ising CFT on $\mathbb{R}\times S^2$, by including effective perturbations of the CFT Hamiltonian with a handful of local operators. This extreme example shows the power of conformal perturbation theory in understanding finite $N$ effects in models on regularized $S^2$. Its ideal arena of application should be the recently proposed models of fuzzy sphere regularization.