The $O(N)$ Free-Scalar and Wilson-Fisher Conformal Field Theories on the Fuzzy Sphere

TL;DR

Using fuzzy-sphere regularization, the authors construct two $O(N)$ models that realize the $O(N)$ Wilson-Fisher and free-scalar CFTs in three dimensions by projecting fermions to the lowest Landau level on $S^2$ and tuning two Hamiltonians $H_1$ and $H_2$. They validate the approach with exact diagonalization for $N=2,3,4$, locating conformal points via a spectral conformal cost function and showing that the operator spectra, conformal multiplet structure, and two-point correlators agree with CFT and bootstrap benchmarks. A striking result is that the CFT ground states on the fuzzy sphere are close to a trivial product state and exhibit unusually small entanglement, indicating a semi-classical character of these regularized CFTs. The work provides a versatile, non-perturbative platform to access conformal data such as OPE coefficients and line defects, and suggests directions for RG flows and large-$N$ analyses within this framework.

Abstract

The fuzzy-sphere regularization is an emerging numerical and theoretical technique for studying conformal field theories (CFTs). In this paper, we apply it to the $O(N)$ vector model, one of the most prominent theories for critical behavior in three space-time dimensions. We construct a model that realizes the $O(N)$ Wilson-Fisher and free-scalar CFTs for general $N$. For $N=2,3,4$, we present numerical evidence including the operator spectra and correlation functions in agreement with conformal symmetry and conformal bootstrap results.

Figures (9)

• Figure 1: A schematic phase diagram of the $O(N)$ WF and $O(N)$ free-scalar model.
• Figure 2: The minimal spectral conformal cost function for the $O(N)$ ($N=2,3,4$) WF and free-scalar CFTs as a function of the inverse linear system size $1/\sqrt{N_\text{orb}}$. The parameters for the miximal size are listed in Table \ref{['tab:params']}. The free-scalar CFTs have a worse conformal cost at finite system sizes and an even-odd effect of $N_\text{orb}$, so we focus on even $N_\text{orb}$.
• Figure 3: Operator spectrum of the $O(N)$ ($N=2,3,4$) WF CFT from state-operator correspondence at different system sizes at their respective optimal conformal point up to $\Delta\lesssim4,\ell\leq 3$. The filled, empty squares and the circles denote respectively operators with improper $\mathbb{Z}_2$-even, odd, and no definite improper $\mathbb{Z}_2$. The bars are bootstrap ChesterO2Chester:2020iyt and Monte Carlo hasenbusch2000eliminatingHasenbusch:2011zwv results and the states in the same multiplet are labeled by the same color.
• Figure 4: Operator spectrum of the $O(N)$ ($N=2,3,4$) free-scalar CFT from state-operator correspondence at different system sizes at their respective optimal conformal point up to $\Delta\lesssim4,\ell\leq 3$ and the rank-2 tensor representations. The filled, empty squares and the circles denote respectively operators with improper $\mathbb{Z}_2$-even, odd and no definite improper $\mathbb{Z}_2$. The bars are theoretical results and the colors label the multiplets.
• Figure 5: Dimensionless two-point correlation functions of UV fields $V_a(\bm x)$, $\omega_a(\bm x)$, and $n_{00}(\bm x)$ in the $O(2)$ fuzzy-sphere model for the WF and free scalar. In the IR, these correlators should converge to the CFT correlators of $\phi_a(\bm x)$, $\pi_a(\bm x)=\partial_\tau \phi_a(\bm x)$ and $S(\bm x)$, shown as the dashed black line.