Quantum Rotors on the Fuzzy Sphere and the Cubic CFT

TL;DR

This paper uses fuzzy sphere regularisation to non-perturbatively study the cubic conformal field theory relevant for Heisenberg magnets with cubic anisotropy, resolving subtle differences from the $O(3)$ model.

Contribution

It introduces a fuzzy sphere approach with a cubic-invariant interaction to isolate and analyze the cubic critical point non-perturbatively.

Findings

Calculated scaling dimensions of key operators.

Resolved the splitting of the $O(3)$ tensor into cubic group representations.

Results agree with Monte Carlo and perturbation theory benchmarks.

Abstract

The three-dimensional cubic conformal field theory governs the critical behaviour of Heisenberg magnets with cubic anisotropy. Studying this theory non-perturbatively is challenging, because its most easily accessible observables are numerically very close to those of the more symmetric $O(3)$ model. In this work, we overcome this difficulty using the fuzzy sphere regularisation method. By adding a cubic-invariant two-body interaction to the quantum rotor Hamiltonian used for the $O(3)$ model, we break the continuous rotational symmetry by construction and unambiguously isolate the cubic critical point. Using exact diagonalisation and the density matrix renormalisation group, we calculate the scaling dimensions of several key operators, including the leading scalar singlets, and resolve the splitting of the $O(3)$ rank-two traceless symmetric tensor into the $E_g$ and $T_{2g}$ representations of the cubic group. Our results are consistent with existing Monte Carlo, conformal perturbation theory, and $\varepsilon$ expansion benchmarks, demonstrating the power of the fuzzy sphere in resolving closely spaced universality classes.