Locating the Ising CFT via the ground-state energy on the fuzzy sphere

Abstract

We locate the phase-transition line for the Ising model on the fuzzy sphere from a finite-size scaling analysis of its ground-state energy. Our strategy is to write the latter as $E_{GS}(N_m)/N_m = E_{0} + E_1 /N_m + E_{3/2}/N_m^{3/2}+ ...$, and to search for a minimum of $ \chi:=E_{3/2}/E_0$ as a function of the couplings. Conformal perturbation theory predicts that around a CFT, $\chi= \chi_{\rm min} + \sum_i \lambda_i^2 N_m^{-\omega_i} + O(\lambda^3)$, where $\lambda_i$ are the couplings associated to perturbations of operators with dimension $\Delta_i$, and $\omega_i = d-\Delta_i$. This procedure finds the critical curve of [PRX 13 (2023) 021009] and their sweet spot with good precision. Varying two coupling constants allows us to extract the correction-to-scaling exponent $\omega$ associated to the two leading scalars $\epsilon$, and $\epsilon'$. We find similar results when normalizing by the gap to the stress tensor $T$ or first parity-odd operator $\sigma$ instead of $E_0$.