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

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 $E_{GS}(N_m)\mathrm{/}{N}_m=E_0+E_1\mathrm{/}{N}_m+E_{3\mathrm{/}2}\mathrm{/}{N}_m^{3\mathrm{/}2}+\mathrm{.}\mathrm{.}\mathrm{.}$, and to search for a minimum of $\chi :=E_{3\mathrm{/}2}\mathrm{/}{E}_0$ as a function of the couplings. Conformal perturbation theory predicts that around a CFT, $\chi ={\chi }_{\mathrm{m}\mathrm{i}\mathrm{n}}+{\sum }_i{\lambda }_i^2{N}_m^{-{\omega }_i}+O({\lambda }^3)$, where ${\lambda }_i$ are the couplings associated to perturbations of operators with dimension ${\mathrm{\Delta }}_i$, and ${\omega }_i=d-{\mathrm{\Delta }}_i$. 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 ${\epsilon }^{\mathrm{\prime }}$. We find similar results when normalizing by the gap to the stress tensor $T$ or first parity-odd operator $\sigma$ instead of $E_0$.