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, $\chi ={\chi }_{\min }+{\sum }_i{\lambda }_i^2{N}_m^{-{\omega }_i}+O({\lambda }^3)$, 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 ${\epsilon }^{\prime }$. We find similar results when normalizing by the gap to the stress tensor $T$ or first parity-odd operator $\sigma$ instead of $E_0$.

10 pages, 17 figures. v2: conformal perturbation theory for correction to scaling added