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

TL;DR

This work advances a GS-energy based route to locate the Ising CFT on the fuzzy sphere by exploiting a finite-size expansion $E_{GS}(N_m)/N_m = E_0 + E_1/N_m + E_{3/2}/N_m^{3/2}+\cdots$ and the near-CFT form $\chi = E_{3/2}/E_0 = \chi_{\min} + \sum_i \lambda_i^2 N_m^{-\omega_i}$. By computing only the ground-state energy with the fuzzifiED package and fitting up to $E_{5/2}$ terms, they identify the critical line and the sweet spot, and extract curvature-scaling exponents $\omega_i$ that agree with bootstrap predictions for $\omega_1 = \Delta_\varepsilon - 3$ and $\omega_2 = \Delta_{\varepsilon'} - 3$. They also test alternative normalizations using the stress-energy and parity-odd gaps, finding consistent results, and discuss the universality of the ${\cal F}$-function and its (current) inaccessibility via this GS-energy approach. The method offers a fast, robust diagnostic for locating 3D CFTs on the fuzzy sphere and provides a stepping stone toward analyzing more complex or nonunitary CFTs in this framework.

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 $ χ:=E_{3/2}/E_0$ as a function of the couplings. Conformal perturbation theory predicts that around a CFT, $χ= χ_{\rm min} + \sum_i λ_i^2 N_m^{-ω_i} + O(λ^3)$, where $λ_i$ are the couplings associated to perturbations of operators with dimension $Δ_i$, and $ω_i = d-Δ_i$. 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 $ω$ associated to the two leading scalars $ε$, and $ε'$. We find similar results when normalizing by the gap to the stress tensor $T$ or first parity-odd operator $σ$ instead of $E_0$.

Figures (16)

• Figure 1: $E_{\rm GS}(x)$ for various values of $h$ and $g$, for $o=4$, using $x^{i/2}$ as a basis, $\{ 1, x, x^{3/2},x^2,x^{5/2}\}$, $N_m^{\rm max}=16$. Gray dots are not used for the fit, but in agreement with it.
• Figure 2: $E_0= E_{\rm GS}(0|h,g)$ for $o=4$, $N_m^{\rm max}=16$; this plot changes little between $N_m^{\rm max}=13$ and $N_m^{\rm max}=16$. In blue the critical line.
• Figure 3: $\chi$ for $N_m^{\rm max}=16$; in blue the critical line. To make the minimum visible, we plot the signed root ($\sqrt{\chi}:=\hbox{sign}(\chi) \sqrt{|\chi|}$). To enhance the resolution of the plot, the interpolation outlined below in Eq. (\ref{['rho(h,g)']}) and Fig. \ref{['f:fit-weights']} was used.
• Figure 4: Heat map of $\chi(h,g)$ given in Eq. (\ref{['K(h,g)']}) for $o=4$, $N_m^{\rm max}=16$. The white dashed line is the critical line of ZhuHanHuffmanHofmannHe2023, with a white shamrock marking their sweat spot (best agreement with a CFT). In dark blue dots the minimum of the valley of ${\chi (h,g)}$. The cyan diamond marks the global minimum of ${\chi (h,g)}$.
• Figure 5: The weights $\rho(h,g)$ defined in Eq. (\ref{['rho(h,g)']}), for $h=2.5$ and $g= 4.75$ (off grid).