Improving 3d Ising OPE Coefficients with Fuzzy Sphere Conformal Generators

TL;DR

This work develops a rigorous method to extract high-dimension CFT data from a fuzzy-sphere realization of the 3d Ising CFT by diagonalizing the square of the special conformal generator $|K|^2$. By tuning and constructing $ ilde{oldsymbol{ abla}}^A$ and isolating primaries with small $|K|^2$, the authors identify primaries up to $oldsymbol{ riangle} oughly 8.7$ and uncover new parity-odd primaries, building their conformal multiplets via the action of $P_z$. These primaries enable more reliable extrapolation of OPE coefficients to the CFT limit, including a comprehensive set of coefficients involving parity-odd operators and the stress tensor, with Ward-identity checks and close agreement with bootstrap results. A comparison with ETH reveals that, at the energies accessible in this framework, diagonal and off-diagonal matrix elements of $oldsymbol{ ilde{oldsymbol{ riangle}}}$ do not fully conform to ETH expectations, indicating residual structure in these high-dimension states. Overall, the approach provides a robust, generalizable path to high-dimension CFT data and offers cross-checks with bootstrap and ETH, with potential applications to Truncated Conformal Space approaches and beyond.

Abstract

We use the $K$ special conformal generator in the Fuzzy sphere setup of the Ising CFT to determine primary states. For $Δ\lesssim 8$, we recover the known primaries and find several new ones, including in the parity-odd sector. We then use these primaries to compute OPE coefficients. We find that using primaries constructed from special-$K$ allows for better extrapolation of OPE coefficients to the CFT limit, because of the existence of an $O(1)$ gap between primaries and descendants in the spectrum of eigenvalues of $|K|^2$ which protects the primaries from strongly mixing with descendants. We compare the CFT data we obtain with the Eigenstate Thermalization Hypothesis.

Figures (22)

• Figure 1: Plots of $\delta V_0$, $\delta h$ and $Z$ for different values of $N$. The leading power $N^{-0.206}$ in the first two plots is due to the descendant $\nabla^2 \epsilon$, and the leading power $1/N$ in the third plot is due to the descendant $\nabla^2 T^0_0$.
• Figure 2: $|K|^2$ values between 0 and 1.5 for increasing $N$; a gap visibly opens up in the range between small $|K|^2$ and $|K|^2=1$. Gray points are for states with $\Delta>7.8$, which have the largest errors since they are closest to the cutoff. The teal horizontal line is our $|K|^2$ threshold for keeping primaries. A few descendants with $|K|^2$ close to 1 have been explicitly identified as descendants of $\sigma$ and $\sigma_{\mu_1\mu_2\mu_3}$, and the (dimension,spin) $(\Delta,\ell)$ is indicated for three states just above our $|K|^2$ threshold at $N=16$ (these discarded states are shown in more detail in Appendix \ref{['app:discarded']}). Only the first 150 eigenvalues in the $j_z=0$ sector are used to make the plot.
• Figure 3: Plot of the $|K|^2$ and $\Delta$ values of the states obtained with our construction. Left is $\mathbb{Z}_2$ odd and right is $\mathbb{Z}_2$ even. Solid horizontal lines indicate the location of states predicted by the bootstrap. Previously unknown primaries are shown in red. The right plot has an inset blowing up the region at $6.75 < \Delta < 8$ near $|K|^2=0$, where many primaries lie close together. Error bars are estimated as described in the text.
• Figure 4: $|K|^2$ eigenvalues, and dimension expectation values $\Delta \equiv \langle H \rangle$, as a function of $N$, and extrapolations to large $N$, for $\mathbb{Z}_2$-odd operators. Stars represent known results from numerical conformal bootstrap. In the language of \ref{['WC']}, the $1/N$ correction corresponds to the $\gamma_{\mathcal{R}}$ term, while the $1/N^{1.25}$ scaling arises from $\mathcal{O} = T^\prime$. We assume $g_{\epsilon}=g_{\mathcal{R}\epsilon}=g_{\epsilon^\prime}\simeq 0$ and approximate $g_C$ by the $1/N$ term, since $(3-\Delta_C)/2 \simeq -1.01$.
• Figure 5: Same as in Fig \ref{['fig:Z2oddPrimaries']} for $\mathbb{Z}_2$-even operators.