Solving the 3D Ising Model with the Conformal Bootstrap

TL;DR

This work demonstrates that 3D conformal symmetry, combined with unitarity, suffices to constrain the 3D Ising CFT data nonperturbatively via the conformal bootstrap. By computing conformal blocks efficiently in arbitrary dimensions and solving a linear-programming problem imposed by crossing symmetry on the $raket{\sigma^4}$ correlator, the authors locate a kink near the Ising point and extract rigorous bounds on operator dimensions, OPE coefficients, and the central charge. They further explore the impact of gaps in the spectrum, obtain bounds on higher-spin primaries, and show that the central charge attains a minimum close to the Ising values, suggesting the Ising CFT lies at a special corner of the allowed space. The results provide a framework for progressively sharpening CFT data from first principles and offer insights into higher-spin constraints and AdS/CFT implications.

Abstract

We study the constraints of crossing symmetry and unitarity in general 3D Conformal Field Theories. In doing so we derive new results for conformal blocks appearing in four-point functions of scalars and present an efficient method for their computation in arbitrary space-time dimension. Comparing the resulting bounds on operator dimensions and OPE coefficients in 3D to known results, we find that the 3D Ising model lies at a corner point on the boundary of the allowed parameter space. We also derive general upper bounds on the dimensions of higher spin operators, relevant in the context of theories with weakly broken higher spin symmetries.

Figures (11)

• Figure 1: The conformal bootstrap condition $=$ associativity of the operator algebra.
• Figure 2: Using conformal freedom, three operators can be fixed at $x_1=0$, $x_3=(1,0,\ldots,0)$, $x_4\to \infty$, while the fourth point $x_2$ can be assumed to lie in the (12) plane. The variable $z$ is then the complex coordinate of $x_2$ in this plane, while $\bar{z}$ is its complex conjugate. Also shown: the conformal block analyticity cut (thick black), and the boundary of the absolute convergence region of the power series representation (\ref{['eq:doublesum']}) (thin red).
• Figure 3: Shaded: the part of the $(\Delta_\sigma,\Delta_\varepsilon)$ plane allowed by the crossing symmetry constraint (\ref{['eq:cross']}). The boundary of this region has a kink remarkably close to the known 3D Ising model operator dimensions (the tip of the arrow). The zoom of the dashed rectangle area is shown in Fig. \ref{['fig:deltaE-closeup']}. This plot was obtained with the algorithm described in Appendix \ref{['app:details']} with $n_{\text{max}}=11$.
• Figure 4: The zoom of the dashed rectangle area from Fig. \ref{['fig:deltaE']}. The small red rectangle is drawn using the $\Delta_\sigma$ and $\Delta_\varepsilon$ error bands given in Table \ref{['tab:dims']}.
• Figure 5: Same as Figs. \ref{['fig:deltaE']}, \ref{['fig:deltaE-closeup']}, but imposing the extra constraints $\Delta_{\varepsilon'}\ge\{3, 3.4, 3.8\}$.