Solving the 3D Ising model with the conformal bootstrap

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 product-expansion 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.

Figure 1

The conformal bootstrap condition $=$ associativity of the operator algebra.

Figure 2

Using conformal freedom, three operators can be fixed at $x_1=0$, $x_3=(1,0,\dots ,0)$, $x_4\to \infty$, 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 $\overline{z}$ is its complex conjugate. The figure also shows the conformal block analyticity cut (thick black line) and the boundary of the absolute convergence region of the power series representation (4.7) (thin red line).

Figure 3

Shaded: the part of the $({\Delta }_{\sigma },{\Delta }_{\epsilon })$ plane allowed by the crossing symmetry constraint (5.3). 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. 4. This plot was obtained with the algorithm described in Appendix  with $n_{\max }=11$.

Figure 4

The zoom of the dashed rectangle area from Fig. 3. The small red rectangle is drawn using the ${\Delta }_{\sigma }$ and ${\Delta }_{\epsilon }$ error bands given in Table .

Figure 5

Same as Figs. 3, 4, but imposing the extra constraints ${\Delta }_{{\epsilon }^{\prime }}\ge \{3,3.4,3.8\}$.

Figure 6

The bound on ${\Delta }_{{\epsilon }^{\prime }}$ under the condition that ${\Delta }_{\epsilon }$ has already been fixed to the maximal value allowed by Fig. 3. Here $n_{\max }=10$ (see Appendix ). The width of the vertical red line marking the 3D Ising value of ${\Delta }_{\sigma }$ is about five times the error band in Table .

Figure 7

Upper bound on the dimension of the second spin 2 operator $T_{\mu \nu }^{\prime }$ from the crossing symmetry constraint (5.3). The algorithm from Appendix  was used with $n_{\max }=10$. The 3D Ising vertical red line is five times wider than the error band in Table . We do not show the region of ${\Delta }_{\sigma }$ close to the unitarity bound, which is subject to numerical instabilities.

Figure 8

Upper bound on the dimension of the first spin 4 operator in the $\sigma \times \sigma$ OPE from the crossing symmetry constraint (5.3). The algorithm from Appendix  was used with $n_{\max }=10$. The tip of the arrow shows the point $({\Delta }_{\sigma },{\Delta }_C)$ with the 3D Ising model values from Table . The dashed line is the Gaussian solution ${\Delta }_4=2{\Delta }_{\sigma }+4$.

Figure 9

Upper bound on the dimension of the first spin 6 operator in the $\sigma \times \sigma$ OPE from the crossing symmetry constraint (5.3). The algorithm from Appendix  was used with $n_{\max }=10$. The dashed line is the Gaussian solution ${\Delta }_6=2{\Delta }_{\sigma }+6$.

Figure 10

Upper bound on the dimension of the first spin 2 operator in the $\sigma \times \sigma$ OPE from the crossing symmetry constraint (5.3) in nonlocal theories without a stress tensor. The algorithm from Appendix  was used with $n_{\max }=10$. The dashed line is the Gaussian solution ${\Delta }_2=2{\Delta }_{\sigma }+2$.

Figure 11

The lower bound on $C_T$ as a function of ${\Delta }_{\sigma }$. The plot was obtained with $n_{\max }=11$. The 3D Ising vertical red line is five times wider than the error band in Table .