Critical exponents of the 3d Ising and related models from Conformal Bootstrap

Abstract

The constraints of conformal bootstrap are applied to investigate a set of conformal field theories in various dimensions. The prescriptions can be applied to both unitary and non unitary theories allowing for the study of the spectrum of low-lying primary operators of the theory. We evaluate the lowest scaling dimensions of the local operators associated with the Yang-Lee edge singularity for $2 \le D \le 6$. Likewise we obtain the scaling dimensions of six scalars and four spinning operators for the 3d critical Ising model. Our findings are in agreement with existing results to a per mill precision and estimate several new exponents.

Figures (11)

• Figure 1: Progressive truncations $Tr_n(z)$ of the sum rule (\ref{['sumrule']}) at $z=\bar{z}$ for the 4-point function of the energy operator in the $4d$$O(n)$-invariant free massless scalar theory. The four curves represent respectively the sum of the contributions of the primaries of scaling dimensions $\Delta\le4,\,\Delta\le6,\,\Delta\le8,\,\Delta\le10$. In this specific example we have chosen $n=1$.
• Figure 2: Progressive truncations $Tr_n(z)$ of the sum rule (\ref{['sumrule']}) for $z=\bar{z}$ of the 4-point function of a free massless scalar theory in $3d$. Few terms are sufficient to well approximate the rhs of the sum rule around the symmetric point $z=\frac{1}{2}$.
• Figure 3: The scalar product $\langle\alpha\vert\Delta\rangle\equiv \langle\alpha\vert 0,\Delta,\Delta_\phi\rangle$ as a function of $\Delta$ in the critical 3d Ising model. $\langle\alpha \vert$ is the left eigenvector corresponding to the null eigenvalue of the $7\times7$ matrix ${\sf f}$ discussed in Section \ref{['Ising']} and $\vert\Delta\rangle$ is a column of ${\sf f}$ associated with conformal blocks of spin 0. The zeros are the scaling dimensions of the $\mathbb{Z}_2$-even scalars $\varepsilon,\varepsilon',\varepsilon",\varepsilon"'$.
• Figure 4: Dispersion of the estimates of $\Delta_4$ versus $\Delta_\phi$ in the 2d Yang-Lee model. Notice the strong correlation between these two quantities.
• Figure 5: Plot of the function $\sigma(D)$ given by eq. (\ref{['YLepsilon']}) compared with the results of our bootstrap analysis for $5\le D\le6$. In this range the different resummations of the epsilon expansion do not give visually different results. The difference between the bootstrap estimates and the epsilon expansion are probably due to the contributions of higher order in $\epsilon$.