Bootstrapping the 3d Ising model at finite temperature

Abstract

We estimate thermal one-point functions in the 3d Ising CFT using the operator product expansion (OPE) and the Kubo-Martin-Schwinger (KMS) condition. Several operator dimensions and OPE coefficients of the theory are known from the numerical bootstrap for flat-space four-point functions. Taking this data as input, we use a thermal Lorentzian inversion formula to compute thermal one-point coefficients of the first few Regge trajectories in terms of a small number of unknown parameters. We approximately determine the unknown parameters by imposing the KMS condition on the two-point functions $\langle \sigma\sigma \rangle$ and $\langle \epsilon\epsilon \rangle$. As a result, we estimate the one-point functions of the lowest-dimension $\mathbb Z_2$-even scalar $\epsilon$ and the stress-energy tensor $T_{\mu \nu}$. Our result for $\langle \sigma\sigma \rangle$ at finite-temperature agrees with Monte-Carlo simulations within a few percent, inside the radius of convergence of the OPE.