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 $\langle σσ\rangle$ and $\langle εε\rangle$. As a result, we estimate the one-point functions of the lowest-dimension $\mathbb Z_2$-even scalar $ε$ and the stress-energy tensor $T_{μν}$. Our result for $\langle σσ\rangle$ at finite-temperature agrees with Monte-Carlo simulations within a few percent, inside the radius of convergence of the OPE.

Figures (13)

• Figure 1: Twists of the double-twist family $[\sigma\sigma]_0$. Here, we plot $\tau=\Delta-\ell$ versus $\overline{h} = \frac{\Delta+\ell}{2}$. The dots show estimates from Simmons-Duffin:2016wlq using the extremal functional method Poland:2010wgElShowk:2012huEl-Showk:2014dwa and the numerical bootstrap. The curve shows the prediction of large-spin perturbation theory with only $\Delta_\sigma,\Delta_\epsilon,f_{\sigma\sigma\epsilon},c_T$ taken from the numerical bootstrap. figure reproduced from Simmons-Duffin:2016wlq.
• Figure 2: Twists of the double-twist families $[\epsilon\epsilon]_0$ (orange) and $[\sigma\sigma]_1$ (blue). Again, we plot $\tau=\Delta-\ell$ versus $\overline{h} = \frac{\Delta+\ell}{2}$. The dots show estimates using the extremal functional method and the numerical bootstrap. The curves are estimates using large-spin perturbation theory and the mixing procedure described in Simmons-Duffin:2016wlq and reviewed in section \ref{['section:mixing between families']}. The dashed curves illustrate the effects of modifying the mixing procedure. Figure reproduced from Simmons-Duffin:2016wlq.
• Figure 3: Diagram of the algorithm which is used to obtain the thermal coefficients in the 3d Ising CFT. Here, "..." represents contributions to the thermal coefficients of families other than $[\sigma\sigma]_0$, $[\sigma\sigma]_1$, and $[\epsilon\epsilon]_0$ that we account for when considering operator mixing.
• Figure 4: Thermal coefficients for the three families $[\sigma\sigma]_0$, $[\sigma\sigma]_{1}$, and $[\epsilon\epsilon]_0$. The orange horizontal lines are obtained by using the KMS condition in combination with the thermal inversion formula and by averaging over several parameter choices. The spread given by the orange error bars is obtained by computing the operator mixing using different sets of $\overline{z}$ values as explained in Section \ref{['section:mixing between families']} and by imposing the KMS conditions in different regions on the thermal cylinder (see figure \ref{['fig:zzbar region for KMS']} for an example). The blue stars are MC estimates for $a_\epsilon^{\langle\sigma\sigma\rangle,\text{~MC}} = 0.711(3)$HasenbuschPrivate and $a_T^{\langle\sigma\sigma\rangle,\text{~MC}} = 2.092(13)$PhysRevE.79.041142PhysRevE.53.4414PhysRevE.56.1642. The blue lines are the estimates for the thermal coefficients of all other operators in $[\sigma\sigma]_0$, $[\sigma\sigma]_{1}$ and $[\epsilon\epsilon]_0$ families using these MC results together with the inversion formula. Note that the spread of the thermal coefficients of higher-spin operators estimated by the bootstrap are too small to be visible on this scale.
• Figure 5: Left: The thermal two-point function obtained by applying the inversion formula and then solving the KMS condition (yellow) compared to that obtained from a MC simulation (red). Note that we restrict the plot to the region of OPE convergence around $x = 0$ and $\tau = 0$. Right: Percentage difference between the two correlators, showing good agreement (within $~5\%$) between the bootstrap and MC predictions. At small values of $\sqrt{|x|^2 + \tau^2}$ we expect the MC results to be inaccurate due to lattice-size effects. As $\sqrt{|x|^2 + \tau^2} \rightarrow \beta$, we exit the region of OPE convergence, and we expect inaccuracies in the bootstrap calculation.