Critical amplitudes in two-dimensional theories

Abstract

We derive exact analytical expressions for the critical amplitudes $A_\psi$, $A_{gap}$ in the scaling laws for the fermion condensate $<\bar \psi \psi> = A_\psi m^{1/3} g^{2/3}$ and for the mass of the lightest state $M_{gap} = A_{gap} m^{2/3} g^{1/3}$ in the Schwinger model with two light flavors, $m \ll g$. $A_\psi$ and $A_{gap}$ are expressed via certain universal amplitude ratios being calculated recently in TBA technique and the known coefficient $A_{\psi\psi}$ in the scaling law $<\bar \psi \psi (x) \bar \psi \psi (0)> = A_{\psi\psi} (g/x)$ at the critical point. Numerically, $A_\psi = -0.388..., A_{gap} = 2.008...$ . The same is done for the standard square lattice Ising model at $T = T_c$. Using recent Fateev's results, we get $<\sigma_{lat}> = 1.058... (H_{lat}/T_c)^{1/15}$ for the magnetization and $M_{gap} = a/\xi = 4.010... (H_{lat}/T_c)^{8/15}$ for the inverse correlation length ($a$ is the lattice spacing). The theoretical prediction for $<\sigma^{lat}>$ is in a perfect agreement with numerical data. Two available numerical papers give the values of $M_{gap}$ which differ from each other by a factor $\approx \sqrt{2}$ . The theoretical result for $M_{gap}$ agrees with one of them.