Corrections to scaling in the 2D phi^4 model: Monte Carlo results and some related problems

Monte Carlo (MC) simulations have been performed to refine the estimation of the correction-to-scaling exponent $\omega$ in the 2D $\varphi^4$ model, which belongs to one of the most fundamental universality classes. If corrections have the form $\propto L^{-\omega}$, then we find $\omega=1.546(30)$ and $\omega=1.509(14)$ as the best estimates. These are obtained from the finite-size scaling of the susceptibility data in the range of linear lattice sizes $L \in [128,2048]$ at the critical value of the Binder cumulant and from the scaling of the corresponding pseudocritical couplings within $L \in [64,2048]$. These values agree with several other MC estimates at the assumption of the power-law corrections and are comparable with the known results of the $\epsilon$-expansion. In addition, we have tested the consistency with the scaling corrections of the form $\propto L^{-4/3}$, $\propto L^{-4/3} \ln L$ and $\propto L^{-4/3} /\ln L$, which might be expected from some considerations of the renormalization group and Coulomb gas model. The latter option is consistent with our MC data. Our MC results served as a basis for a critical reconsideration of some earlier theoretical conjectures and scaling assumptions. In particular, we have corrected and refined our previous analysis by grouping Feynman diagrams. The renewed analysis gives $\omega \approx 4-d-2 \eta$ as some approximation for spatial dimensions $d<4$, or $\omega \approx 1.5$ in two dimensions.