Corrections to scaling in the 2D <i>φ</i> ⁴ model: Monte Carlo results and some related problems

Abstract Monte Carlo (MC) simulations have been performed to refine the estimation of the correction-to-scaling exponent ω in the 2D φ 4 model, which belongs to one of the most fundamental universality classes. If corrections have the form ∝ L − ω , then we find ω = 1.546(30) and ω = 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 ∈ [128, 2048] at the critical value of the Binder cumulant and from the scaling of the corresponding pseudocritical couplings within L ∈ [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 ϵ -expansion. In addition, we have tested the consistency with the scaling corrections of the form ∝ L −4/3 , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo>∝</mml:mo> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mi>ln</mml:mi> <mml:mi>L</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo>∝</mml:mo> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>/</mml:mo> <mml:mi>ln</mml:mi> <mml:mi>L</mml:mi> </mml:math> , 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 ω ≈ 4 − d − 2 η as some approximation for spatial dimensions d &lt; 4, or ω ≈ 1.5 in two dimensions.