Critical thermodynamics of the two-dimensional systems in five-loop renormalization-group approximation

Abstract

The RG functions of the 2D $n$-vector $ϕ^4$ model are calculated in the five-loop approximation. Perturbative series for the $β$ function and critical exponents are resummed by the Pade-Borel and Pade-Borel-Leroy techniques, resummation procedures are optimized and an accuracy of the numerical results is estimated. In the Ising case $n = 1$ as well as in the others ($n = 0$, $n = -1$, $n = 2, 3,...32$) an account for the five-loop term is found to shift the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the corresponding lattice calculations; even error bars of the RG and lattice estimates do not overlap in the most cases studied. This is argued to reflect the influence of the singular (non-analytical) contribution to the $β$ function that can not be found perturbatively. The evaluation of the critical exponents for $n = 1$, $n = 0$ and $n = -1$ in the five-loop approximation and comparison of the numbers obtained with their known exact counterparts confirm the conclusion that non-analytical contributions are visible in two dimensions.