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

Abstract

The RG functions of the 2D $n$-vector ${\varphi }^4$ model are calculated in the five-loop approximation. Perturbative series for the $\beta$ 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 $\beta$ 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.

Comments: 8 pages, 3 tables, as published