Helmholtz free energy of finite spin systems near criticality

Abstract

We present a quantitative comparison between 4-ε expansion and Monte Carlo estimates of critical finite-size properties. Specifically we consider the Helmholtz free energy (where the overall order parameter rather than its conjugate field is kept fixed) at the bulk critical temperature for a cube with periodic boundary conditions within a 4-ε expansion to one-loop order. An estimate for the complete asymptotic scaling function obtained from renormalization flow equations as well as systematic ε-expansion estimates for several amplitude ratios agree well with corresponding Monte Carlo results in three dimensions. The nonequivalence of thermodynamic ensembles for critical finite-size properties is discussed in some detail.