Scaling and nonscaling finite-size effects in the Gaussian and the mean spherical model with free boundary conditions

TL;DR

This paper investigates finite-size effects and deviations from finite-size scaling in the Gaussian and mean spherical models with free boundary conditions across dimensions 2<d<4, revealing logarithmic and power-law violations at critical dimensions.

Contribution

It provides a detailed analysis of finite-size effects, including logarithmic and power-law deviations from scaling, in Gaussian and mean spherical models with free boundaries across various dimensions.

Findings

Finite-size scaling valid for d<3 and d>3, with logarithmic deviations at d=3.

Non-logarithmic violation of scaling in susceptibility at d=3 in film geometry.

Power-law violation of scaling for d>3 and universal finite-size scaling for 2<d<3.

Abstract

We calculate finite-size effects of the Gaussian model in a L\times \tilde L^{d-1} box geometry with free boundary conditions in one direction and periodic boundary conditions in d-1 directions for 2<d<4. We also consider film geometry (\tilde L \to \infty). Finite-size scaling is found to be valid for d<3 and d>3 but logarithmic deviations from finite-size scaling are found for the free energy and energy density at the Gaussian upper borderline dimension d* =3. The logarithms are related to the vanishing critical exponent 1-\alpha-\nu=(d-3)/2 of the Gaussian surface energy density. The latter has a cusp-like singularity in d>3 dimensions. We show that these properties are the origin of nonscaling finite-size effects in the mean spherical model with free boundary conditions in d>=3 dimensions. At bulk T_c in d=3 dimensions we find an unexpected non-logarithmic violation of finite-size scaling for the susceptibility \chi \sim L^3 of the mean spherical model in film geometry whereas only a logarithmic deviation \chi\sim L^2 \ln L exists for box geometry. The result for film geometry is explained by the existence of the lower borderline dimension d_l = 3, as implied by the Mermin-Wagner theorem, that coincides with the Gaussian upper borderline dimension d*=3. For 3<d<4 we find a power-law violation of scaling \chi \sim L^{d-1} at bulk T_c for box geometry and a nonscaling temperature dependence \chi_{surface} \sim \xi^d of the surface susceptibility above T_c. For 2<d<3 dimensions we show the validity of universal finite-size scaling for the susceptibility of the mean spherical model with free boundary conditions for both box and film geometry and calculate the corresponding universal scaling functions for T>=T_c.