Finite-size effects in film geometry with nonperiodic boundary conditions: Gaussian model and renormalization-group theory at fixed dimension

TL;DR

This paper provides exact analytical results for finite-size effects, including free energy and Casimir force, in the Gaussian model with various boundary conditions in film geometry, highlighting scaling behaviors and dimensional crossover.

Contribution

It offers the first exact solutions for finite-size effects with nonperiodic boundary conditions in the Gaussian model across different dimensions.

Findings

Finite-size scaling valid for Casimir force in all boundary conditions.

Logarithmic deviations from scaling at d=3 for certain boundary conditions.

Exact description of dimensional crossover near critical points.

Abstract

Finite-size effects are investigated in the Gaussian model with isotropic and anisotropic short-range interactions in film geometry with nonperiodic boundary conditions (b.c). We have obtained exact results for the free energy and the Casimir force for antiperiodic, Neumann, Dirichlet, and Neumann-Dirichlet mixed b.c. in 1<d<4 dimensions. For the Casimir force, finite-size scaling is found to be valid for all b.c.. For the free energy, finite-size scaling is valid in 1<d<3 and 3<d<4 dimensions for antiperiodic, Neumann, and Dirichlet b.c., but logarithmic deviations from finite-size scaling exist in d=3 dimensions for Neumann and Dirichlet b.c.. This is explained in terms of the borderline dimension d*=3, where the critical exponent of the Gaussian surface energy density vanishes. For Neumann-Dirichlet b.c., finite-size scaling is strongly violated above T_c for 1<d<4. Our results include an exact description of the dimensional crossover between the d-dimensional finite-size critical behavior near bulk T_c and the (d-1)-dimensional critical behavior near T_c,film(L). This dimensional crossover is illustrated for the critical behavior of the specific heat. For 2<d<4, the Gaussian results are reformulated as one-loop contributions of the phi^4 theory at fixed dimension and are compared with the epsilon=4-d expansion results as well as with d=3 Monte Carlo data. For d=2, the Gaussian results for the Casimir force scaling function are compared with those for the Ising model; unexpected exact relations are found between the Gaussian and Ising scaling functions. For both the Gaussian and the Ising model it is shown that anisotropic couplings imply nonuniversal scaling functions of the Casimir force. Our Gaussian results provide the basis for the investigation of finite-size effects of the mean spherical model with nonperiodic b.c..