Finite-size effects in the spherical model of finite thickness

TL;DR

This paper investigates finite-size effects in the spherical model with finite thickness, analyzing boundary conditions and their impact on critical behavior and Casimir amplitudes, revealing conditions where finite-size scaling holds or fails.

Contribution

It systematically evaluates finite-size corrections for various boundary conditions in the spherical model, providing explicit formulas and critical amplitude estimates.

Findings

Finite-size scaling holds for periodic and antiperiodic boundary conditions.

Casimir amplitudes are computed for different boundary conditions.

Surface interactions can induce attraction or repulsion between confining surfaces.

Abstract

A detailed analysis of the finite-size effects on the bulk critical behaviour of the $d$-dimensional mean spherical model confined to a film geometry with finite thickness $L$ is reported. Along the finite direction different kinds of boundary conditions are applied: periodic $(p)$, antiperiodic $(a)$ and free surfaces with Dirichlet $(D)$, Neumann $(N)$ and a combination of Neumann and Dirichlet $(ND)$ on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary $d$. It is found, for $2<d<4$, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for $(p)$ and $(a)$. For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At $d=3$, the critical amplitude of the singular part of the free energy (related to the so called Casimir amplitude) is estimated. We obtain $\Delta^{(p)}=-2\zeta(3)/(5\pi)=-0.153051...$, $\Delta^{(a)}=0.274543...$ and $\Delta^{(ND)}=0.01922...$, implying a fluctuation--induced attraction between the surfaces for $(p)$ and repulsion in the other two cases. For $(D)$ and $(N)$ we find a logarithmic dependence on $L$.