On the finite-size behaviour of systems with asymptotically large critical shift

Abstract

Exact results of the finite-size behavior of the susceptibility in threedimensional mean spherical model films under Dirichlet-Dirichlet, Dirichlet-Neumann and Neumann-Neumann boundary conditions are presented. The corresponding scaling functions are explicitly derived and their asymptotics close to, above and below the bulk critical temperature Tc are obtained. The results can be incorporated in the framework of the finite-size scaling theory where the exponent λ characterizing the shift of the finitesize critical temperature with respect to Tc is smaller than 1/ν, with ν being the critical exponent of the bulk correlation length.

Finite-size effects in the spherical model of finite thickness

“…Consequently our result disagrees with that of (Chen & Dohm 2003), where it has been argued that equation (4.69) of (Barber & Fisher 1973) was incorrect far from K c,3 + . At the bulk critical point, solving (5.6), with the assumption L √ φ ≪ 1, we find φ ∼ L −3 in agreement with (Chen & Dohm 2003, Dantchev & Brankov 2003. It is worth mentioning that (5.6) cannot be put in a scaling form because of the logarithmic dependence on L. However, if one considers a shifted critical point that absorbs the term proportional to ln L and the surface contributions according to (Barber & Fisher 1973)…”

“…The free energy density and the equation for the spherical field were computed for a film with arbitrary dimension d subject to the different boundary conditions. In the particular case d = 3, our general expressions for (D), (N) and (ND) reduce to those obtained by (Barber & Fisher 1973) and (Danchev et al 1997, Dantchev & Brankov 2003. It is found that the singular part of the free energy density has the standard finite-size scaling form for 2 < d < 4 only in the cases (p) and (a) i.e.…”

“…Furthermore, the finite-size contributions to the free energy density are obtained directly form the corresponding general expression rather than through integration of the equation for the spherical field. We anticipate here that from our results we recover the particular cases of Barber & Fisher (1973), Barber (1974), Danchev et al (1997) and Dantchev & Brankov (2003) but not the results of Chen & Dohm (2003). We will return to these points later in the paper.…”

“…At the bulk critical temperature K c,3 the spherical field behaves as φ ∼ L −1 to the leading order assuming L √ φ ≪ 1. In the limit L √ φ ≫ 1 one would expect a logarithmic behaviour as pointed out by (Dantchev & Brankov 2003). At the shifted critical temperature, defined through (Dantchev & Brankov 2003) (6.4) equation (6.3) may be written in a scaling form.…”

“…In the limit L √ φ ≫ 1 one would expect a logarithmic behaviour as pointed out by (Dantchev & Brankov 2003). At the shifted critical temperature, defined through (Dantchev & Brankov 2003) (6.4) equation (6.3) may be written in a scaling form. An inspection of the expression (6.4) shows that the predictions of the theory of finite-size scaling for the shifted critical temperature is not fulfilled due to the presence of the term proportional to ln L. For 3 < d < 4 the scaling form of the equation for the spherical field (6.2) is given by…”