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Surface critical exponents for a three-dimensional modified spherical model

Daniel M. Danchev, J.G. Brankov, Magdy E. Amin

Published Nov 19, 2012• 6 citations
OpenAlex

Abstract

Abstract. A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann-Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some ρ> 0, and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals one). The surface susceptibility χ1,1 has been evaluated exactly. For ρ = 1 we find that χ1,1 is finite at the bulk critical temperature Tc, in contrast with the recently derived value γ1,1 = 1 in the case of just one global spherical constraint. The result γ1,1 = 1 is recovered only if ρ = ρc = 2 − (12Kc) −1, where Kc is the dimensionless critical coupling. When ρ> ρc, χ1,1 diverges exponentially as T → T + c. An effective hamiltonian which leads to an exactly solvable model with γ1,1 = 2, the value for the n → ∞ limit of the corresponding O(n) model, is proposed too.

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Journal of Physics C Solid State Physics•Oct 7, 1974•20 citations
Abstract. A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann-Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some ρ> 0, and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals one). The surface susceptibility χ1,1 has been evaluated exactly. For ρ = 1 we find that χ1,1 is finite at the bulk critical temperature Tc, in contrast with the recently derived value γ1,1 = 1 in the case of just one global spherical constraint. The result γ1,1 = 1 is recovered only if ρ = ρc = 2 − (12Kc) −1, where Kc is the dimensionless critical coupling. When ρ> ρc, χ1,1 diverges exponentially as T → T + c. An effective hamiltonian which leads to an exactly solvable model with γ1,1 = 2, the value for the n → ∞ limit of the corresponding O(n) model, is proposed too.
A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann - Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some , and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals 1). The surface susceptibility has been evaluated exactly. For we find that is finite at the bulk critical temperature , in contrast to the recently derived value of in the case of just one global spherical constraint. The result is only recovered if , where is the dimensionless critical coupling. When , diverges exponentially as . An effective Hamiltonian is also proposed which leads to an exactly solvable model with , the value for the limit of the corresponding model.
The three-dimensional mean spherical model with a L-layer film geometry, under Neumann - Neumann and Neumann - Dirichlet boundary conditions is considered. Surafce fields and are supposed to act at the surfaces bounding the system. In the case of Neumann boundary conditions a new surface critical exponent is found. It is argued that this exponent corresponds to a special (surface - bulk) phase transition in the model. The Privman - Fisher scaling hypothesis for the free energy is verified and the corresponding scaling functions for both the Neumann - Neumann and Neumann - Dirichlet boundary conditions are explicitly derived. If the layer field is applied at some distance from the Dirichlet boundary, a family of critical exponents emerges: their values depend on the exponent defining how the distance scales with the finite size of the system, and interpolate continuously between the extreme cases and .
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 Δ ( p ) = −2ζ(3)/(5π) = −0.153 051..., Δ ( a ) = 0.274 543... and Δ ( ND ) = 0.019 22..., 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 .
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 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 (related to the so called Casimir amplitude) is estimated. We obtain
Exact results of the finite-size behaviour of the susceptibility in three-dimensional 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 T c 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 finite-size critical temperature with respect to T c is smaller than 1/ν, with ν being the critical exponent of the bulk correlation length.
The local surface susceptibility chi 1,1(T) is calculated and analysed in detail for a ferromagnetic spherical model on a d-dimensional hypercubic half-space bounded by a free surface in which the exchange coupling, J(1+ Delta ) with Delta >0, is enhanced over the bulk value J. For Delta > Delta c=(2d-2)-1, chi 1,1(T) is found to diverge at a temperature Tcx( Delta ) above the bulk critical temperature, while for Delta < Delta c, chi 1,1(T) behaves as for a surface without enhancement ( Delta =0). The nature of the crossover phenomena is explored; it is shown that for Delta near Delta c and T near Tcx( Delta ), chi 1,1(T) can be represented in a scaled form in accord with the general theory of crossover phenomena.