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Physics Letters A

Surface critical behavior and scaling functions for the three-dimensional mean spherical model

Magdy E. Amin

Published May 12, 2006• 0 citations
Publisher
DOI
10.1016/j.physleta.2006.05.005

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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 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.
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.