New surface critical exponents in the spherical 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 h(1) and h(L). are supposed to act at the surfaces bounding the system. In the case of Neumann boundary conditions a new surface critical exponent Delta(1)(sb) = 3/2 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 Delta(1)(o) = 1/2 and Delta(1)(sb) = 3/2.