Surface critical exponents for a three-dimensional modified spherical 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 $χ_{1,1}$ has been evaluated exactly. For $ρ=1$ we find that $χ_{1,1}$ is finite at the bulk critical temperature $T_c$, in contrast with the recently derived value $γ_{1,1}=1$ in the case of just one global spherical constraint. The result $γ_{1,1}=1$ is recovered only if $ρ=ρ_c= 2-(12 K_c)^{-1}$, where $K_c$ is the dimensionless critical coupling. When $ρ> ρ_c$, $χ_{1,1}$ diverges exponentially as $T\to T_c^{+}$. An effective hamiltonian which leads to an exactly solvable model with $γ_{1,1}=2$, the value for the $n\to \infty $ limit of the corresponding O(n) model, is proposed too.