Surface critical exponents for a three-dimensional modified spherical model

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 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 − (12K c ) −1 , where K c is the dimensionless critical coupling. When ρ > ρ c , χ 1,1 diverges exponentially as T → T + c . An effective hamiltonian which leads to an exactly solvable model with γ 1,1 = 2, the value for the n → ∞ limit of the corresponding O(n) model, is proposed too.

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

“…In order to understand the above facts, let us note that the observed behavior is in agreement with the length of the spins near the boundary. This length is reduced near a Dirichlet boundary and enlarged near a Neumann one [23] (there < σ 2 >≃ 1.34 [23]; similar estimation for the Dirichlet boundary gives < σ 2 >≃ 0.83). Then, since the total length of all the spins is fixed, that leads to spins in the main part of the system being larger than 1 under Dirichlet and smaller than 1 under Neumann boundary conditions.…”

“…∆ o 1 = 1 for d = 3) for ordinary and ∆ sb 1 = 2/(d − 2) (i.e. ∆ sb 1 = 2 for d = 3) for special phase transitions, while ∆ o 1 = 1/2 and ∆ sb 1 = 3/2 [10], [23] for the three-dimensional spherical model. It is believed that the corresponding equivalence will be recovered if one imposes spherical constraints in a way which ensures that the mean square value of each spin of the system is the same [28] (unfortunately such a model is rather untractable analytically).…”

“…The finite-size scaling behavior of the mean spherical model has been studied so far under periodic, antiperiodic, Dirichlet, [8], [9], [21], [22], Neumann [10], [23] and Neumann-Dirichlet [10] boundary conditions (for a review see, e.g., [5]). For lattice systems under Dirichlet boundary conditions we mean that…”

Out-of-equilibrium properties of the semi-infinite kinetic spherical model

“…The present work is on the one hand meant to close a gap in the study of kinetic spherical models [31], which up to now have been restricted to systems with periodic boundary conditions. On the other hand our intention is also to extend towards dynamics earlier investigations of the surface criticality of the spherical model [21,22,23,24,25,26]. It has to be noted in this context that the static properties of the critical semi-infinite spherical model with one spherical field have been shown to differ from those of the O(N) model with N −→ ∞ [23,24], even so both models are strictly equivalent in the bulk system [27,28].…”

Semi-infinite anisotropic spherical model: Correlations atT>~Tc

“…[29,30,31]). Very recently, a compromise model was proposed [32], which uses a constraint for the spins at the surface in addition to that for the bulk ones. The properties of such a model are closer, in a sense, to those of the SM with the local spin constraint and, hence, to those of the D = ∞, or O(∞), model.…”