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Journal of Statistical Physics

The influence of external boundary conditions on the spherical model of a ferromagnet. I. Magnetization profiles

A. E. Patrick

Published Apr 1, 1994• 11 citations
Publisher
DOI
10.1007/bf02186289

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The ferromagnetic mean spherical model with a layer geometry of thickness L and Neumann-Dirichlet boundary conditions is investigated in the presence of a step-like external field which changes sign at distance from the Neumann boundary. The amplitude of the field can be taken to vanish in the thermodynamic limit as an inverse power of L . Exact expressions for the magnetization profile are derived and studied in three different temperature and field regimes: high-temperature bulk limit, critical finite-size scaling regime, and low-temperature moderate-field regime. It is found that in the critical finite-size scaling regime there exist two special values of x , denoted by , , which depend on the scaled temperature and field variables, and have the property that the magnetization changes sign only when . The magnetization is everywhere negative when and everywhere positive when . In the low-temperature moderate-field regime we establish that the field-induced critical point, in the case of periodic boundary conditions and a step-like field with , appears at .
The analysis of a previous study (Allen and Pathria. Can. J. Phys. 67, 952 (1989)) on finite-size effects in systems with O(n) symmetry [Formula: see text], confined to geometry Ld−d′ × ∞d′ (where d and d′ are continuous variables such that 2 < d′ < d < 4) and subjected to periodic boundary conditions, is extended (i) to include the region of first-order phase transition (T < Tc) as well as the region of second-order phase transition [Formula: see text] and (ii) to allow the presence of an external field H > 0. Predictions, involving both amplitudes and exponents, are made on the magnetization m and susceptibility χ in different regimes of the variables T, H, and L. Analytical verification of the predicted results is carried out in the case of the spherical model of ferromagnetism (n = ∞), and complete agreement is found.