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Journal of Physics A Mathematical and General

New effects in the magnetization profile of the spherical model in a step-like field

Magdy E. Amin, J.G. Brankov

Published May 29, 1998• 3 citations
Publisher

Abstract

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 .

DOI
10.1088/0305-4470/31/21/003

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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 validity of finite-size scaling in the presence of an inhomogeneous external field vanishing in the thermodynamic-limit is studied using a fully finite three-dimensional mean spherical model. The external field is chosen to change sign stepwise in one space dimension and to be translationally invariant in the other two dimensions, in which the lattice is assumed periodic. The boundary conditions in the direction of broken translational invariance are (i) periodic, and (ii) free, and (iii) fixed. Exact expressions for the magnetization profile are derived and studied. An extended, coordinate-dependent finite-size scaling is found to hold near the shifted critical temperature. Different scaling forms hold near the bulk critical temperature: in case (ii) the distance from the boundary scales with the finite-size correlation length, and in case (iii) with the linear size of the system.
A detailed analysis of the finite-size effects on the bulk critical behaviour of the d -dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different kinds of boundary conditions are applied: periodic ( p ), antiperiodic ( a ) and free surfaces with Dirichlet ( D ), Neumann ( N ) and a combination of Neumann and Dirichlet ( ND ) on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d . It is found, for 2 < d < 4, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for ( p ) and ( a ). For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At d = 3, the critical amplitude of the singular part of the free energy (related to the so-called Casimir amplitude) is estimated. We obtain Δ ( p ) = −2ζ(3)/(5π) = −0.153 051..., Δ ( a ) = 0.274 543... and Δ ( ND ) = 0.019 22..., implying a fluctuation-induced attraction between the surfaces for ( p ) and repulsion in the other two cases. For ( D ) and ( N ) we find a logarithmic dependence on L .
A detailed analysis of the finite–size effects on the bulk critical behaviour of the d–dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different boundary conditions are applied: periodic (p), antiperiodic (a) and free surfaces with Dirichlet (D), Neumann (N) and a combination of Neumann and Dirichlet (ND) on both surfaces. A systematic method for the evaluation of the finite–size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d. It is found, for 2 < d < 4, that the singular part of the free energy has the required finite–size scaling form at the bulk critical temperature only for (p) and (a). For the remaining boundary conditions the standard finite–size scaling hypothesis is not valid. At d = 3, the critical amplitude (related to the so called Casimir amplitude) is estimated. We obtain
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 .
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.
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.