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Physica A Statistical Mechanics and its Applications

The influence of three external fields on the mean spherical model: magnetization profile

Magdy E. Amin

Published Mar 1, 2001• 2 citations
Publisher
DOI
10.1016/s0378-4371(00)00538-0

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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.
The coherent quantum dynamics of a single bosonic spin variable, subject to a constraint derived from the quantum spherical model of a ferromagnet, and coupled to an external heat bath, is studied through the Lindblad equation for the reduced density matrix. Closed systems of equations of motion for several quantum observables are derived and solved exactly. The relationship to the single-mode Dicke model from quantum optics is discussed. The analysis of the interplay of the quantum fluctuation and the dissipation and their influence on the relaxation of the time-dependent magnetisation leads to the distinction of qualitatively different regimes of weak and strong quantum couplings. Considering the model's behaviour in an external field as a simple mean-field approximation of the dynamics of a quantum spherical ferromagnet, the magnetic phase diagram appears to be re-entrant and presents a quantum analogue of well-established classical examples of fluctuation-induced order.