PHYSICAL REVIEW B VOLUME 50, NUMBER 5 1 AUGUST 1994-I Finite-size scaling in the presence of an inhomogeneous external field: An analytical-model treatment J. G. Brankov Institute of Mechanics, Bulgarian Academy of Sciences, 1113So+a, Bulgaria N. S. Tonchev Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 So+a, Bulgaria (Received 4 March 1994) 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 exter- nal 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 expres- sions 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 correla- tion length, and in case (iii) with the linear size of the system. I. INTRODUCTION Vanishing external fields are usually used to break the symmetry of the Hamiltonian and to single out pure Gibbs phases in the low-temperature region. For that purpose the amplitude H of the field is set to zero after the thermodynamic limit is taken. This approach, ap- plied to classical and quantum systems with various symmetry-breaking sources, constitutes the basis of Bogoliubov's definition of quasiaverages. ' A generalized version of Bogoliubov's quasiaverages makes use of symmetry-breaking fields with a size- dependent amplitude H' ', which tends to zero simul- taneously with the unlimited increase of the linear size L of the system. For example, one may set H' '~L as L ~ ~, with some a) 0. When the thermodynamic limit is taken at fixed density of the number N of particles, i.e. , for a d-dimensional system in a space domain L, the ra- tio X /L"=c osnt as L~ao, one may set alternatively H'~' ~ N ". The generalized quasiaverage approach has been suggested in Ref. 2 and used to explore the set of zero-field-limit Gibbs states of some exactly solvable models: the Curie-Weiss-Ising ferromagnet, the n-vector Curie-Weiss model, and the spherical model with nearest-neighbor interaction. It has been realized ' that the generalized quasiaver- age approach, with field amplitudes vanishing according to a suitably chosen power law, provides a constructive procedure for the explicit calculation of finite-size scaling functions at both second-order and first-order phase tran- sitions. In the case of a second-order phase transition, one has to consider the system at temperatures T' which approach the critical temperature T, simultane- ously with L ~ ~; the appropriate choice, predicted by the finite-size scaling theory, is T' '/T, — 1~L where v is the critical exponent of the correlation length. Most of the works on finite-size scaling (see Ref. 8 and references therein) have focused on the case of uniform external fields (sources) which break rotational (gauge) symmetries. Hypotheses are most readily tested on the example of the mean spherical model, for which a variety of analytical results has been obtained (see the re- cent Refs. 10 — 12 and references therein). Vanishing uni- form fields have been applied in that model to study the scaling behavior with respect to the magnetic vari- able. ' ' Bulk and surface fields have been used in a de- tailed investigation of the surface properties, in particular the variation of the susceptibility with the distance from the surface. ' Inhomogeneous fields, switched off after the thermo- dynamic limit, appeared to be useful tools for investigat- ing phase separation and surface and interface phenome- na. For example, a steplike inhomogeneous external field, which breaks the translational invariance in one space dimension, has been applied in a study of the prob- lem of phase separation in the mean spherical model. ' The spherical model in a magnetic field with the same spatial dependence, but with an amplitude vanishing simultaneously with L ~ ~, has been considered by Pa- trick. ' Interesting phenomena have been found to occur in the regime of moderate rate of decrease of the ampli- tude H' '~L: the leading O(L ) correction term of the free-energy density exhibits a new singularity with respect to the temperature at T= T, ', where 0& T, * & T„ below T, there appears a "frozen" (temperature- independent), smooth magnetization profile. Qualitative- ly similar phenomena were shown to appear under the ac- tion of surface magnetic fields. ' The aim of the present work is to study the validity of finite-size scaling in systems subjected to inhomogene- ous external fields. Obviously, the problem is a part of the more general investigations on finite-size effects in spatially inhomogeneous systems. In the present work we consider the mean spherical 0163-1829/94/50(5)/2970(8)/$06. 00 50 2970 1994 The American Physical Society