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arXiv:cond-mat/0305470 · Journal of Physics A Mathematical and General

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

Daniel Dantchev, J.G. Brankov

Published Aug 12, 2003• 5 citations
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Abstract

Exact results of the finite-size behaviour of the susceptibility in three-dimensional mean spherical model films under Dirichlet–Dirichlet, Dirichlet–Neumann and Neumann–Neumann boundary conditions are presented. The corresponding scaling functions are explicitly derived and their asymptotics close to, above and below the bulk critical temperature T c are obtained. The results can be incorporated in the framework of the finite-size scaling theory where the exponent λ characterizing the shift of the finite-size critical temperature with respect to T c is smaller than 1/ν, with ν being the critical exponent of the bulk correlation length.

DOI
10.1088/0305-4470/36/34/301

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Exact results of the finite-size behaviour of the susceptibility in three-dimensional mean spherical model films under Dirichlet–Dirichlet, Dirichlet–Neumann and Neumann–Neumann boundary conditions are presented. The corresponding scaling functions are explicitly derived and their asymptotics close to, above and below the bulk critical temperature T c are obtained. The results can be incorporated in the framework of the finite-size scaling theory where the exponent λ characterizing the shift of the finite-size critical temperature with respect to T c is smaller than 1/ν, with ν being the critical exponent of the bulk correlation length.
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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 .
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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.
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 χ1,1 has been evaluated exactly. For ρ = 1 we find that χ1,1 is finite at the bulk critical temperature Tc, in contrast with the recently derived value γ1,1 = 1 in the case of just one global spherical constraint. The result γ1,1 = 1 is recovered only if ρ = ρc = 2 − (12Kc) −1, where Kc is the dimensionless critical coupling. When ρ> ρc, χ1,1 diverges exponentially as T → T + c. An effective hamiltonian which leads to an exactly solvable model with γ1,1 = 2, the value for the n → ∞ limit of the corresponding O(n) model, is proposed too.
We calculate finite-size effects of the Gaussian model in a Lx(d-1) box geometry with free boundary conditions in one direction and periodic boundary conditions in d-1 directions for 2<d<4. We also consider film geometry (L--> infinity ). Finite-size scaling is found to be valid for d<3 and d>3 but logarithmic deviations from finite-size scaling are found for the free energy and energy density at the Gaussian upper borderline dimension d*=3. The logarithms are related to the vanishing critical exponent 1-alpha-nu=(d-3)/2 of the Gaussian surface energy density. The latter has a cusplike singularity in d>3 dimensions. We show that these properties are the origin of nonscaling finite-size effects in the mean spherical model with free boundary conditions in d > or =3 dimensions. At bulk T(c), in d=3 dimensions we find an unexpected nonlogarithmic violation of finite-size scaling for the susceptibility chi approximately L3 of the mean spherical model in film geometry, whereas only a logarithmic deviation chi approximately L2 ln L exists for box geometry. The result for film geometry is explained by the existence of the lower borderline dimension d(l)=3, as implied by the Mermin-Wagner theorem, that coincides with the Gaussian upper borderline dimension d*=3. For 3<d<4 we find a power-law violation of scaling chi approximately L(d-1) at bulk T(c) for box geometry and a nonscaling temperature dependence chi(surface) approximately xi(d) of the surface susceptibility above T(c). For 2<d<3 dimensions we show the validity of universal finite-size scaling for the susceptibility of the mean spherical model with free boundary conditions for both box and film geometry and calculate the corresponding universal scaling functions for T > or =T(c).