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Journal of Physics A Mathematical and General

Finite-size logarithmic corrections in the free energy of the mean spherical model

J.G. Brankov, Daniel M. Danchev

Published Sep 21, 1993• 2 citations
Publisher

Abstract

The validity of the finite-size scaling prediction about the existence of logarithmic corrections in the free energy due to corners is studied by the example of the mean spherical model. The general case of a hypercubic lattice of arbitrary dimensionality d>2, under boundary conditions which are periodic in d'>or=0 dimensions and free in the remaining d-d' dimensions is considered. The critical regime, as the size of the system L to infinity , is specified by the asymptotic behaviour of the ratio L/ xi L , where xi L is the correlation length fo the finite system. New results are the double-logarithmic corrections due to corners and logarithmic corrections due to one-dimensional edges in the regime L/ xi L varies as ln L which takes place at the bulk critical point.

DOI
10.1088/0305-4470/26/18/016

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The validity of the finite-size scaling prediction about the existence of logarithmic corrections in the free energy due to corners is studied by the example of the mean spherical model. The general case of a hypercubic lattice of arbitrary dimensionality d>2, under boundary conditions which are periodic in d'>or=0 dimensions and free in the remaining d-d' dimensions is considered. The critical regime, as the size of the system L to infinity , is specified by the asymptotic behaviour of the ratio L/ xi L , where xi L is the correlation length fo the finite system. New results are the double-logarithmic corrections due to corners and logarithmic corrections due to one-dimensional edges in the regime L/ xi L varies as ln L which takes place at the bulk critical point.
A constrained monomer-dimer (CMD) model on decorated hypercubic lattices, which is exactly solvable in any dimensionality d, is studied. The model is critical in the limit of maximal packing of the dimer configurations allowed by the constraint. For d=2 it is equivalent to the rooted-tree model on the square lattice considered by Duplantier and David (1988). The validity of two finite-size scaling predictions in higher dimensionalities is studied: (1) the existence of logarithmic corrections in the free energy due to corners, and (ii) the amplitude-exponent relation for the pair correlation function. The logarithmic finite-size corrections are obtained for arbitrary d, for boundary conditions which are periodic for d'>or=0 dimensions and free in the remaining d-d' dimensions. Dimer-dimer correlation functions are studied at d=3 for a system finite in two dimensions and infinite in the third. It is shown that the amplitude-exponent relation for the CMD model holds for d=2 and breaks down for d=3.
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 .
Explicit expressions are derived for the free energy, the specific heat, and the magnetic susceptibility of a spherical model of spins on a d-dimensional hypercubical lattice, of size L1{L}_{1}L1​\ifmmode\times\else\texttimes\fi{}L2{L}_{2}L2​\ifmmode\times\else\texttimes\fi{}...\ifmmode\times\else\texttimes\fi{}Ld{L}_{d}Ld​, under antiperiodic boundary conditions. The relevant scaling functions that govern the critical behavior of the system are obtained and, with the use of the asymptotic properties of these functions, various predictions of the Privman-Fisher hypothesis [Phys. Rev. B 30, 322 (1984)] on the hyperuniversality of finite systems are verified. Approach towards standard critical behavior, both for TTc{T}_{c}Tc​(\ensuremath{\infty}) and T>Tc{T}_{c}Tc​(\ensuremath{\infty}), is examined. In the former case, the approach generally takes place through a power law; only in some special situations does one obtain an exponential law instead. In the latter case, the approach is generally exponential, except for the susceptibility of the system which (somewhat surprisingly) displays a finite-size effect dominantly determined by the surface-to-volume ratio of the lattice.
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).
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
Abstract Logarithmic finite-size scaling of the O(n) universality class at the upper critical dimensionality (dc = 4) has a fundamental role in statistical and condensed-matter physics and important applications in various experimental systems. Here, we address this long-standing problem in the context of the n-vector model (n = 1, 2, 3) on periodic four-dimensional hypercubic lattices. We establish an explicit scaling form for the free-energy density, which simultaneously consists of a scaling term for the Gaussian fixed point and another term with multiplicative logarithmic corrections. In particular, we conjecture that the critical two-point correlation g(r, L), with L the linear size, exhibits a two-length behavior: follows r2−dcr^{2-d_c}r2−dc​ governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances whose height decays as L−dc/2(lnL)p^L^{-d_c/2}({\rm ln}L)^{\hat{p}}L−dc​/2(lnL)p^​ with p^=1/2\hat{p}=1/2p^​=1/2 a logarithmic correction exponent. Using extensive Monte Carlo simulations, we provide complementary evidence for the predictions through the finite-size scaling of observables, including the two-point correlation, the magnetic fluctuations at zero and nonzero Fourier modes and the Binder cumulant. Our work sheds light on the formulation of logarithmic finite-size scaling and has practical applications in experimental systems.