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Physical review. B, Condensed matter

Finite-size effects in the spherical model of ferromagnetism: Antiperiodic boundary conditions

Surjit Singh, Raj K. Pathria

Published Oct 1, 1985• 21 citations
Publisher

Abstract

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 ${L}_{1}$\ifmmode\times\else\texttimes\fi{}${L}_{2}$\ifmmode\times\else\texttimes\fi{}...\ifmmode\times\else\texttimes\fi{}${L}_{d}$, 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 T${T}_{c}$(\ensuremath{\infty}) and T>${T}_{c}$(\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.

DOI
10.1103/physrevb.32.4618

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Physical review. B, Condensed matter•Aug 1, 1988•16 citations
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.
The Privman-Fisher hypothesis on the singular part of the free-energy density of a finite system, near the bulk critical point T=Tc{T}_{c}Tc​, is examined in the context of the spherical model of ferromagnetism. A d-dimensional hypercubical lattice (of size N1{N}_{1}N1​a\ifmmode\times\else\texttimes\fi{}N2{N}_{2}N2​a\ifmmode\times\else\texttimes\fi{}...\ifmmode\times\else\texttimes\fi{}Nd{N}_{d}Nd​a) is considered and, subject to periodic boundary conditions, explicit expressions are derived for the free energy, the specific heat, and the magnetic susceptibility of the system at temperatures close to Tc{T}_{c}Tc​. The relevant scaling functions governing 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 are verified. By implication, the passage of the given system towards standard bulk behavior, as Nj{N}_{j}Nj​\ensuremath{\rightarrow}\ensuremath{\infty}, is also elucidated.
The finite-size scaling hypothesis of Privman and Fisher (Phys. Rev. B, 30, 322 (1984)) is applied to 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. Predictions, involving amplitudes as well as exponents, are made on the singular part of the specific heat c(s), spontaneous magnetization m0, magnetic susceptibility χ0, and correlation length ξ0 in the region of the second-order phase transition [Formula: see text]. Analytical verification of these predictions is carried out in the case of the spherical model of ferromagnetism (n = ∞), which includes evaluation of the shift in the critical temperature of the system.
The overall zero-field susceptibility \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{} of a finite-sized spherical model of spins under various antiperiodic boundary conditions is reexamined with a view to explaining the finite-size effects of an algebraic nature found recently by Singh and Pathria. The cause of this ``unexpected'' behavior at temperatures above the bulk critical temperature Tc{T}_{c}Tc​(\ensuremath{\infty}) is seen to lie in the spatial variation of the local susceptibility which, on averaging over the system, leads precisely to the effects found previously. Below Tc{T}_{c}Tc​(\ensuremath{\infty}), the influence of antiperiodic conditions is even more severe, in that not only are the finite-size amplitudes for \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{} modified but, for the local susceptibility, new exponents also appear.
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 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 .
We propose a finite-size scaling hypothesis for the ``singular'' part of the free-energy density, f(s){f}^{(s)}f(s)(T,H;L), and the correlation function, G(R,T,H;L), of a system with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (d>4, d'\ensuremath{\le}2) and subjected to periodic boundary conditions. On the basis of this hypothesis, we predict finite-size effects in the zero-field susceptibility, \ensuremath{\chi}(T;L), the correlation length, \ensuremath{\xi}(T;L), and the ``singular'' part of the specific heat, c(s){c}^{(s)}c(s)(T;L), in the region of first-order phase transition (TTc{T}_{c}Tc​) as well as the region of second-order phase transition (T\ensuremath{\simeq}Tc{T}_{c}Tc​). It turns out that in both these regions the system in zero field can be described by a single, unifying variable v\ensuremath{\sim}${L}^{2(d\mathrm{\ensuremath{-}}d\mathcal{'})/(4\mathrm{\ensuremath{-}}d\mathcal{'})}$t\ifmmode \tilde{}\else \~{}\fi{}, where t\ifmmode \tilde{}\else \~{}\fi{} is an appropriate temperature parameter of the system. To test the various predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions are seen to be fully borne out. For general n, we compare our findings with those of the previous authors, wherever possible, and in each case the comparison turns out to be perfect.