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Physical review. B, Condensed matter

Finite-size effects in the spherical model of ferromagnetism: Zero-field susceptibility under antiperiodic boundary conditions

Surjit Singh, Raj K. Pathria, Michael Ellis Fisher

Published May 1, 1986• 10 citations
Publisher

Abstract

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 ${T}_{c}$(\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 ${T}_{c}$(\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.

DOI
10.1103/physrevb.33.6415

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Phase transitions in finite systems with O(n) symmetry: Magnetization and susceptibility in the presence of an external field

Surjit Singh, Raj K. Pathria

A scaling hypothesis is set up for the magnetization m(T,H;L) and susceptibility \ensuremath{\chi}(T,H;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${\mathit{L}}^{\mathit{d}\mathrm{\ensuremath{-}}\mathit{d}\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathit{d}\mathcal{'}}$(2d4,d'\ensuremath{\le}2) and subjected to periodic boundary conditions. Finite-size effects in both m and \ensuremath{\chi} are predicted, in the region of the first-order phase transition (T${T}_{c}$) as well as in the region of the second-order phase transition (T\ensuremath{\simeq}${T}_{c}$), for different regimes of the parameters H and L. To test these predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions based on the scaling hypothesis are seen to be fully borne out.

Physical review. B, Condensed matter•May 1, 1988•9 citations
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.
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 analysis of a previous study (Allen and Pathria. Can. J. Phys. 67, 952 (1989)) on finite-size effects in 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, is extended (i) to include the region of first-order phase transition (T < Tc) as well as the region of second-order phase transition [Formula: see text] and (ii) to allow the presence of an external field H > 0. Predictions, involving both amplitudes and exponents, are made on the magnetization m and susceptibility χ in different regimes of the variables T, H, and L. Analytical verification of the predicted results is carried out in the case of the spherical model of ferromagnetism (n = ∞), and complete agreement is found.
We present the results of the first analytical study of finite-size effects in a random-field model. In particular we consider the random-field spherical model in partially infinite geometries (infinite in ${d}^{\ensuremath{'}}$ dimensions and finite in $d\ensuremath{-}{d}^{\ensuremath{'}}$ dimensions) under periodic and antiperiodic boundary conditions in the finite directions. We find a simultaneous dimensional reduction by 2 in ddd and ${d}^{\ensuremath{'}}$ for 4<d<64<d<64<d<6 and $2<~{d}^{\ensuremath{'}}<~4$. For $0<~{d}^{\ensuremath{'}}<2$, we find new power laws describing the approach to bulk in different geometries.
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.
We report exact results on 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){\mathrm{c}}^{(\mathrm{s})}c(s) (T;L) of a finite-sized spherical model of ferromagnetism subjected to periodic boundary conditions. We take the total dimensionality of the system d to be 4 and deal with the geometry ${\mathrm{L}}^{4\mathrm{\ensuremath{-}}\mathrm{d}\ensuremath{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\ensuremath{'}}$ (d\ensuremath{'}\ensuremath{\leqslant}2). In the region of first-order phase transition (TTc{\mathrm{T}}_{\mathrm{c}}Tc​), our results are formally the same as in other cases with d>2. The ``core'' region (T\ensuremath{\simeq}Tc{\mathrm{T}}_{\mathrm{c}}Tc​), however, is characterized by the appearance of factors involving lnL, which appear only when d=4. The relationship between these results and the corresponding ones following from the hyperscaling regime as d\ensuremath{\rightarrow}4- or from the mean-field regime as d\ensuremath{\rightarrow}4+ is explored, and a formulation in terms of the finite-size scaling theory is presented.
A scaling hypothesis is set up for the magnetization m(T,H;L) and susceptibility \ensuremath{\chi}(T,H;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${\mathit{L}}^{\mathit{d}\mathrm{\ensuremath{-}}\mathit{d}\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathit{d}\mathcal{'}}$(2d4,d'\ensuremath{\le}2) and subjected to periodic boundary conditions. Finite-size effects in both m and \ensuremath{\chi} are predicted, in the region of the first-order phase transition (TTc{T}_{c}Tc​) as well as in the region of the second-order phase transition (T\ensuremath{\simeq}Tc{T}_{c}Tc​), for different regimes of the parameters H and L. To test these predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions based on the scaling hypothesis are seen to be fully borne out.