Spherical Model Of Ferromagnetism Research Articles

Interface formation and a structural phase transition for the spherical model of ferromagnetism

A detailed analysis is reported examining the local magnetic susceptibility χ(r), in relation to the correlation functionG(R) and correlation length ζ, of a spherical model ferromagnet confined to geometryΩ =Ld −d′ × ∞d′ (d′≤2,d>2) under a continuous set oftwisted boundary conditions. The “twist” parameter\(\underline \tau \) in this problem may be interpreted as a measure of the geometry-dependent doping level of interfacial impurities (or antiferromagnetic seams) in theextended system at various temperatures. For τj →0, ∀j∈d-d′, no seams are present except at infinity, whereas if τj = 1/2, impurity saturation occurs. For 0 L), defining the region between seams containing the origin, depends on temperature above a certain threshold (T>T0). Below that temperature (T>T0), seams are frozen at the same position (D≈L/2τ,d-d'=1), revealing a smoothly varying largescale structural phase transition.

Explicit results for the correlation length of the finite-sized spherical model of ferromagnetism

We report explicit results for the zero-field correlation length, xi (T;L), of a spherical-model ferromagnet confined to geometry Ld-d'* infinity d' (d>2, d'<or=2) and subjected to 'twisted' boundary conditions. In the region of first-order phase transition (T<Tc), xi under 'twisted' boundary conditions obeys the same scaling law as under periodic boundary conditions, though with a different amplitude. In the 'core' region (T approximately=Tc), the scaling behaviour changes radically as one moves from one set of boundary conditions to the other-affected greatly by the 'pinning' of the ground-state wavevector k0 of the system.

Exact results for a finite-sized spherical model of ferromagnetism at the borderline dimensionality 4.

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 ${\mathrm{c}}^{(\mathrm{s})}$ (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 (T${\mathrm{T}}_{\mathrm{c}}$), our results are formally the same as in other cases with dg2. The ``core'' region (T\ensuremath{\simeq}${\mathrm{T}}_{\mathrm{c}}$), 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.

Finite-size scaling of O(n) models with singular behaviour: Magnetization and susceptibility in the presence of an external field

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 &lt; d′ &lt; d &lt; 4) and subjected to periodic boundary conditions, is extended (i) to include the region of first-order phase transition (T &lt; 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 &gt; 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.

Finite-size scaling in arbitrary dimensions

The authors present the results of an analytical study of certain thermodynamic properties of an ideal relativistic Bose gas confined to the curved space S3*Rd-3, with 3<or=d<or=4. As d approaches the marginal dimensionality 4, the amplitudes pertaining to these properties display a singularity similar to that displayed by the correlation length xi c of a spherical model of ferromagnetism in the flat-space geometry Ld-d'* infinity d', with 2(d<or=4 and d'(2. These results are in sharp contrast with Cardy's (1985) generalization of xi c for a system in the space Sd-1*R1, with 2(d(4.

Finite-size scaling of O(n) models with long-range interactions.

A scaling hypothesis is set up for the magnetization m(T,H;L), susceptibility \ensuremath{\chi}(T,H;L), and correlation length \ensuremath{\xi}(T,H;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2) and long-range interactions decaying as 1/${r}^{d+\ensuremath{\sigma}}$ (0l\ensuremath{\sigma}l2), confined to geometry ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (\ensuremath{\sigma}ldl2\ensuremath{\sigma},d'\ensuremath{\le}\ensuremath{\sigma}) and subjected to periodic boundary conditions. Finite-size effects are predicted, in the region of a first-order phase transition (T${T}_{c}$) as well as in the region of a 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. In situations where L\ensuremath{\gg}\ensuremath{\xi}, finite-size corrections to standard bulk values of the various physical quantities pertaining to these models are found to vary as (\ensuremath{\xi}/L${)}^{d+\ensuremath{\sigma}}$, rather than as ${e}^{\mathrm{\ensuremath{-}}L/\ensuremath{\xi}}$, which is characteristic of models with short-range interactions.

Finite-size scaling of O(n) models with singular behaviour

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 &lt; d′ &lt; d &lt; 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.

Finite-size scaling of O(n) models in higher dimensions.

We propose a finite-size scaling hypothesis for the ``singular'' part of the free-energy density, ${f}^{(s)}$(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{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (dg4, 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, ${c}^{(s)}$(T;L), in the region of first-order phase transition (T${T}_{c}$) as well as the region of second-order phase transition (T\ensuremath{\simeq}${T}_{c}$). 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.

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

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.

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

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

Phase transitions in finite systems: Influence of geometry on approach towards bulk critical behavior.

We predict the manner in which a physical system, of size ${\mathrm{L}}^{\mathrm{d}\mathrm{\ensuremath{-}}\mathrm{d}\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\mathcal{'}}$, subject to periodic boundary conditions, approaches bulk critical behavior as L\ensuremath{\rightarrow}\ensuremath{\infty}. While for Tg${\mathrm{T}}_{\mathrm{c}}$(\ensuremath{\infty}) the approach is exponential, for T${\mathrm{T}}_{\mathrm{c}}$(\ensuremath{\infty}) it is generally governed by power laws whose indices are determined by the bulk exponents for the corresponding d- and d'-dimensional systems. Specific predictions on the spherical model of ferromagnetism and the relativistic Bose gas with pair production are verified by analytical results.