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Physical review. B, Condensed matter

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

Surjit Singh, Raj K. Pathria

Published Aug 1, 1988• 16 citations
Publisher

Abstract

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{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (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, ${c}^{(s)}$(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.

DOI
10.1103/physrevb.38.2740

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Physical review. B, Condensed matter•May 1, 1992•11 citations
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{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (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}L2(d\ensuremath−d′)/(4\ensuremath−d′){L}^{2(d\mathrm{\ensuremath{-}}d\mathcal{'})/(4\mathrm{\ensuremath{-}}d\mathcal{'})}L2(d\ensuremath−d′)/(4\ensuremath−d′)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.
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/${r}^{d+\ensuremath{\sigma}}$ (0<\ensuremath{\sigma}<2), confined to geometry ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (\ensuremath{\sigma}<d<2\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 (TTc{T}_{c}Tc​) as well as in the region of a 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. 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.
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.
A scaling hypothesis is set up for the correlation function G(R,T;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (2d4, d'\ensuremath{\le}2) and subjected to periodic boundary conditions. Finite-size effects in the correlation length \ensuremath{\xi}(T;L) are predicted 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​); for the correlation function, the predictions relate mostly to the regime TTc{T}_{c}Tc​. To test these predictions, a detailed analytical study is carried out in the case of the spherical model and all predictions are seen to be fully borne out. In the process we find that, though a single correlation length \ensuremath{\xi} governs the scaling of the variable R in all directions, the functional dependence of the correlation function on the scaled parameter R/\ensuremath{\xi} is highly anisotropic, in that it depends very differently on the components, ${\mathrm{R}}_{\ensuremath{\perp}}$ and R?{\mathrm{R}}_{?}R?​, of R pertaining to the finite and the infinite dimensions, respectively. Most importantly, while long-range order at temperatures below Tc{T}_{c}Tc​ prevails all the way in the direction of ${\mathrm{R}}_{\ensuremath{\perp}}$ it is severely limited in the direction of R?{\mathrm{R}}_{?}R?​. In fact, for R?{R}_{?}R?​\ensuremath{\gg}L, the qualitative features of the correlation function become characteristic of a d'-dimensional bulk system, though \ensuremath{\xi} still pertains to the actual, finite-sized system; the net result is that long-range order in the direction of R?{\mathrm{R}}_{?}R?​ persists only to distances small in comparison with \ensuremath{\xi} which, for d'2, is found to be O(L(L/a${)}^{(d\mathrm{\ensuremath{-}}2)/(2\mathrm{\ensuremath{-}}d\mathcal{'})}$), a being the lattice constant.
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.
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 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.