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Physical review. B, Condensed matter

Spin-spin correlations in a finite-sized spherical model under twisted boundary conditions

Scott Allen, Raj K. Pathria

Published Sep 1, 1994• 10 citations
Publisher

Abstract

Finite-size effects in the correlation function G(R,T;L) of a spherical-model ferromagnet, confined to geometry ${\mathrm{L}}^{\mathrm{d}\mathrm{\ensuremath{-}}\mathrm{d}\ensuremath{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\ensuremath{'}}$ (2d4,d\ensuremath{'}\ensuremath{\leqslant}2) and subjected to twisted boundary conditions, are analyzed. Focusing our attention on the region of first-order phase transition (T${\mathrm{T}}_{\mathrm{c}}$), we examine the influence of the twist parameter \ensuremath{\tau} on the function G(R,T;L) in different regimes of the distance parameter \ensuremath{\varepsilon} (=R/L). We find that the functional dependence of the function G on the parameter \ensuremath{\varepsilon} is highly anisotropic, in that it depends very differently on the components, ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\perp}}}$ and ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\parallel}}}$, of \ensuremath{\varepsilon} pertaining to the finite and the infinite directions, respectively. While the long-range order in the direction of ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\perp}}}$ is significantly altered by the parameter \ensuremath{\tau}, it is curtailed very seriously in the direction of ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\parallel}}}$. In fact, for ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\parallel}}}$\ensuremath{\gg}1, the qualitative features of the correlation function acquire characteristics of a d\ensuremath{'}-dimensional bulk system, though the correlation length \ensuremath{\xi} appearing therein still pertains to the actual, finite-sized system. The net result is that long-range order in the direction of ${\mathbf{R}}_{\mathrm{\ensuremath{\parallel}}}$ persists only to distances small in comparison with \ensuremath{\xi} which, for d\ensuremath{'}2, is known to be O{L(L/a${)}^{(\mathrm{d}\mathrm{\ensuremath{-}}2)/(2\mathrm{\ensuremath{-}}\mathrm{d}\ensuremath{'})}$}, a being the lattice constant.

DOI
10.1103/physrevb.50.6765

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Finite-size effects in the correlation function G(R,T;L) of a spherical-model ferromagnet, confined to geometry Ld\ensuremath−d\ensuremath′{\mathrm{L}}^{\mathrm{d}\mathrm{\ensuremath{-}}\mathrm{d}\ensuremath{'}}Ld\ensuremath−d\ensuremath′\ifmmode\times\else\texttimes\fi{}\ensuremath∞d\ensuremath′{\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\ensuremath{'}}\ensuremath∞d\ensuremath′ (2d4,d\ensuremath{'}\ensuremath{\leqslant}2) and subjected to twisted boundary conditions, are analyzed. Focusing our attention on the region of first-order phase transition (TTc{\mathrm{T}}_{\mathrm{c}}Tc​), we examine the influence of the twist parameter \ensuremath{\tau} on the function G(R,T;L) in different regimes of the distance parameter \ensuremath{\varepsilon} (=R/L). We find that the functional dependence of the function G on the parameter \ensuremath{\varepsilon} is highly anisotropic, in that it depends very differently on the components, \ensuremathε\ensuremath⊥{\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\perp}}}\ensuremathε\ensuremath⊥​ and \ensuremathε\ensuremath∥{\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\parallel}}}\ensuremathε\ensuremath∥​, of \ensuremath{\varepsilon} pertaining to the finite and the infinite directions, respectively. While the long-range order in the direction of \ensuremathε\ensuremath⊥{\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\perp}}}\ensuremathε\ensuremath⊥​ is significantly altered by the parameter \ensuremath{\tau}, it is curtailed very seriously in the direction of \ensuremathε\ensuremath∥{\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\parallel}}}\ensuremathε\ensuremath∥​. In fact, for \ensuremathε\ensuremath∥{\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\parallel}}}\ensuremathε\ensuremath∥​\ensuremath{\gg}1, the qualitative features of the correlation function acquire characteristics of a d\ensuremath{'}-dimensional bulk system, though the correlation length \ensuremath{\xi} appearing therein still pertains to the actual, finite-sized system. The net result is that long-range order in the direction of R\ensuremath∥{\mathbf{R}}_{\mathrm{\ensuremath{\parallel}}}R\ensuremath∥​ persists only to distances small in comparison with \ensuremath{\xi} which, for d\ensuremath{'}2, is known to be O{L(L/a)(d\ensuremath−2)/(2\ensuremath−d\ensuremath′){)}^{(\mathrm{d}\mathrm{\ensuremath{-}}2)/(2\mathrm{\ensuremath{-}}\mathrm{d}\ensuremath{'})})(d\ensuremath−2)/(2\ensuremath−d\ensuremath′)}, a being the lattice constant.
Finite-size effects in the correlation function G(R,T;L)\ensuremath{\equiv}〈\ensuremath{\Phi}(0)\ensuremath{\cdot}\ensuremath{\Phi}(R)〉 of a spherical-model ferromagnet, confined to geometry ${\mathit{L}}^{\mathit{d}\mathrm{\ensuremath{-}}\mathit{d}\ensuremath{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathit{d}\ensuremath{'}}$ (d\ensuremath{\ge}4, ${\mathit{d}}^{\ensuremath{'}}$\ensuremath{\le}2) and subjected to twisted boundary conditions, are analyzed. Focusing attention on the region of first-order phase transition (TTc{\mathit{T}}_{\mathit{c}}Tc​), we examine the influence of the twist parameter \ensuremath{\tau} on the function G(R,T;L) in different regimes of the distance parameter \ensuremath{\varepsilon}(=R/L). Two complementary methodologies are employed: (i) dimensional regularization involving the limit d\ensuremath{\rightarrow}4 as approached from the mean-field regime at short separations \ensuremath{\varepsilon}\ensuremath{\ll}1, and (ii) \ensuremath{\zeta}-function regularization at comparably larger separations \ensuremath{\varepsilon}=O(1) using generators from more simplified classes of sums so as to properly handle the singular features of the correlation function at d=4. Results following from the two methodologies are found to be completely consistent with one another.
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.
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We study the correlation function G(R,T;L) and the correlation length \ensuremath{\xi}(T;L) in a finite spherical model if size ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ under periodic boundary conditions and emphasize the role of the variable L/\ensuremath{\xi}(L), rather than L/\ensuremath{\xi}(\ensuremath{\infty}), as a natural scaling variable of the system throughout the transition region, including temperatures below Tc{T}_{c}Tc​(\ensuremath{\infty}). We obtain a variety of finite-size effects, some of which may have validity for all O(n) models with n>2.
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