The finite-size scaling functions of thermodynamic functions in anisotropic systems have been shown to be dependent on the spatial anisotropy [X.S. Chen and V. Dohm, Phys. Rev. E 70, 056136 (2004)]. Here we extend this study to the correlation length ξ‖ of the anisotropic O (n) symmetric φ4 model in an Ld−1 × ∞ cylindric geometry with periodic boundary conditions. We calculate the exact finite-size scaling function of correlation length ξ‖ for T ≥ Tc in 2 < d < 4 dimensions and in the limit n → ∞. The finite-size scaling function of ξ‖ is dependent on a normalized symmetric (d − 1) × (d − 1) matrix defined by the anisotropy matrix of anisotropic systems.The finite-size scaling functions of thermodynamic functions in anisotropic systems have been shown to be dependent on the spatial anisotropy [X.S. Chen and V. Dohm, Phys. Rev. E 70, 056136 (2004)]. Here we extend this study to the correlation length ξ‖ of the anisotropic O (n) symmetric φ4 model in an Ld−1 × ∞ cylindric geometry with periodic boundary conditions. We calculate the exact finite-size scaling function of correlation length ξ‖ for T ≥ Tc in 2 < d < 4 dimensions and in the limit n → ∞. The finite-size scaling function of ξ‖ is dependent on a normalized symmetric (d − 1) × (d − 1) matrix defined by the anisotropy matrix of anisotropic systems.

International Journal of Modern Physics BSep 30, 20076 citations

Finite-size scaling of correlation functions in finite systems

Xin Zhang, Gaoke Hu, Yongwen Zhang +2 more

Science China Physics Mechanics and AstronomyOct 8, 201815 citations

Finite-size effects at critical points with anisotropic correlations: Phenomenological scaling theory and Monte Carlo simulations

Kurt Binder, Jian‐Sheng Wang

Journal of Statistical PhysicsApr 1, 1989162 citations

Nonuniversal finite-size scaling in anisotropic systems

Xiaofang Chen, V. Dohm

We study the bulk and finite-size critical behavior of the O(n) symmetric phi4 theory with spatially anisotropic interactions of noncubic symmetry in d<4 dimensions. In such systems of a given (d,n) universality class, two-scale factor universality is absent in bulk correlation functions, and finite-size scaling functions including the Privman-Fisher scaling form of the free energy, the Binder cumulant ratio, and the Casimir amplitude are shown to be nonuniversal. In particular it is shown that, for anisotropic confined systems, isotropy cannot be restored by an anisotropic scale transformation.We study the bulk and finite-size critical behavior of the O(n) symmetric phi4 theory with spatially anisotropic interactions of noncubic symmetry in d<4 dimensions. In such systems of a given (d,n) universality class, two-scale factor universality is absent in bulk correlation functions, and finite-size scaling functions including the Privman-Fisher scaling form of the free energy, the Binder cumulant ratio, and the Casimir amplitude are shown to be nonuniversal. In particular it is shown that, for anisotropic confined systems, isotropy cannot be restored by an anisotropic scale transformation.

arXiv:cond-mat/0408511Physical Review ENov 30, 200465 citations

Spin-spin correlations in finite systems with O(n) symmetry: Scaling hypothesis and corrections to bulk behavior

Surjit Singh, Raj K. Pathria

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 (T ${T}_{c}$ ) as well as the region of second-order phase transition (T\ensuremath{\simeq} ${T}_{c}$ ); for the correlation function, the predictions relate mostly to the regime T ${T}_{c}$ . 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 ${\mathrm{R}}_{?}$ , of R pertaining to the finite and the infinite dimensions, respectively. Most importantly, while long-range order at temperatures below ${T}_{c}$ prevails all the way in the direction of ${\mathrm{R}}_{\ensuremath{\perp}}$ it is severely limited in the direction of ${\mathrm{R}}_{?}$ . In fact, for ${R}_{?}$ \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 ${\mathrm{R}}_{?}$ 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 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 (T${T}_{c}$) as well as the region of second-order phase transition (T\ensuremath{\simeq}${T}_{c}$); for the correlation function, the predictions relate mostly to the regime T${T}_{c}$. 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 ${\mathrm{R}}_{?}$, of R pertaining to the finite and the infinite dimensions, respectively. Most importantly, while long-range order at temperatures below ${T}_{c}$ prevails all the way in the direction of ${\mathrm{R}}_{\ensuremath{\perp}}$ it is severely limited in the direction of ${\mathrm{R}}_{?}$. In fact, for ${R}_{?}$\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 ${\mathrm{R}}_{?}$ 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.

Physical review. B, Condensed matterSep 1, 198733 citations

Finite Size Corrections in Quantum Correlated Chains

IMPERIAL COLLEGE PRESS eBooksMay 1, 20050 citations

{T}_{c}

) as well as the region of second-order phase transition (T\ensuremath{\simeq}

{T}_{c}

); for the correlation function, the predictions relate mostly to the regime T

{T}_{c}

. 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

{\mathrm{R}}_{?}

, of R pertaining to the finite and the infinite dimensions, respectively. Most importantly, while long-range order at temperatures below

{T}_{c}

prevails all the way in the direction of ${\mathrm{R}}_{\ensuremath{\perp}}$ it is severely limited in the direction of

{\mathrm{R}}_{?}

. In fact, for

{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

{\mathrm{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.