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Journal of Henan University

Finite-size Scaling and Scaling Functions of Cylindrical System in 3 Dimensions

Zhao Qiu-yu, Xiaoyu Li, Zhang De-xi

Published Jan 1, 2006• 0 citations
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Abstract

This paper investigates the finite-size scaling of a infinite long cylindrical system in 3 dimensions which is a domain between 2 dimensions and 3 dimensions.Using the exact solution to a O(n) symmetrical φ~4 field model while n→∞,we prove that finite-size scaling is still tenable in the 3-dimension cylindrical system.Then we give the critical exponent of the magnetic susceptibility and the specific heat capacity and confirm the universal class to which our 3-dimension cylindrical system belongs.Finally we get the scaling functions of the magnetic susceptibility and the specific heat capacity and compare them with the corresponding results in 3-dimension cubic system.

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This paper investigates the finite-size scaling of a infinite long cylindrical system in 3 dimensions which is a domain between 2 dimensions and 3 dimensions.Using the exact solution to a O(n) symmetrical φ~4 field model while n→∞,we prove that finite-size scaling is still tenable in the 3-dimension cylindrical system.Then we give the critical exponent of the magnetic susceptibility and the specific heat capacity and confirm the universal class to which our 3-dimension cylindrical system belongs.Finally we get the scaling functions of the magnetic susceptibility and the specific heat capacity and compare them with the corresponding results in 3-dimension cubic system.
Domains with a transitional dimension are very interesting in the research of finite-size effects.This paper investigates the finite-size effects of a infinite long cylindrical system in 3 dimensions which is a domain between 2 dimensions and 3 dimensions,studies this problem by using an O(n)-symmetrical φ~4 field model and gets the exact solation when n→∞,gives the exact expressions of magnetic susceptibility and specific heat capacity to this cylindrical system.These results can be used to study the finite-size scaling in the future work.
We have performed extensive studies of finite-size scaling in the three-dimensional Ising model, using a specially designed Ising processor. We show results for the susceptibility, the energy and specific heat, and the connected part of the fourth moment of the magnetization. Corrections to finite-size scaling are found to be small only for lattices of linear dimension greater than about L=24. This effect severely limits the accuracy to which critical parameters can be estimated. We estimate Kc{K}_{c}Kc​=0.221 650\ifmmode\pm\else\textpm\fi{}0.000 005 for the critical coupling, while exponent estimates are consistent with those obtained by other methods.
We study the sign distribution of generalized magnetic susceptibilities in the temperature-external magnetic field plane using the three-dimensional three-state Potts model. We find that the sign of odd-order susceptibility is opposite in the symmetric (disorder) and broken (order) phases, but that of the even-order one remains positive when it is far away from the phase boundary. When the critical point is approached from the crossover side, negative fourth-order magnetic susceptibility is observable. It is also demonstrated that non-monotonic behavior occurs in the temperature dependence of the generalized susceptibilities of the energy. The finite-size scaling behavior of the specific heat in this model is mainly controlled by the critical exponent of the magnetic susceptibility in the three-dimensional Ising universality class.
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.