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Journal of Xinyang Normal University

Exact Solutions to O(n)-symmetrical φ~4 Field Model of a Cylindrical System in 3 Dimensions

Zhao Qiu-yu

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

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.

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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.
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.
Abstract A number of numerical calculations, like Monte Carlo or transfer matrix calculations, are performed with systems in which the size in several or all dimensions is finite. To extrapolate the results to the infinite system, it is thus necessary to understand how the infinite size limit is reached. In particular in a system in which the forces are short range, no phase transition can occur in a finite volume, or in a geometry in which the size is infinite only in one dimension. This indicates that the infinite-size extrapolation is somewhat non-trivial. In this chapter, the problem is analysed in the case of second-order phase transitions, in the framework of the N-vector model. The existence of a finite-size scaling is established, extending renormalization group (RG) arguments to this new situation. Then, finite volume geometry and cylindrical geometry, in which the size is finite in all dimensions except one, are distinguished. It is explained how to adapt the methods used in the case of infinite systems to calculate the new universal quantities appearing in finite-size effects, for example, in d = 4−ϵ or d = 2+ϵ dimensions. Special properties of the commonly used periodic boundary conditions are emphasized.
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.
We demonstrate that the standard O(n) symmetric φ4 field theory does not correctly describe the leading finite-size effects near the critical point of spin systems with periodic boundary conditions on a d-dimensional lattice with d>4. We show that these finite-size effects require a description in terms of a lattice Hamiltonian. For n →∞ and n=1, explicit results are given for the susceptibility and for the Binder cumulant. They imply that these quantities do not have the universal properties predicted previously and that recent analyses of Monte Carlo results for the five-dimensional Ising model are not conclusive.
Abstract The one-dimensional Ising model with various boundary conditions is considered. Exact expressions for the thermodynamic and magnetic properties of the model using different kinds of boundary conditions [Dirichlet (D), Neumann (N), and a combination of Neumann–Dirichlet (ND)] are presented in the absence (presence) of a magnetic field. The finite-size scaling functions for internal energy, heat capacity, entropy, magnetisation, and magnetic susceptibility are derived and analysed as function of the temperature and the field. We show that the properties of the one-dimensional Ising model is affected by the finite size of the system and the imposed boundary conditions. The thermodynamic limit in which the finite-size functions approach the bulk case is also discussed.