An Investigation of Finite Size Scaling

Ramon Villanova, Nelson A. Alves, Bernd A. Berg

E. Brézin

Calculations on finite (or partially finite) systems, such as Monte-Carlo simulations or the diagonalization of a finite transfer matrix, are often extrapolated to the thermodynamic limit by the use of scaling assumptions involving the finite sizes L of the system. It is shown here that these finite size scaling laws follow from the fact that : i) in the renormalization group flow equations the length L is not renormalized ; ii) there is no singularity at the fixed point. This second property fails in four (and above four) dimensions and finite size scaling does not hold there. An illustration of these points is presented, with detailed calculations of the large N limit of the N- vector model. As a consequence, the ε-expansion of finite size scaling properties is singular.

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

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.

E. Brézin

A. P. Young