II Critical finite-size scaling functions

[69.1.1.1] We consider the two-dimensional Ising model on a square lattice of side length L. The N=L2 spins σi=±1 interact according to the Hamiltonian H=-J⁢∑i,jσi⁢σj-H⁢∑i=1Nσi where J>0 is the ferromagnetic coupling strength and H is an external field. [69.1.1.2] The first summation ∑i,j runs over all nearest neighbour pairs on the lattice. [69.1.1.3] The order parameter is the magnetization per spin

whose value fulfills -1≤m≤1. [69.1.1.4] In the following we set J=1 and also the Boltzmann constant to unity. [69.1.1.5] We denote the temperature by T, and write h=H/kB⁢T for the magnetic field.

In this paper we focus on the probability density p⁢m of the order parameter defined as

where β=1/kB⁢T, δ⁢i,j=δi⁢j is a Kronecker δ, and where m is such that N⁢m/2 is an integer not larger than N/2. [69.1.1.6] The probability density p⁢m depends parametrically on temperature T, field h and system size N=L2,

It is also called order parameter distribution. [69.1.1.7] In the following we limit ourselves to the case h=0, and hence p⁢m=p⁢m;T,L. [69.1.1.8] The critical order parameter distribution is obtained in the limit L→∞ and T→Tc where Tc=2/arsinh⁢1≈2.2691853... is the critical temperature. [69.1.1.9] There are different ways of taking this limit (see [23] for an overview). [69.1.1.10] Traditionally this limit is understood as the finite-size scaling limit defined by

where ξ=ξ⁢T is the temperature dependent spin-spin-correlation length for the infinite system. [69.1.1.11] Note that in an infinite system ξ⁢T→∞ as T→Tc. [69.1.1.12] A second way to take the limit is the finite ensemble scaling limit defined through [23]

[69.2.0.1] All other possibilities for taking the limits are discussed in [23]. [69.2.0.2] It is often postulated that ξ fulfills the finite size scaling hypothesis

in the finite-size scaling limit. [69.2.0.3] Here t=T-Tc/Tc is the reduced temperature, ξ~⁢x is a universal scaling function and ν is the correlation length exponent. For the two-dimensional Ising model ν=1.

[69.2.1.1] The traditional finite-size scaling hypothesis [13, 24] for the critical order parameter distribution assumes that

where p~⁢x is the universal scaling function of the order parameter distribution and β is the order parameter exponent. [69.2.1.2] For the two-dimensional Ising model [25]

Our scaling variable is then

Using the scaling assumption (7) one obtains the absolute moments of the critical order parameter distribution

where

[69.2.1.3] From these one calculates the so called renormalized coupling constant g=m~4/m~22 or the Binder cumulant

which are often used in studies of critical behaviour because they are independent of L at criticality, if all the assumptions are valid.