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# II Critical finite-size scaling functions

[69.1.1.1] We consider the two-dimensional Ising model on a square lattice of side length . The spins interact according to the Hamiltonian where is the ferromagnetic coupling strength and is an external field. [69.1.1.2] The first summation runs over all nearest neighbour pairs on the lattice. [69.1.1.3] The order parameter is the magnetization per spin

 (1)

whose value fulfills . [69.1.1.4] In the following we set and also the Boltzmann constant to unity. [69.1.1.5] We denote the temperature by , and write for the magnetic field.

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

 (2)

where , is a Kronecker , and where is such that is an integer not larger than . [69.1.1.6] The probability density depends parametrically on temperature , field and system size ,

 (3)

It is also called order parameter distribution. [69.1.1.7] In the following we limit ourselves to the case , and hence . [69.1.1.8] The critical order parameter distribution is obtained in the limit and where 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

 (4)

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

 (5)

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

 (6)

in the finite-size scaling limit. [69.2.0.3] Here is the reduced temperature, is a universal scaling function and is the correlation length exponent. For the two-dimensional Ising model .

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

 (7)

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

 (8)

Our scaling variable is then

 (9)

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

 (10)

where

 (11)

[69.2.1.3] From these one calculates the so called renormalized coupling constant or the Binder cumulant

 (12)

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