[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.