Chemistry Reference
In-Depth Information
Figure 7
(
Upper panel
) Binder cumulant of the classical Hamiltonian, Eq. [29], at the
critical point as a function of
L
t
for various
L
and impurity concentration
p
5
.(
Lower
panel
) Power-law scaling plot of the Binder cumulant. (
Inset
) Activated scaling plot of
the Binder cumulant. (Taken with permission from Ref. 71.)
¼
1
Sknepnek, Vojta and Vojta
71
and Vojta and Sknepnek
72
have performed
large-scale Monte Carlo simulations of the classical Hamiltonian, Eq. [29], by
means of the Wolff cluster algorithm.
51
Because of the disorder, all simula-
tions involve averages over a large number (up to several 10,000) of disorder
configurations. Let us first discuss the generic transition (
p
p
p
). As explained
above, the scaling behavior of the Binder cumulant can be used to self-
consistently find the critical point and the dynamical exponent
z.
A typical
result of these calculations is presented in Figure 7. It shows the Binder cumu-
lant at the critical point for a system with impurity concentration
p
<
5. As
seen in the main panel of this figure, the data scale very well when analyzed
according to power-law scaling while the inset shows that they do not fulfill
activated (exponential) scaling. Analogous data were obtained for impurity
concentrations
¼
1
=
7
,and
3
. The dynamical exponent of the generic transition
now follows from a power-law fit of the maximum position
L
ma
t
vs.
L
,
as shown in Figure 8. Taking the leading corrections to scaling into account
gives a universal value
z
1
2
8
,
31. The correlation length exponent can be deter-
mined from the off-critical finite-size scaling of the Binder cumulant, giving
n
1
:
2 as required for a
sharp transition in a disordered system (see discussion on quenched disorder
in the section on Phase Transitions and Critical Behavior). Analyzing the mag-
netization and susceptibility data at criticality yields
1
:
16. Note that this value fulfills the inequality
d
n
>
b
=
n
0
:
53,
g
=
n
2
:
26.
