Chemistry Reference
In-Depth Information
analytical and numerical, gave values between 0.07 and 0.35 for the critical
impurity concentration, suggesting a transition driven by quantum fluctua-
tions.
Sandvik
107
performed quantum Monte Carlo simulations of
the
Heisenberg Hamiltonian on the critical infinite percolation cluster
p
p
Þ
using the stochastic series expansion method with operator loop update.
100
He computed the staggered structure factor and from it the staggered
ground-state magnetization of the cluster. Figure 13 shows the extrapolation
of this quantity to infinite system size. The data demonstrate that the ground
state is magnetically ordered, with a sizable staggered magnetization of about
m
s
ð
p
¼
150 (roughly half the value of the undiluted system). This means that
even right at the percolation threshold
p
p
, the quantum fluctuations are not
strong enough to destroy the magnetic long-range order. The phase transition
to a paramagnetic ground state occurs right at
p
p
. It is driven by the geometry
of the underlying lattice and is thus of percolation type. More recently, Wang
and Sandvik
108
studied the dynamical quantum critical behavior of this transi-
tion (the static behavior is given by classical percolation). They found a
dynamical critical exponent of
z
¼
0
:
3
:
7, which is much larger than the value
91
48
found for the dimer diluted bilayer
72,73
z
¼
D
f
¼
(see discussion in the
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.00
0.05
0.10
0.15
1/
L
d
/2
, 1/
N
1/2
Figure 13
Squared staggered ground-state magnetization of the Heisenberg model on a
site-diluted lattice at
p
p
p
. The two curves correspond to two different ways of
constructing the percolation clusters in the simulation. (
Solid circles
) Largest cluster on
L
¼
L
lattices. (
Open Circles
) Clusters with a fixed number
N
c
sites. (Taken with
permission from Ref. 107.)
