Chemistry Reference
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0and
h
i
>
where both
J
ij
>
0 are random functions of the lattice site. In one
space dimension, the critical behavior of the quantum phase transition can be
determined exactly
22,23
by means of the Ma-Dasgupta-Hu ''strong-disorder''
renormalization group.
24
This calculation predicts an exotic infinite-randomness
critical point, characterized by the following unusual properties: (1) the effective
disorder increases without limit under coarse graining (i.e., with increasing
length scale), (2) instead of the usual power-law dynamical scaling one now
has activated scaling, ln
x
t
/
x
c
,with
1
2
, (3) the probability distributions
of observables become very broad, even on a logarithmic scale, with their widths
diverging when approaching the critical point, (4) average and typical correla-
tions behave differently: At criticality, the average correlation function
C
av
c
¼
Þ
falls off with a power of the distance
r
, while the typical correlations decay
much faster, as a stretched exponential ln
C
typ
ð
r
r
c
. These results have
been confirmed by extensive and efficient numerical simulations
74,75
based on
mapping the spin systems onto free fermions.
76
In dimensions
d
ð
Þ/
r
1, an exact solution is not available because the strong
disorder renormalization group can be implemented only numerically.
19
Moreover, mapping the spin system onto free fermions is restricted to one
dimension. Therefore, simulation studies have mostly used Monte Carlo
methods. The quantum-to-classical mapping for the Hamiltonian in Eq. [30]
can be performed analogously to the clean case. The result is a disordered
classical Ising model in
d
>
þ
1 dimensions with the disorder perfectly correla-
ted in one dimension (in 1
1 dimensions, this is the famous McCoy-Wu
model
77,78
). The classical Hamiltonian reads
þ
X
X
H
cl
k
B
T
¼
n
ð
e
J
ij
Þ
S
i
;
n
S
j
;
n
K
i
S
i
;
n
S
i
;
n
þ1
½
31
h
i
;
j
i;
i
;
n
1
with
J
ij
>
0 being independent random variables.
Pich et al.
79
performed Monte Carlo simulations of this Hamiltonian in
0and
K
i
¼
2
ln coth
ð
e
h
i
Þ >
1 dimensions using the Wolff cluster algorithm.
51
As in the two examples
above, they used the scaling behavior of the Binder cumulant to find the criti-
cal point and to analyze the dynamical scaling. The resulting scaling plot is
shown in Figure 9. The figure shows that the curves do not scale when ana-
lyzed according to the usual power-law dynamical scaling,
2
þ
z
, but rather
get broader with increasing system size. In the inset, the data for
L
x
t
/
x
12 scale
x
t
/
x
c
,with
quite well according to activated scaling,
42. Pich
et al.
79
also studied the behavior of the correlation function at criticality.
They found a power-law decay of the average correlations and a stretched ex-
ponential decay of the typical correlations, as in one dimension. These results
provide strong simulational evidence for the quantum critical point in the two-
dimensional random transverse-field Ising model being of exotic infinite ran-
domness type. This agrees with the prediction of the numerically implemented
strong-disorder renormalization group
19
and with a general classification of
ln
c
0
:
