Chemistry Reference
In-Depth Information
quantitative answers for nonuniversal observables such as the critical value of
the ratio
J
, which can only be obtained by a true quantum algorithm.
Sandvik and Scalapino
105
have performed quantum Monte Carlo simulations
of the bilayer Heisenberg quantum antiferromagnet employing the stochastic
series expansion method. By analyzing the staggered structure factor and the
staggered susceptibility, they found a critical ratio of
?
=
J
k
02
(see the vertical axis in Figure 6). More recently, Wang and co-workers
106
performed a high-precision study of the same model using the stochastic series
expansion algorithm with operator loop update.
100
Using the Binder cumu-
lant, the spin stiffness and the uniform susceptibility, they obtained a value
of
ð
J
?
=
J
k
Þ
c
¼
2
:
51
0
:
0001 for the critical coupling. In addition, they
computed the correlation length exponent and found
ð
J
?
=
J
k
Þ
c
¼
2
:
5220
0
:
0009,
which agrees within error bars with the best value of the three-dimensional
classical Heisenberg exponent
62
n
¼
0
:
7106
0
:
(as expected from the quantum-to-classical
mapping).
Diluted Heisenberg Magnets
In the example above, we saw that increased quantum fluctuations (as
induced by the interlayer coupling
J
in the bilayer system) can cause a quan-
tum phase transition in the two-dimensional Heisenberg quantum antiferro-
magnet. Another way to increase the fluctuations is by dilution, i.e., by
randomly removing spins from the lattice. The phases and phase transitions
of diluted Heisenberg quantum antiferromagnets have been studied exten-
sively during the last few years and many interesting features have emerged.
Consider the site-diluted square lattice Heisenberg model given by the
Hamiltonian
?
J
X
h
i
;
j
i
H
m
i
m
j
S
i
S
j
¼
½
41
where the
m
i
are independent random variables that can take the values 0 and 1
with probability
p
and 1
p
, respectively. As discussed above, the ground
state of the clean system
is aniferromagnetically ordered. It is clear
that the tendency toward magnetism decreases with increasing impurity
concentration
p
, but the location and nature of the phase transition toward
a nonmagnetic ground state remained controversial for a long time. The
most basic question to ask is whether the magnetic order vanishes before
the impurity concentration reaches the percolation threshold of the lattice
p
p
ð
p
¼
0
Þ
4072 (the transition would then be caused by quantum fluctuations)
or whether it survives up to
p
p
(in which case the transition would be of per-
colation type). Magnetic long-range order is impossible above
p
p
because the
lattice is decomposed into disconnected finite-size clusters. Early studies, both
0
:
