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section on Classical Monte Carlo Approaches). This difference is most likely
caused by unpaired spins (uncompensated Berry phases) that exist in the
site-diluted single layer (but not in the dimer-diluted bilayer) and prevent
the quantum-to-classical mapping onto a classical Heisenberg model.
Because the ground state of the diluted Heisenberg model remains long-
range ordered up to the percolation threshold, one has to increase the quantum
fluctuations to induce a quantum phase transition for
p
p
p
. One way to
achieve this is by going to the dimer-diluted bilayer, as in the clean system,
and tuning the fluctuations with the interlayer coupling
J
?
. (The quantum
phase transitions in this system were discussed in the section on Classical
Monte Carlo Approaches.) Yu et al.
109
found a different way of increasing
the quantum fluctuations. They suggested introducing an
inhomogeneous
bond dilution, which is a scenario where not all bonds (interactions) are
removed with the same probability. If the occupation probabilities for
different types of bonds are chosen in such a way that the system preferably
forms dimers and ladders, a nontrivial quantum phase transition to a para-
magnetic ground state can be achieved while the underlying lattice is still in
the percolating phase.
<
Superfluid-Insulator Transition in an Optical Lattice
Having considered several examples of magnetic quantum phase
transitions, we now turn to the superfluid-insulator transition in many-boson
systems. In the section on Classical Monte Carlo Approaches we discussed
how the universal critical behavior of this transition can be determined
by mapping the Bose-Hubbard model, Eq. [32], onto the classical
-
dimensional link current Hamiltonian, Eq. [35], which can then be simulated
using classical Monte Carlo methods.
It is now possible to observe this transition experimentally in ultracold
atomic gases. For instance, in the experiments by Greiner et al.,
110
a gas of
87
Rb atoms was trapped in a simple cubic optical lattice potential. This system
is well described by the Bose-Hubbard Hamiltonian, Eq. [32], with an addi-
tional overall harmonic confining potential, and, the particle density as well as
the interparticle interactions can be controlled easily. To study the properties
of the gas in the experiment, the trapping and lattice potential were switched
off, and absorption images of the freely evolving atomic cloud were taken to
give direct information about the single-particle momentum distribution.
To provide quantitative predictions about how to detect the superfluid-
insulator transition in these experiments, Kashurnikov, Prokof'ev and
Svistunov
111
performed quantum Monte Carlo simulations of the single-
particle density matrix
ð
d
þ
1
Þ
r
ij
¼h
^
i
^
. They used the Bose-Hubbard model
with harmonic confining potential and carried out world-line Monte Carlo
simulations with the continuous-time Worm algorithm.
97
The diagonal
elements of the density matrix provide the real-space particle density, and
j
i
