Chemistry Reference
In-Depth Information
Here,
i
and
are the site indices in space- and the timelike direction, respec-
tively. The dynamic variable
J
¼ð
t
J
x
J
y
J
t
Þ
is a three-dimensional ''current''
with integer-valued components. It must be divergenceless, i.e., the sum over
all currents entering a particular site must vanish;
;
;
m
and
~
v
i
represent the che-
mical and random potentials, renormalized by
U.
To perform Monte Carlo simulations of the classical Hamiltonian, one
must construct updates that respect the zero divergence condition for the cur-
rents. This prevents using the usual type of cluster algorithms.
50,51
For this rea-
son, early simulations
81
used algorithms with local updates that suffered from
significant critical slowing down. Alet and Sorensen
84
developed a cluster
algorithm in which the link currents are updated by moving a ''worm''
through the lattice. This algorithm is efficient and performs comparably to
the Wolff algorithm
51
for classical spin systems. Alet and Sorensen first con-
firmed the three-dimensional
XY
universality class for the clean case at integer
boson density using this algorithm. In the presence of the random potential,
they found a different universality class with exponents
n
1
:
15 and
z
2.
QUANTUM MONTE CARLO APPROACHES
If one is only interested in the universal critical behavior of a quantum
phase transition, then the quantum-to-classical mapping method discussed in
the last section (if available) is usually the most efficient approach. If one is
also interested in nonuniversal quantities such as critical coupling constants
or numerical values of observables, however, the quantum system has to be
simulated directly. This can be done, for example, by quantum Monte Carlo
methods that are the topic of this section.
The name
quantum Monte Carlo
refers to a diverse class of algorithms
used for simulating quantum many-particle systems by stochastic means (for
an overview see Ref. 47). Some of these algorithms, such as variational
Monte Carlo
85,86
and diffusion Monte Carlo,
87,88
aim at computing the
ground-state wave function (and are thus zero-temperature methods). Other
algorithms including path-integral (world-line) Monte Carlo
89,90
sample the
density matrix at finite temperatures. Before discussing quantum phase tran-
sitions, it is useful to illustrate the wide spectrum of problems that can be
attacked by quantum Monte Carlo methods today along with the challenges
involved.
One branch of quantum Monte Carlo research aims at providing a quan-
titative first-principle description of atoms, molecules, and solids beyond the
accuracy of density functional theory.
48,49
If the basic physics and chemistry
of the material in question is well understood at least
qualitatively
, as is the
case for many bulk semiconductors, for example, good trial wave functions
such as the Jastrow-Slater type can be constructed. These functions can
