Chemistry Reference
In-Depth Information
elements of each exponential term
e
e
H
i
can be calculated easily. Even if the
H
i
do not commute, we can use the Trotter product formula to decompose the
canonical density operator
!
N
Y
e
H
=
k
B
T
e
e
H
i
¼
lim
N
½
36
!1
i
with
k
B
TN
. Inserting complete sets of states between the different
factors leads to a representation of the Boltzmann factor in terms of matrix
elements of the
e
e
H
i
. If all of these matrix elements are positive, their product
can be interpreted as a statistical weight, and Monte Carlo algorithms can be
constructed to sample this weight. (If some of the matrix elements are
negative, we have an instance of the notorious sign problem in quantum
Monte Carlo.) The
N
factors of the Trotter decomposition can be interpreted
as
N
time slices in the imaginary time direction, and the spin or particle con-
figurations form ''world lines'' in the resulting
e
¼
1
=
-dimensional space-
time. This gives the method its name. A specific implementation of the
world-line Monte Carlo method will be discussed in our first example later
in this tutorial. More details of this technique can also be found in Chapter 3
of Ref. 47.
Applications of the world-line algorithm to quantum phase transitions
require three extrapolations: (1) to infinite system size, (2) to temperature
T
ð
d
þ
1
Þ
0. The first two extrapolations
can be handled conveniently by finite-size scaling in the space and time direc-
tions, respectively. The systematic error of the Trotter decomposition arising
from a finite
!
0, and (3) to imaginary time step
e
!
was originally controlled by an explicit extrapolation from
simulations with different values of
e
. In 1996, Prokofev et al. showed that
(at least for quantum lattice models) the algorithm can be formulated in con-
tinuous time, taking the limit
e
0 from the outset.
93
World-line Monte
Carlo algorithms with local updates of spin or particle configurations suffer
from critical slowing down close to quantum critical points. This problem is
overcome by using the loop algorithm
94
and its continuous time version.
95
These algorithms, which are generalizations of the classical cluster algo-
rithms
50,51
to the quantum case, have been reviewed in detail in Ref. 96.
Further improvements for systems without spin-inversion or particle-hole
symmetry include the worm algorithm
97
and the directed loop method.
98
e
!
Stochastic Series Expansion
The stochastic series expansion (SSE) algorithm
91,92
is a generalization
of Handscomb's power-series method
99
for the Heisenberg model. To derive
an SSE representation of the partition function, we start from a Taylor
expansion in powers of the inverse temperature. We then decompose the
