Chemistry Reference
In-Depth Information
¼
P
i
H
i
such that the matrix elements
with respect to some basis can be calculated easily, giving
Hamiltonian into two or more terms
H
!
n
n
n
1
1
X
1
n
1
k
B
T
1
n
1
k
B
T
Tr
e
H
=
k
B
T
ð
H
n
H
i
Z
¼
¼
Tr
Þ
¼
Tr
!
!
n
¼0
n
¼0
i
½
37
Inserting complete sets of basis states between the different
H
i
factors then
leads to a similar representation of the partition function and a similar
world-line picture as in the world-line Monte Carlo method. Because there
is no Trotter decomposition involved, the method is free of time discretization
errors from the outset. Early applications of the SSE method employed local
updates, but more efficient cluster-type updates have been developed more
recently to overcome the critical slowing down. They include the operator-
loop update
100
and the previously mentioned directed-loop algorithm.
98
The source code for some of the algorithms discussed above is available
on the Internet as part of the ALPS (Algorithms and Libraries for Physics
Simulations) project.
101
SSE programs for the Heisenberg model can also be
found on the homepage of A. Sandvik.
102
Spin-
2
Quantum Heisenberg Magnet
We will use our first example, the spin-
2
quantum Heisenberg magnet, to
further illustrate the world-line quantum Monte Carlo method. The model we
present is the quantum XXZ model:
X
X
J
x
H
J
x
ð
S
i
S
j
þ
S
i
S
j
Þþ
J
z
S
i
S
j
¼
2
ð
S
i
S
j
þ
S
i
S
j
Þþ
J
z
S
i
S
j
¼
½
½
38
h
i
;
j
i
h
i
;
j
i
where
S
i
,
S
i
,
S
i
are the components of the quantum spin-
2
operator at site
i
,
S
i
;
S
i
are the associate raising and lowering operators, and the sum is over
all pairs of nearest neighbors. We now divide the Hamiltonian into pieces
such that the matrix elements of each piece can be evaluated easily. For
the XXZ Hamiltonian, a convenient choice is the so-called checkerboard
decomposition.
103
We illustrate it by considering one space dimension (see
also Figure 10). We then write
H
¼
H
1
þ
H
2
where
H
1
contains the bonds
Figure 10
Checkerboard decomposition of the one-dimensional XXZ Hamiltonian.
