Chemistry Reference
In-Depth Information
1 for all even
i
while
H
2
contains those for all odd
i.
To
find a world-line representation we now insert this decomposition into the
Trotter formula [36]. Since
H
1
and
H
2
each consist of independent two-
site terms, the matrix elements in a
S
z
basis of
e
e
H
i
factorize completely
into terms of the type
between sites
i
and
i
þ
J
x
e
e
J
z
S
i
;
n
S
i
þ1;
n
S
i
;
n
S
i
þ;
n
exp
2
ð
S
i
S
i
þ1
þ
S
i
S
i
þ1
Þ
S
i
;
n
þ1
S
i
þ;
n
þ1
e
½
39
where
n
is the Trotter index. The remaining matrix elements are easily calcu-
lated. They read [with
h
J
x
ð
S
i
S
i
þ1
þ
S
i
S
i
þ1
Þ=
¼
2) as
e
e
h
e
e
h
hþ þ j
j þ þi ¼ h j
j i ¼
1
e
J
x
2
e
e
h
e
e
h
hþ j
j þ i ¼ h þ j
j þi ¼
cosh
½
40
e
J
x
2
e
e
h
e
e
h
hþ j
j þi ¼ h þ j
jþi¼
sinh
All other matrix elements are zero. The only nonvanishing matrix elements are
those between states with the same total spin in the two Trotter slices, reflect-
ing the spin conservation of the Hamiltonian. Note that the off-diagonal
matrix elements are negative if
J
x
is antiferromagnetic
. This prevents
interpreting the matrix elements as a statistical weight and is indicative of the
sign problem. However, for our one-dimensional chain, or more generally, on
any bipartite lattice, we can eliminate the sign problem by rotating every other
spin by 180
, which changes the sign of
J
x
.
The allowed spin configurations can be easily visualized in a
ð
J
x
>
0
Þ
-
dimensional space-time picture by drawing lines connecting space-time points
where the
z
component of the spin points up (see Figure 11). Because the num-
ber of such sites is conserved, the resulting ''world lines'' are continuous.
Moreover, the periodic boundary conditions implied by the trace require
that the world lines also connect continuously from the last imaginary time
slice to the first.
As the last ingredient for the Monte Carlo algorithm, we have to specify
the Monte Carlo moves within the restricted class of allowed spin configura-
tions. Single spin flips are not allowed, as they break the continuous world
lines. Instead, the simplest Monte Carlo moves consist of proposing a local
deformation of the world line (an example is shown in Figure 11) and accept-
ing or rejecting it with a suitable (Metropolis) probability that is determined
by the changes in the matrix elements involved. As discussed above,
algorithms based on such local moves suffer from critical slowing down
near a quantum critical point. In the more efficient loop algorithm,
94-96
one
ð
1
þ
1
Þ
