Chemistry Reference
In-Depth Information
Figure 11
World-line configuration for the XXZ Hamiltonian of [38]. The world lines
(thick lines) connect space-time points where the
z
component of the spin points up.
They can be either straight or cross the shaded squares, which show where the imaginary
time evolution operators
e
e
H
1
and
e
e
H
2
act. The dotted line shows the configuration
change after a local Monte Carlo update.
builds large world-line loops and then changes the spin direction along the
entire loop.
Let us now focus on the special case of the isotropic
Þ
Heisenberg quantum antiferromagnet on the square lattice (see also the
Hamiltonian of Eq. [19]). This model has played an important role in the his-
tory of high-temperature superconductivity studies because it describes the
magnetic properties of the copper oxide planes in the undoped parent cuprate
perovskites. An important early problem was establishing, beyond doubt, that
the ground state of the square lattice Heisenberg model is antiferromagneti-
cally (N´el) ordered and finding the value of the staggered magnetization.
Reger and Young
104
performed world-line Monte Carlo simulations of the
square lattice Heisenberg antiferromagnet using a two-dimensional version
of the algorithm described above. Since the ground state of the Heisenberg
model for any finite system size is rotationally invariant, the expectation value
of the staggered magnetization vanishes. To determine the macroscopic value,
which assumes that the symmetry has been broken spontaneously, Reger and
Young
104
computed both the (staggered) structure factor
S
ð
J
x
¼
J
z
>
0
ð
Q
Þ
at the ordering
wave vector
Q
¼ð
p
;
p
Þ
and the correlation function
C
L
=2
between spins as far
apart as possible on the lattice. Both quantities reduce to
m
s
=
3, where
m
s
is the
staggered magnetization in the thermodynamic limit. Figure 12 shows the
extrapolation of
S
ð
Q
Þ
and
C
L
=2
to infinite system size (the extrapolations
T
0 have already been carried out). From the intercept with
the vertical axis, Reger and Young found
m
s
¼
!
0 and
e
!
0
:
30
0
:
02 establishing
