Chemistry Reference
In-Depth Information
where the second index of the spin operator distinguishes the two layers. For
J
, the corresponding spins in the two layers form singlets that are mag-
netically inert (i.e.,
J
?
J
k
increases the fluctuations away from the classical N´el
state). Thus, the system is in the paramagnetic phase. In contrast, for
J
?
,
each layer orders antiferromagnetically, and the weak interlayer coupling
establishes global antiferromagnetic long-range order. There is a quantum
phase transition between the two phases at some
J
?
?
J
k
.
We now map this quantum phase transition onto a classical transition.
Chakravarty and co-workers
60
showed that the low-energy behavior of two-
dimensional quantum Heisenberg antiferromagnets is generally described by a
ð
J
k
2
þ
1
Þ
-dimensional quantum rotor model with the Euclidean action
"
X
#
1
Z
=
k
B
T
X
1
2
g
2
S
¼
d
t
ð
q
t
n
i
ð
t
ÞÞ
n
i
ð
t
Þ
n
j
ð
t
Þ
½
21
i
h
i
;
j
i
0
or by the equivalent continuum nonlinear sigma model. Here
n
i
ð
t
Þ
is a three-
dimensional unit vector representing the
staggered
magnetization. For the
bilayer Hamiltonian Eq. [20], the rotor variable
n
i
ð
t
Þ
represents
S
i
;1
S
i
;2
while the conjugate angular momentum represents
S
i
;1
þ
S
i
;2
(see Chapter 5
of Ref. 10). The coupling constant
g
is related to the ratio
J
J
?
. By reinter-
preting the imaginary time direction as an additional space dimension, we
can now map the rotor model Eq. [21] onto a three-dimensional classical Hei-
senberg model with the Hamiltonian
k
=
K
X
h
H
cl
k
B
T
¼
n
i
n
j
½
22
i
;
j
i
Here the value of
K
is determined by the ratio
J
and tunes the phase tran-
sition. (Since the interaction is short ranged in space and time directions, the
anisotropy of Eq. [21] does not play a role in the critical behavior.)
As in the first example, the classical system arising from the quantum-to-
classical mapping is a well-known model of classical statistical mechanics.
While it is not exactly solvable, its properties are known with high precision
from classical Monte Carlo simulations.
61,62
The critical exponents of the
phase
k
=
J
?
transition
are
a
0
:
133,
b
0
:
369,
g
1
:
396,
d
4
:
783,
n
037. Because space and time directions enter Eq. [22]
symmetrically, the dynamical exponent is
z
0
:
711, and
Z
0
:
1. The critical behavior of
observables can be obtained from the homogeneity relation of Eq. [13] as
before. Note that the field
h
appearing in the homogeneity relation is
not
a uni-
form magnetic field but rather the field conjugate to the antiferromagnetic
order parameter, i.e., it is a staggered magnetic field. Including a uniform mag-
netic field in the quantum-to-classical mapping procedure leads to complex
¼
