Chemistry Reference
In-Depth Information
Quantum Scaling and Quantum-to-Classical Mapping
In classical statistical mechanics, static and dynamic behaviors decouple.
Consider a classical Hamiltonian
H
consisting of
a kinetic part
H
kin
that depends only on the momenta
p
i
and a potential part
H
pot
that depends only on the coordinates
q
i
. The classical partition function
of such a system,
Z
ð
p
i
;
q
i
Þ¼
H
kin
ð
p
i
Þþ
H
pot
ð
q
i
Þ
¼
Ð
dp
i
e
H
kin
=
k
B
T
Ð
dq
i
e
H
pot
=
k
B
T
, factorizes into kinetic and
potential parts that are independent of each other. The kinetic contribution to
the free energy generally does not display any singularities since it derives from
a product of Gaussian integrals. One can therefore study the thermodynamic
critical behavior in classical systems using time-independent theories such as
the Landau-Ginzburg-Wilson theory discussed above. The dynamical critical
behavior can be found separately.
The situation is different in quantum statistical mechanics. Here, the
kinetic and potential parts of the Hamiltonian do not commute with each
other. Consequently, the partition function
Z
Tr
e
H
=
k
B
T
does not factorize,
and one must solve for the dynamics together with the thermodynamics. The
canonical density operator
e
H
=
k
B
T
takes the form of a time-evolution opera-
tor in imaginary time,
¼
h
. Thus, quantum
mechanical analogs of the LGW theory, Eq. [3], need to be formulated in
terms of space- and time-dependent fields. A simple example of such a quan-
tum LGW action takes the form
if one identifies 1
=
k
B
T
¼
it
=
1
=
ð
k
B
T
ð
d
d
x
2
2
S
½
f
¼
d
t
½
a
ð
q
t
f
ð
x
;
t
ÞÞ
þ
c
ðr
f
ð
x
;
t
ÞÞ
þ
F
L
ð
f
ð
x
;
t
ÞÞ
h
f
ð
x
;
t
Þ
½
12
0
with
being the imaginary time variable. This action describes, for example,
the magnetization fluctuations of an Ising model in a transverse field.
This LGW functional also illustrates another remarkable feature of
quantum statistical mechanics. The imaginary time variable
t
effectively
acts as an additional coordinate whose extension becomes infinite at zero
temperature. A quantum phase transition in
d
-space dimensions is thus equi-
valent to a classical transition in
d
t
1 dimensions. This property is called
the
quantum-to-classical mapping.
In general, the resulting classical system
is anisotropic because space and time coordinates do not enter in the same
fashion. A summary of the analogies arising from the quantum-to-classical
mapping is given in Table 2.
The homogeneity law, Eq. [5], for the free energy can be generalized easily
to the quantum case (see, e.g., Ref. 10). For the generic case of power-law dyna-
mical scaling, it takes the form
þ
b
ð
d
þ
z
Þ
f
rb
1=
n
;
hb
y
h
Tb
z
f
ð
r
;
h
;
T
Þ¼
ð
;
Þ
½
13
