Chemistry Reference
In-Depth Information
To understand this remarkable observation, it is useful to distinguish
fluctuations with predominantly thermal or quantum character (depending
on whether their thermal energy
k
B
T
is larger or smaller than the quantum
energy scale
o
c
is the typical frequency of the fluctuations). As dis-
cussed in the last section, the typical time scale
h
o
c
, where
x
t
of the fluctuations generally
diverges as a continuous transition is approached. Correspondingly, the typi-
cal frequency scale
o
c
goes to zero and with it the typical energy scale
j
n
z
h
o
c
/j
r
½
11
Quantum fluctuations will be important as long as this typical energy scale is
larger than the thermal energy
k
B
T
. If the transition occurs at some finite tem-
perature
T
c
, quantum mechanics will therefore become unimportant if the dis-
tance
r
from the critical point is smaller than a crossover distance
r
x
given by
r
x
T
1=
n
c
. Consequently, we find that the critical behavior asymptotically
close to the transition is always classical if the transition temperature
T
c
is
nonzero. This justifies calling all finite-temperature phase transitions ''classical
transitions,'' even if they occur in an intrinsically quantum mechanical system.
Consider, as an example, the superconducting transition of mercury at 4.2K.
Here, quantum mechanics is obviously important on microscopic scales for
establishing the superconducting order parameter, but classical thermal fluc-
tuations dominate on the macroscopic scales that control the critical behavior.
In other words, close to criticality the fluctuating clusters become so big (their
typical size is the correlation length
/
) that they behave classically.
In contrast, if the transition occurs at zero temperature as a function of a
nonthermal parameter such as pressure or magnetic field, the crossover dis-
tance
r
x
vanishes; in this situation quantum mechanics is important for the cri-
tical behavior. Consequently, transitions at zero temperature are called
quantum phase transitions as described earlier. In Figure 2, we show the
resulting schematic phase diagram close to a quantum critical point. Suffi-
ciently close to the finite-temperature phase boundary, the critical behavior
is purely classical, as discussed above. However, the width of the classical cri-
tical region vanishes with vanishing temperature. Thus, an experiment along
path (a) at sufficiently low temperatures will mostly observe quantum beha-
vior, with a very narrow region of classical behavior (which may be unobser-
vable) right at the transition. The disordered phase comprises three regions,
separated by crossover lines. In the quantum disordered region at low tem-
peratures and when
B
x
B
c
, quantum fluctuations destroy the ordered phase,
and the effects of temperature are unimportant. In contrast, in the thermally
disordered region, the ordered phase is destroyed by thermal fluctuations while
the corresponding ground state shows long-range order. Finally, the so-called
quantum critical region is located at
B
>
B
c
and extends (somewhat counter-
intuitively) to comparatively high temperatures. In this regime, the system is
critical with respect to
B
, and the critical singularities are cut-off exclusively
