Chemistry Reference
In-Depth Information
.
If the transition is continuous, will the ''dirty'' system show the same
critical behavior as the clean one or will the universality class change?
.
Will only the transition itself be influenced or will the behavior also be
changed in its vicinity?
An important early step toward answering these questions came from the
work of Harris
18
who considered the stability of a critical point against disor-
der. He showed that if a clean critical point fulfills the exponent inequality
d
n
>
2
½
10
now called the Harris criterion, it is perturbatively stable against weak disor-
der. Note, however, that the Harris criterion only deals with the average beha-
vior of the disorder at large length scales; effects due to qualitatively new
behavior at finite length scales (and finite disorder strength) are not covered.
Thus, the Harris criterion is a necessary condition for the stability of a clean
critical point, not a sufficient one.
The Harris criterion can serve as the basis for a classification of critical
points with quenched disorder according to the behavior of the average disor-
der strength with increasing length scale. Three classes can be distinguished:
19
(1) The first class contains critical points fulfilling the Harris criterion. At these
phase transitions, the disorder strength
decreases
under coarse graining, and
the system becomes homogeneous at large length scales. Consequently, the cri-
tical behavior of the dirty system is identical to that of the clean system.
Macroscopic observables are self-averaging at the critical point, i.e., the rela-
tive width of their probability distributions vanishes in the thermodyna-
mic limit.
20,21
A prototypical example is the three-dimensional classical
Heisenberg model whose clean correlation length exponent is
711 ful-
filling the Harris criterion. (2) In the second class, the system remains inhomo-
geneous at all length scales with the relative strength of the disorder
approaching a finite value for large length scales. The resulting critical point
still displays conventional power-law scaling, but it is in a new universality
class with exponents that differ from those of the clean system (and fulfill
the inequality
d
n
0
:
2). Macroscopic observables are
not
self-averaging, but
in the thermodynamic limit, the relative width of their probability distribu-
tions approaches a size-independent constant. An example in this class is the
classical three-dimensional Ising model. Its clean correlation length exponent,
n
n
>
629, does not fulfill the Harris criterion. Introduction of quenched dis-
order, for example, via dilution, thus leads to a new critical point with an
exponent of
0
:
684. (3) At critical points in the third class, the relative
magnitude of the disorder counterintuitively
increases
without limit under
coarse graining. At these so-called infinite-randomness critical points, the
power-law scaling is replaced by activated (exponential) scaling. The probabil-
ity distributions of macroscopic variables become very broad (even on a loga-
rithmic scale) with their width diverging with system size. Infinite-randomness
n
0
:
