As before, several approaches to the problem are possible[13]. I will present the RG approach, which so far has only been developed for the CDW case. I believe that the extension to vortex lattices is do-able, but may involve some non-trivial mathematics.
In the RG approach, we again search for a zero temperature fixed
point. However, because the potential 
is periodic, if we intend
to find such a fixed point, we must choose 
to preserve the
period of 
. Rescaling as
this automatically implies
if such a fixed point exists. This is again an exact result due to rotational invariance, and implies that the lower critical dimension
Under this rescaling, the random vector term becomes
Assuming Gaussian correlations,
we find the renormalization
Near d=4, where the T=0 RG works, this term is therefore strongly irrelevant. We will therefore neglect it, at least until we discuss d=2, where this form suggests it may become important.
Having disposed of the unfamiliar term, the mechanics of the RG
reduces to a problem we have worked out before. We define the
correlator 
by
It obeys the RG equation
This equation was worked out first by D. S. Fisher in the context of the random-field XY model.
The substantive differences between the CDW and the random manifold
(with N=1) are that here and that we are looking for a
-periodic fixed point for . We can do this in a simple
way by defining 
. Differentiating
Eq. 182 twice gives
This equation has a fixed point with
The solution on the full real axis is obtained by simply repeating
Eq. 184 periodically in . This fixed point
solution was obtain by O. Narayan and D. S. Fisher in the context of
driven CDWs.
Using the fixed point solution, we may calculate the displacement correlations. By summing the scale-dependent contributions to the T=0 correlator at O(R), we obtain the result
At higher order in 
, we expect logarithmic correlations to
persist, but that the coefficient will be corrected. This result was
first obtained by T. Giamarchi and P. le Doussal.
It is natural to think that these logarithmic phase correlations lead to power-law order-parameter behavior. For a Gaussian field, we have
However, the distribution of 
is clearly non-Gaussian, and
correlations of exponentials can be highly sensitive to the form of
the distribution function. Particularly away from 
, we
must take a power-law form for the 
correlations as only a
conjecture. Another likely possibility is 
.
Within the RSB framework, these correlations can also be obtained and agree quite well with Eq. 185. The RSB calculations have also been extended to models of vortex lattices, where again logarithmic correlations are predicted.
The result for has an extremely nice physical picture. It
may be obtained as the two-point correlation function of a periodic
random potential composed of piecewise parabolic segments. The
location (in ) of the matching point is an independent,
uniformly distributed random variable at each point is space. The cusp
in 
then corresponds to the discontinuity of the random
force at this value of .
This construction allows for a nice physical picture of metastability.
At each scale, we are minimizing over the position of the modes at
scale L, for a given set of modes at longer scales. The effective
potential seen by each mode at scale L is a sum of the parabolic
elastic term (which is O(1)) and of a small ( 
)
randomly displaced piecewise parabolic potential. For most values of
the larger scale modes, the minimum is unique, since the parabolic
piece dominate the effective potential. If, however, it happens that
the joining point of two parabolas falls very near (within
) to 
, then the cusp leads to two local
minima. It can be shown that the RG picks the lower energy
state. The presence of these minima, and the overall energy scale 
implies the existence of metastable states
with energy differences and barriers of that order.
We can view these two local minima as a sort of splitting of the
states of the CDW. That is, given a particular choice of modes at
larger L, the system can make two choices at this scale. But
similar choices obtain as well at larger scales. This implies a kind
of tree structure of the metastable states in the system. In the
region of applicability of the RG, the probability of splitting is
very small, of at any scale.
This cusp singularity in is in fact a general property
of the RG results for random manifolds as well. A recent point of
comparison has been made with the RSB approach with J.-P. Bouchaud and
M. Mezard. One finds that this singularity can be obtained within the
RSB approach as well, and furthermore, that the picture of a piecewise
parabolic effective potential is in fact equivalent to the RSB
variational solution.
