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Zero temperature fixed point

 

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 tex2html_wrap_inline3644 is periodic, if we intend to find such a fixed point, we must choose tex2html_wrap_inline3188 to preserve the period of tex2html_wrap_inline2792 . Rescaling as

eqnarray1266

this automatically implies

equation1269

if such a fixed point exists. This is again an exact result due to rotational invariance, and implies that the lower critical dimension

equation1271

Under this rescaling, the random vector term becomes

equation1274

Assuming Gaussian correlations,

equation1280

we find the renormalization

equation1287

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 tex2html_wrap_inline3658 by

equation1292

It obeys the RG equation

  equation1298

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 tex2html_wrap_inline3188 and that we are looking for a tex2html_wrap_inline2792 -periodic fixed point for tex2html_wrap_inline3658 . We can do this in a simple way by defining tex2html_wrap_inline3668 . Differentiating Eq. 182 twice gives

equation1304

This equation has a fixed point with

  equation1307

The solution on the full real axis is obtained by simply repeating Eq. 184 periodically in tex2html_wrap_inline2792 . 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

  equation1315

At higher order in tex2html_wrap_inline3212 , 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

eqnarray1322

However, the distribution of tex2html_wrap_inline3438 is clearly non-Gaussian, and correlations of exponentials can be highly sensitive to the form of the distribution function. Particularly away from tex2html_wrap_inline3680 , we must take a power-law form for the tex2html_wrap_inline3050 correlations as only a conjecture. Another likely possibility is tex2html_wrap_inline3684 .

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 tex2html_wrap_inline3658 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 tex2html_wrap_inline3438 ) of the matching point is an independent, uniformly distributed random variable at each point is space. The cusp in tex2html_wrap_inline3690 then corresponds to the discontinuity of the random force at this value of tex2html_wrap_inline3438 .

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 ( tex2html_wrap_inline3700 ) 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 tex2html_wrap_inline3700 ) to tex2html_wrap_inline3704 , 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 tex2html_wrap_inline3706 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 tex2html_wrap_inline3700 at any scale.

This cusp singularity in tex2html_wrap_inline3690 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.


next up previous contents
Next: Behavior in d=2 Up: Topological Glasses Previous: Inclusion of disorder

Leon Balents
Thu May 30 08:21:44 PDT 1996