Let us begin with the first question. For *d*>2, the topological
glass state is a true ordered phase, described by a zero temperature
fixed point. The issue then is to determine whether this *T*=0 fixed
point is stable to the introduction of topological defects. To answer
this, we must determine if the total ground state energy of the system
is reduced relative to its elastic value in a configuration containing
such defects *at large scales*. By present at large scales, we
mean unbound defects, so that their presence is felt in long-distance
(or small *q*) correlation functions.

To get an idea of how to think about this question, let us consider a similar problem that arises in the pure two-dimensional XY model, i.e. a two-dimensional CDW with zero disorder. In the elastic approximation, the system has logarithmic correlations at finite temperature, since

The factor of *T* in from of the expression indicates that the
long-distance physics in this case is governed by a finite temperature
fixed line. In this case, one would like to know if this fixed line
is stable to the introduction of vortices. One can formulate a
precise RG calculation, but a simple scaling argument due to
Kosterlitz and Thouless gives the correct answer and a good physical
picture. Let us ask if a system of size *L* increases or lowers its
free energy by introducing a vortex. The energy cost of a single
vortex can be easily calculated. If we place the vortex at the
origin, a convenient choice for the phase configuration is , the polar angle. Then

which gives the elastic energy cost

where *a* is a short-distance cutoff (e.g. lattice size). There is
also an entropic contribution, which is just the logarithm of the
number of positions the vortex may take, or

The total free energy change is thus

Thus, for , a vortex costs free energy, and the elastic state is stable. For , it is favorable for the vortex to enter the system, and in fact one expects vortices to proliferate and destroy the elastic state.

This argument suggests we should consider the effect of introducing a
single topological defect into a system of size *L*. However, because
we are considering the stability of a zero-temperature fixed point,
it is simply the *energy* (and not the free energy) change which
must be negative for an instability. This is possible because the
system without topological defects is constrained. By releasing the
constraint, the system may be able to lower its energy.

In fact, I believe it is guaranteed that the total energy will
decrease. This is because there will always be rare regions of the
sample in which it is favorable to put local phase slips. However,
these are really *bound* topological defects, which do not destroy
the elasticity of the system at long distances. To destabilize the
elastic state, one needs to introduce extensive, i.e. unbound *L*
defects.

Instead of a balance between energy and entropy, therefore, we must
consider a balance between strain energy and disorder energy. Let us
consider these two contributions. The dimensionality of vortex or
dislocation defects depends upon *d*. For *d*=2, 3, and 4, they
are point, line, and surface defects, respectively. In general, the
defect dimensionality is . In each case, there is
at least a core energy cost, because the phases must have large
gradients in this core region (one may think of this core region in
the continuum as one in which is reduced to zero). This core
energy is always positive and scales as

What about the energy change from the rest of the region? Consider
placing the defect in the center of the size *L* region. In *d*=2,
place a vortex in the center of a circle; in *d*=3, anchor two ends of
a vortex line in the centers of two ends of a cylinder; etc. Now we
can draw a hyper-surface from the core to the boundary of the region.
In *d*=2, this is a radial line, in *d*=3 a fin-like surface. Now we
should calculate the ground state energy of this system given that we
allow to change discontinuously by a multiple of across
this hypersurface. In fact, we should also allow the position of this
surface to be optimized.

For *d*>4, we can answer this question very solidly. Recall that in
this case in the elastic limit, the strain energies dominated over the
disorder, and we had only bounded displacements of *u*. The random
contribution to the ground state energy therefore simply scales as
, because the medium cannot take advantage of spatially
variations in the potential. Now let us suppose we place a vortex in
the system. We can go to radial coordinates, in which are parallel to the defect, and *r* and are
perpendicular to it. Then let . Making
this change of variables gives

where *c* is a constant. Because elasticity dominates for *d*>4, one
can safely perturb in . To a very good approximation, then,
, and one has

The energy difference between the vortex ground state and the elastic one is thus

where *c*' and *c*'' are other constants. The first term is just the
elastic strain energy, the second is the core energy, and the third is
the disorder energy. For large *L*, the sum is always positive, since
*d*>4. So for *d*>4, the pinned state is certainly stable, at least
for weak disorder, to the introduction of topological defects.

As is apparent from Eq. 201, this energy balance
changes once *d*<4. This of course reflects the non-trivial nature of
the fixed point in low dimensions. What can we say about this most
important limit?

Very little can really be said persuasively. A rough scaling argument
suggests that the problem is a marginal one, and therefore leads to no
definite conclusion! To see how this works, consider our size *L*
system. We would like to know how the energy changes if we allow the
system to include the defect. As we have seen, this is equivalent to
allowing ``twisted'' boundary conditions across a surface connecting the
defect core and the system boundary. As before, there is a core
contribution to the energies of twisted systems. If we include this
explicitly, we should be able to ``remove'' the core and replace the
system by an annulus with twisted boundary conditions. If one unrolls
this annulus into a strip, the energy we are asking for is closely
related to the usual stiffness. This should be controlled by
the energy exponent , so that

This scales in the same way as the core energy, Eq. 198.
This means that vortices are *marginal*. Are they marginally
relevant or marginally irrelevant? Naively, if the disorder is weak,
then we would expect the core energy to dominate, and vortices to be
excluded.

However, this naive argument is suspect in the marginal case. Often
times, there are extremely subtle logarithmic effects which favor one
phase over another. It is easy to see that this may be the case here.
Indeed, by making the same change of variables as for *d*>4, one may show
that

with *c*>0 as before. This is another consequence of statistical rotational
invariance. It means that Eq. 202 cannot be quite
correct. Several possibilities exist:

1. has a mean value of , with much smaller sample-to-sample variations of order .

2. has a mean value of , with comparable or larger (in prefactor) sample-to-sample variations which scale in the same way (i.e. with the same logarithmic factor).

3. has a mean value of , with symmetric sample-to-sample variations which are larger in a scaling sense (perhaps logarithmically larger?).

4. has a mean value of . In a typical sample, however, is actually of order . The logarithm arises due to extremely rare samples with anomalous large (positive) coefficients, whose contribution grows somehow slowly with system size.

5. Our argument is completely incorrect, and the typical energy differences scale in a way unrelated to .

In case 1, the elastic state would be stable to disorder, while cases 2 and 3 would be unstable. Cases 4 would probably indicate instability, while case 5 indicates only our ignorance!

There are several other attacks at this issue in the literature.
Kierfeld, Natterman, and Hwa use various arguments to argue that case
5 obtains in three dimensions, and that vortices are *strongly
irrelevant*. I find their arguments somewhat unconvincing. Gingras
and Huse have attempted to study the problem numerically, using a
lattice random-field XY model. In two dimensions, they show
convincingly that vortices unbind for arbitrarily weak disorder. This
is not surprising, since vortices are marginal in the pure system in
*d*=2, and disorder should only make them more relevant. They also
find some evidence for the existence of the topologically ordered
phase in *d*=3, but cannot rule out a weak instability (i.e. that the
minimum size of unbound vortices diverges rapidly with decreasing
disorder, and thus quickly exceeds their system size). Clearly, this
is an important open problem, which would profit from some new
insights.

Thu May 30 08:21:44 PDT 1996