We have been spending most of our time discussing the nature of the ordered phases of these elastic models. One can also consider the phase transitions to the high-temperature disordered phases. These would seem a priori less relevant, since the critical point occupies only a small portion of the phase diagram, while the ordered phase fills an extensive region.
In these glassy systems, however, when extremely long time-scales develop in the ordered phase, it can be very difficult to observe reproducible equilibrium behavior at low temperatures. The critical regime, where time-scales are only starting to grow large, then can prove a useful test of our theoretical understanding. Probably the most fundamental question we would like to answer in this regard is: does an ordered phase exist? One way to probe this question is to look for a demonstration of critical behavior. If the critical point exists, it must separate the system into two phases.
What sorts of critical points will arise here? At this point we have very little controlled understanding of their properties. In the CDW or random-field XY model, if the topologically ordered phase is stable, we expect a finite temperature transition to the liquid or paramagnetic state. Formal RG treatments have been made of the phase transition in a random field O(N) model. One finds that the critical fixed point is actually driven to T=0 for d<6, and an -expansion may be developed [see. e.g. Boyanovsky and Cardy, Phys. Rev. B 27, 4447 (1983)]. The physical cases of interest are, however, rather far for six dimensions, so that this expansion is expected to be rather poor. In fact, it is unclear to me whether the transition might not be governed by a more prosaic finite temperature fixed point in low dimensions.
The vortex glass phase is closer to a conventional spin glass, and is therefore expected to have a conventional finite temperature critical fixed point. Simple scaling therefore makes fairly strong predictions for the behavior near , the glass temperature. Provided that the elastic vortex glass-liquid transition is also conventional in this sense, the same scaling predictions (albeit with different exponents) should hold there as well.
In the critical regime, , of a continuous transition, both the correlation length
and correlation time
grow large. Here I have assumed that the critical behavior scales isotropically, so that correlation lengths in all directions diverge with the same power. An anisotropic generalization can be found in Nelson and Vinokur, Phys. Rev. B 48, 13060 (1993). The scaling assumption, which holds for most quantities near the critical point, is that these large length scales dominate over all other shorter length scales, and lead to universal physics as functions of these quantities. In the critical regime, there is generally a third exponent describing the power-law decay of correlations in the critical regime.
Without an explicit theory (which requires some small control parameter), we cannot compute the exponents. However, the general predictions of scaling theory do apply. As an illustration, let us consider the scaling prediction for the current-voltage relation in the superconductor near such a putative glass transition. First, we need to relate the current and electric field to fields in the Ginzburg-Landau theory. As usual, the current density may be defined
where f is the free energy density, and is the vector potential. From Maxwell's equation, the electric field is
Gauge invariance provides an essential simplification to the theory. It requires that all gradients be taken in the form of covariant derivatives,
as in the Ginzburg-Landau model. Because this gauge invariance is an exact symmetry, it must be preserved by scaling. This implies that the vector potential scales like
The so-called ``hyperscaling'' relation determines the scaling of the free energy density. One expects that in the critical regime, there will be a fluctuation energy of order in each critical volume of size . The density is therefore
The definitions therefore imply
On general grounds, one therefore expects the universal relation
in the critical regime. Here the subscripts refer to and , respectively.
What do we expect for the properties of ? It is universal but different for different types of transitions, e.g. vortex glass to liquid and elastic glass to liquid. Some properties of its limits are constrained by physical considerations. For instance, for , we expect a finite conductivity in the normal (liquid) phase. Therefore as , which implies that the resistivity
In the glass phase, we expect exponentially small voltages (i.e. vortex velocities) from a small applied current, so that , as . At the critical point, we should have a well-defined IV curve, which requires as . This fixes the power law
for . These general scaling predictions have indeed been observed in many different high-temperature superconductors. In the vortex glass phase, numerics predict values of the exponents in agreement with the measured values.