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Glass transitions


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 tex2html_wrap_inline3212 -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 tex2html_wrap_inline3916 , 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, tex2html_wrap_inline3918 , 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 tex2html_wrap_inline3756 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 tex2html_wrap_inline3924 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 tex2html_wrap_inline3916 in each critical volume of size tex2html_wrap_inline3928 . The density is therefore


The definitions therefore imply




On general grounds, one therefore expects the universal relation


in the critical regime. Here the tex2html_wrap_inline3930 subscripts refer to tex2html_wrap_inline3932 and tex2html_wrap_inline3934 , respectively.

What do we expect for the properties of tex2html_wrap_inline3936 ? 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 tex2html_wrap_inline3932 , we expect a finite conductivity in the normal (liquid) phase. Therefore tex2html_wrap_inline3940 as tex2html_wrap_inline3942 , 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 tex2html_wrap_inline3944 , as tex2html_wrap_inline3942 . At the critical point, we should have a well-defined IV curve, which requires tex2html_wrap_inline3948 as tex2html_wrap_inline3950 . This fixes the power law


for tex2html_wrap_inline3952 . 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.

next up previous contents
Next: Driven dynamics Up: Beyond the elastic approximation Previous: Plastic phases

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