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Plan of the notes

The notes are divided into two sections. The first half focuses on the systems I calling random manifolds, which comprise various topological defects embedded in larger disordered environments. These are the most studied and best understood of elastic systems, yet nevertheless are quite rich physically and conceptually. The second half describes the even richer physics of random elastic media, which are bulk ``lattices'' of various types pinned by disorder. There are numerous important experimental examples and considerable new physics arises beyond the manifold models.

The notes begin In section 3 with a discussion of the ferromagnetic Ising model. First, the ordered and disordered equilibrium phases are described and the concepts of the correlation length, order parameter, and susceptibility are reviewed and reinterpreted in terms of the renormalization group (RG). Scaling ideas and the general concepts of the RG play an important role in our understanding of these systems, and here the basic ideas are presented in an elementary way in order to familiarize the reader with some of the terminology. Next, to motivate the study of elastic models, we consider the behavior of the magnet after a rapid quench from high temperatures into the ordered phase. This coarsening dynamics, i.e. growth of the ordered state, is governed by the motion of domain walls. In a pure system, only the curvature forces drive the coarsening, and a simple argument determines how the characteristic length of ferromagnetic correlations grows with time. Next, we consider the addition to the Ising model of a small fraction of random bonds. Because the interfaces are almost flat in the long-time regime, however, the curvature forces become extremely small. One therefore expects that, in a random magnet, impurity forces then dominate over the curvature forces.

We are thus faced with the interesting problem of the behavior of such a domain wall in the presence of impurities. This is in fact only a single example of the more general elastic manifold model, which is introduced in section 4. It can be written as a continuum field theory, parameterized by the number of longitudinal and transverse dimensions of the manifold, d and N, respectively. The case N=1 corresponds to the interface already described, while d=1 is the so-called directed polymer (DP) problem, which will be returned to in section 9.

Given the manifold model, we would like to first understand how many phases it may have. This is considered in section 5, using primarily RG ideas. In particular, we consider the possibility of a stable high-temperature phase, in which thermal fluctuations ``average out'' the effects of impurities. This provides a second excercise in RG ideas, as the problem may be posed as the determination of the stability of the pure ``thermal'' fixed point. This stability analysis is carried out using standard momentum-shell RG techniques. One finds that the thermal fixed point is in fact unstable in all the physical instances, and a natural hypothesis is instead that the system is ordered, i.e. governed by a zero temperature fixed point at long distances.

If this hypothesis is correct, it should be sufficient to consider the behavior of a manifold at T=0 to determine most long-distance properties of the system at any temperature. In section 6, the simplest possible picture of such a zero-temperature manifold is described. The essential ingredients of the picture are power-law scaling of energy variations and manifold roughness, characterized by two universal exponents and a number of scaling functions. We will see that the ``strong'' scaling assumptions made here for simplicity appear to fit all the known results, and also arises quite naturally in several approximate treatments to follow. In section 7, we consider the application of these ideas to the original problem of coarsening in a random bond magnet. The picture is seen to lead to a dramatic slowdown of phase ordering.

Next, to justify the picture, we turn to the two systematic approaches developed for the problem. These are the replica symmetry breaking (RSB) method and the RG. Due to lack of space, I have only presented a sketch of the the RSB method and its results (section 8.1). With the help of Marc's lectures and the references cited here, the interested reader should be able to pursue the ideas further. The results summarized there anticipate those that are obtained in section 8.2 using the RG. Both approaches point to the existence of a ``renormalized'' probability distribution for a single low-momentum (large-scale) mode, which can be characterized by an overall energy scale and an effective (random) potential. The existence of a length-scale dependent energy landscape is the essential justification of the assumptions of the scaling picture. The methods also provide approximate values for the scaling exponents.

Aside from these systematic approaches, a number of additional results are known for the DP model (d=1), which we turn to specifically in section 9. These are of particular interest because of applications to magnetic flux lines in type II superconductors (section 9.1) and to the growth of surfaces by random deposition (section 9.2). The new results are described in section 9.3. They include the beautiful numerical results of Kardar and Zhang demonstrating the tree structure of constrained ground states of DPs, exact results for the thermal fluctuations around the ground state at low temperatures, and special ``fluctuation-dissipation theorem'' results for N=1, which determine exactly the scaling exponents in that case.

In the second half, we turn to elastic media, which are periodic structures (i.e. lattices) acted upon by impurities. These are true thermodynamic systems, and have the additional features of symmetry breaking and lattice defects.

We begin with a discussion of some interesting physical realizations and the corresponding theoretical models (section 10). I have included charge density waves (CDWs), vortex lattices, and Wigner crystals. Of these, the CDW is simplest, but all may be described using elastic theories, given in 10.1. The ordered phases break translational and sometimes other symmetries, and appropriate order parameters are defined in section 10.3. By defining such order parameters, one implicitly allows for the presence of topological defects, which go beyond the elastic description. Several types exist, and are described in section 10.4.

A first attack on these systems is to consider their equilibrium behavior within elasticity, i.e. neglecting topological defects. If a phase exists in which this approach is valid, i.e. defects are bound, it will be called a ``topological glass''. Section 11.1 discusses how disorder is included in the elastic models. It is shown that the CDW with disorder is equivalent to a random field XY model. In section 11.2, the RG technique of section 8.2 is applied to solve this problem for tex2html_wrap_inline2658 . It leads to an exact result for the energy scaling, an approximate one for order parameter correlations, and a rather complete picture of the energy landscape in the model. The lower critical dimension for the model is d=2, and I briefly summarize a current controversy surrounding the behavior in two dimensions in section 11.3.

In section 12, we consider the new physics which may arise beyond the elastic approximation. The first problem is to determine the stability of the topological glass phase to defects. A simple argument is given in section 12.1, suggesting that it is indeed marginally stable if the disorder is weak. This conclusion cannot, however, be regarded as firm, and this remains as an important open problem. If the disorder is not weak, elasticity will definately break down. Several distinct new phases may arise in the vortex case, including the vortex glass, discussed in section 12.2. While the topological glass to liquid transition is probably first order, the vortex-glass to liquid critical behavior may well be continuous. A very successful scaling theory to describe the transport in this regime in described in section 12.3.

The final section (13) deals with depinning phenomena in driven elastic media. We consider the motion of an elastic system Under the influence of a driving force, such as, e.g. an electric field for a CDW, a current for a superconductor, or a magnetic field for a ferromagnetic domain wall. In section 10.2, I discuss recent theoretical progress in understanding such depinning behavior in purely elastic models. Once topological defects are allowed, much less is understood. In section 13.2, I present arguments that if the density involved (i.e. the number of particles in the lattice) is not conserved, the depinning transition is rounded out by plastic motion even at T=0. In systems with conservative dynamics (e.g. the vortex lattice), the situation is less clear. Some rough arguments and various numerical results suggest that one may have either elastic depinning or a new type of plastic depinning transition known as channel flow. These possibilities are outlined in section 13.3, though these phenomena largely constitute open problems.

next up previous contents
Next: Random Manifolds Up: Introduction Previous: General remarks

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