Now that we are faced with a physical example, I'll step back and try and put this problem in a more general perspective. There are a whole class of models which I will call elastic manifold models[3, 4]. The fundamental dynamical object in these models is the manifold itself, which I will characterize by its dimensionality N. It is assumed to be elastic in that it is characterized by a non-zero surface tension, or energy cost per unit length, area, or volume (for N=1,2,3). For these elastic manifold models, we will assume that the object is embedded in a larger space, with dimension . The manifold can then fluctuate within this space to take advantage of impurities, etc.
I will also assume that the manifold is not fractal, so that it may be asymptotically defined as an oriented manifold, i.e. one without overhangs. There are other elastic objects which are not oriented. These include ordinary polymers and fluid membranes. While such systems are, of course, very interesting, it is generally difficult to prepare them in an environment with quenched disorder, and we will not speak much about them here.
Figure 2: Examples of elastic manifolds: (a) interface in two dimensions, (b) directed polymer in three dimensions (c) interface in three dimensions.
To describe the fluctuations of our elastic object within the larger space, we need an analytic description of the configurations of the manifold. We'll do this by writing the transverse coordinates of the manifold as functions of the parallel or internal coordinates , i.e. . In the notation I just described, the vector , while the coordinate (see Fig. 2). In the case of the magnetic interface, we have N=1, and u is simply the height of the interface, as a function of d= D-1 coordinates perpendicular to the interface. We can also consider the case of an elastic line, in which case N=D-1 (d=1), and the field gives the location of the line as a function of x, the linear distance along the axis of extension. This kind of elastic line is known as a directed polymer, because it is like a polymer which always points along a particular direction, in this case the x axis.
Like the domain wall, the directed polymer arises very naturally as a topological defect. In two dimensions D = 1+1, the directed polymer is of course identical to a domain wall. In three dimensions (D=2+1), it is the appropriate defect for a complex or vector order parameter. In such a defect, the phase (or vector angle) winds by as one goes around the defect line. The most common examples of this type are vortex lines in superfluid helium and in type II superconductors. I'll come back to the latter later, where this kind of physics is actually playing a crucial role.
Including disorder, the Hamiltonian for the random manifold model is then
I have added a random potential , representing the impurities in the full D-dimensional space. We'll take a specific distribution for V later to do some calculations. The general requirements are that it should be narrowly distributed and short-range correlated.
As usual, the probability distribution for is obtained using the Boltzmann form,
where the partition function is a functional integral