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
