A second approach to the equilibrium behavior of random manifolds is
the renormalization group treatment[4]. It is
complimentary to the replica approach, in that it is valid for
arbitrary N but only for 
. The methodology is
quite different, but agreement has nevertheless been found between the
two approaches wherever a valid and careful comparison has been made.
The idea of the RG is to construct an explicit field theory for the
T=0 fixed point governing the low-temperature phase. The model may
be consistently formulated at zero temperature. The problem of
computing the partition function then reduces to finding the minimum
of the Hamiltonian, Eq.13. Functionally
differentiating with respect to 
(i.e. using the calculus of
variations) gives the extremal condition (Euler-Lagrange equation)
Now let us note a general trend. As the dimension d of the system
increases, the manifold tends to become less rough. This is simply
because of the strong increase (like 
) of the elastic energy
with d. Recall that in the pure system, the thermal roughness
vanishes for d>2 ( 
). According to the replica
solution, 
for d>4 even in the low temperature pinned
phase.
Let us then attempt to recover this result. We will assume that
the manifold is flat, and then check self-consistency. This means
that, in the zeroth order approximation 
,
and the leading correction is obtained by simply setting
in the second term. This gives, in Fourier space,
We may then calculate the roughness at T=0,
, where
We see that our assumption is indeed self-consistent for d>4, so we
have recovered the first result of the replica calculation. For
d<4, however, we have arrived at a contradiction, and a more careful
treatment is required. In particular, we should not trust the naive
prediction 
.
We do expect, however, to find 
for . From this we can hope to derive an 
expansion for the
scaling exponents (see Fig. 6). To do so, let us
first consider the simple power-counting considerations, as we did for
the thermal fixed point in section 5.
Figure 6: Schematic picture of the RG flows in the space of temperature
T and disorder. The pure T=0 fixed point (at the origin) is
unstable to a new, disordered zero temperature fixed point (solid
circle). If , this fixed point is close
( 
) to the pure fixed point, and can thus be accessed
using a perturbative RG.
At zero temperature, our goal is only to find the minimum of
H. As such, we do not require that the Hamiltonian itself be
invariant under the RG. Instead, it is permissible that H be
multiplied by a constant after the rescaling transformation. This is
because the same configuration 
which minimized H
also minimizes 
, where 
is any positive constant.
Let us then proceed as before, rescaling
Under this rescaling, the Hamiltonian becomes
We have indeed picked up an overall rescaling of energies, and
identified the rescaling exponent as 
. We
could make the identification as the (negative) RG eigenvalue of
temperature more explicit by considering instead the partition
function,
so we see that
Indeed, for 
, the temperature is an irrelevant variable.
At this point, we may make a simple argument to understand why the
power-counting result for 
is exact. This can be seen by
considering the change in the free energy corresponding to a uniform
tilt of the manifold (or equivalently a change in boundary
conditions). If the fields are shifted by a linear function of the
coordinates
the Hamiltonian is shifted by an additive constant
The random potential is also changed. However, the new random potential has an identical distribution to the old one, i.e.
due to the delta-function correlations in 
. This implies
that the mean ground-state energy is shifted exactly by the term
in Eq. 95.
This is an exact statement about the
model, and must therefore be true at all stages of the RG; it requires
that T (i.e. the coefficient of the stiffness term) only be
renormalized by the scale changes (see, e.g. U. Schulz, J. Villain,
E. Brézin, and H. Orland, J. Stat. Phys. 51, 1 (1988)). This
proves the desired exponent identity.
Now we would like to continue and pursue the RG. In the thermal case
at this point, we were able to make an ``ultra-local'' expansion of
the disorder correlator 
in derivatives of
delta-functions. Higher derivatives of delta functions were strongly
irrelevant. We see immediately that this expansion will fail in this
case, since each derivative of the delta-functions is suppressed only
by the infinitesimal factor 
. This implies that we need
to really keep the full function R.
At linear order in V, this is trivial. We simply calculate the variance of the rescaled potential
which gives
For infinitesimal rescaling, 
, this gives the linear RG flow equation
where we have expanded the infinitesimal dl inside R.
We have found a weak linear instability, which we hope will be stabilized by the quadratic terms in the RG equation to describe the fixed point. To calculate them, we need to consider the other part of the RG, where we remove the ``fast'' modes in momentum space. This is done by splitting the field according to
where
We wish to minimize H over 
and arrive at a renormalized
Hamiltonian which is only a function of 
. This is
accomplished perturbatively in V from Eq. 85:
Defining the Fourier transform
the approximate solution is
We next insert his into the Hamiltonian to obtain the renormalized random part of H,
It is straightforward to show that if 
is constant
over regions of size L, this can be rewritten as an integral of a
local potential, up to small errors of order 1/L. Thus, in
the long wavelength limit, the renormalized Hamiltonian is
well-described simply by a renormalized potential. It's correlations
can be calculated from the expression
Straightforward manipulations give
Combining this renormalization with the linear rescaling transformation gives the final differential flow equation,
This equation has various fixed points, where 
. For
short-range correlated disorder, for which we are interested here,
they are highly non-trivial. Each fixed point is characterized by a
non-trivial (eigen)value of 
. They can be obtained numerically
for any N, and asymptotically for 
. One finds
and
These results hold to leading order in . Note that, as
, Eq. 111 agrees with the RSB result
for 2<d<4, believed to hold in that limit.
