A second approach to the equilibrium behavior of random manifolds is the renormalization group treatment. 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,
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
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
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
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.