Ilya Gruzberg

Much of the interest of today's condensed matter physics lies in the area of disordered systems. Properties specific to the presence of randomness in a broad class of such systems may be determined by the so called supersymmetry method. This method allows one to deal with the quenched disorder averages, and applies whenever the problem at hand has a representation in terms of free fermions or bosons. In this thesis we apply the supersymmetry method to two problems of this kind.

First we consider the directed network model describing chiral edge states on the surface of a cylindrical three-dimensional quantum Hall system. Combining a mapping of this network to a ferromagnetic superspin chain, scaling analysis, and perturbation theory, we give an essentially complete solution to the problem of finding the mean and variance of the conductance along the cylinder in various regimes and crossovers between them.

The second problem addressed in the thesis is a classical random bond Ising model (RBIM) on the square lattice at finite temperature. On the phase diagram of this model there is the so called Nishimori line which intersects the ferro-para phase boundary at a multicritical point. The RBIM along the Nishimori line has many Jemarkable properties. In the supersymmetry approach the special property of the Nishimori line is that along it the RBIM has an enhanced supersymmetry as compared to the rest of the phase diagram.

We also study a problem on Bethe lattice, related to localization problems studied before, by the supersymmetry and replica methods. While the replica method is widely believed to be only a perturbative tool, we show that when used carefully, it can faithfully reproduce results of the supersymmetry method showing non-perturbative features.