On a Network Model of Localization in a Random Magnetic Field

We consider a network model of snake states to study the localization problem of non-interacting fermions in a random magnetic field with zero average. After averaging over the randomness, the network of snake states is mapped onto $M$ coupled SU$(2N)$ spin chains in the $N \rightarrow 0$ limit. The number of snake states near the zero-field contour, $M$, is an even integer. In the large conductance limit $g = M {e^2 \over 2 \pi \hbar}$ ($M \gg 2$), it turns out that this system is equivalent to a particular representation of the ${\rm U}(2N) / {\rm U}(N) \times {\rm U}(N)$ sigma model ($N \rightarrow 0$) {\it without} a topological term. The beta function $\beta (1/M)$ of this sigma model in the $1/M$ expansion is consistent with the previously known $\beta (g)$ of the unitary ensemble. These results and further plausible arguments support the conclusion that all the states are localized.