Network model of localization in a random magnetic field

We consider a multichannel network model to study the localization problem of noninteracting fermions in a random magnetic field with zero average. We argue that the number of channels M is even. After averaging over the randomness, the network is mapped onto M coupled SU(2N) spin chains in the N→0 limit. In the large conductance limit g=M(${\mathit{e}}^2$/2πħ) (M≫2), it turns out that this system is equivalent to a particular representation of the U(2N)/U(N)×U(N) sigma model (N→0) without a topological term. The beta function β(1/M) of this sigma model in the 1/M expansion is consistent with the previously known β(g) of the unitary ensemble. These results and further arguments support the conclusion that all the states are localized.