On a Network Model of Localization in a Random Magnetic Field

TL;DR

This paper models localization of non-interacting fermions in a random magnetic field using a network of snake states, mapping it onto sigma models, and concludes that all states are localized.

Contribution

It introduces a network model of snake states mapped onto sigma models, providing new insights into localization in random magnetic fields.

Findings

System maps onto a U(2N)/U(N)×U(N) sigma model without topological term.

Beta function analysis supports all states being localized.

Results align with known beta function of the unitary ensemble.

Abstract

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.