Localization from $\sigma$-model geodesics

We use a method based on the semiclassical analysis of $\sigma$ models to describe the phenomenon of strong localization in quasi one-dimensional conductors, obtaining the density of transmission eigenvalues. For several symmetry classes, describing random superconducting and chiral Hamiltonians, the target space of the appropriate $\sigma$ model is a (super)group manifold. In these cases our approach turns out to be exact. The results offer a perspective on localization.

Figure 1

Transmission eigenvalue densities for the classes $C\mathrm{I}$ and $D\mathrm{III}$ at $T=0.01$ (weak localization—dashed line) and $T=20$ (strong localization—solid line). Note that, although eigenvalue crystallization occurs in both classes, localization in class $D\mathrm{III}$ is absent due to the persistence of an eigenvalue near $\varphi =0$ (or $\mathcal{T}=1$).