Topology vs. Anderson localization: non-perturbative solutions in one dimension

We present an analytic theory of quantum criticality in quasi one-dimensional topological Anderson insulators. We describe these systems in terms of two parameters $(g,\chi )$ representing localization and topological properties, respectively. Certain critical values of $\chi$ (half-integer for $\mathbb{Z}$ classes, or zero for ${\mathbb{Z}}_2$ classes) define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two parameter flow of the integer quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given in terms of the transfer-matrix solution of corresponding supersymmetric non-linear sigma-models. In ${\mathbb{Z}}_2$ classes we uncover a hidden supersymmetry, present at the quantum critical point.