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Multi-channel spin-orbit quantum wires, when subjected to a magnetic fieldand proximity coupled to s-wave superconductor, may support Majorana states. Westudy what happens to these systems in the presence of disorder. Inspired bythe widely established theoretical methods of mesoscopic superconductivity, wedevelop a la Eilenberger a quasiclassical approach to topological nanowiresvalid in the limit of strong spin-orbit coupling. We find that the "Majorananumber", distinguishing between the state with Majorana fermion (symmetry classB) and no Majorana (class D), is given by the product of two Pfaffians ofgapped quasiclassical Green's functions fixed by right and left terminalsconnected to the wire. A numerical solution of the Eilenberger equationsreveals that the class D disordered quantum wires are prone to the formation ofthe zero-energy anomaly (class D impurity spectral peak) in the local densityof states which shares the key features of the Majorana peak. In this way weconfirm the robustness of our previous conclusions [Phys. Rev. Lett. 109,227005 (2012)] on a more restrictive system setup. Generally speaking, we findthat the quasiclassical approach provides a highly efficient means to addressdisordered class D superconductors both in the presence and absence oftopological structures.

Quasiclassical theory of disordered multi-channel Majorana quantum wires