Thermodynamic limit and topological invariants in the presence of disorder

Topological phases (and appropriate invariants characterizing them) can be defined rigorously in the thermodynamic limit and operationally in large-enough systems (as compared to the characteristic length scales of the localized low-energy modes). However, in certain experimentally relevant conditions the system is effectively "small" and, consequently, the topological phase is ill defined, while the topological "invariants" exhibit significant finite size effects. Focusing on disordered semiconductor-superconductor hybrid nanowires capable of hosting Majorana zero modes, we introduce a well-defined topological invariant characterizing an infinite system obtained by joining together copies of a finite length (possibly "short") wire. We study in detail the dependence of the topological phase diagram characterizing this infinite wire with "periodic disorder" on the system parameters, disorder strength, and segment length. We also characterize the variations of the topological phase diagram associated with different disorder realizations within the finite length segment and provide comparisons with predictions based on commonly used invariants.