  We consider theoretically the physics of bulk topological superconductivity
accompanied by boundary non-Abelian Majorana zero modes in
semiconductor-superconductor (SM-SC) hybrid systems consisting of finite wires
in the presence of correlated disorder arising from random charged impurities.
We find the system to manifest a highly complex behavior due to the subtle
interplay between finite wire length and finite disorder, leading to copious
low-energy in-gap states throughout the wire and considerably complicating the
interpretation of tunneling spectroscopic transport measurements used
extensively to search for Majorana modes. The presence of disorder-induced
low-energy states may lead to the non-existence of end Majorana zero modes even
when tunneling spectroscopy manifests zero bias conductance peaks in local
tunneling and signatures of bulk gap closing/reopening in the nonlocal
transport. In short wires within the intermediate disorder regime, apparent
topology may manifest in small ranges ("patches") of parameter values, which
may or may not survive the long wire limit depending on various details.
Because of the dominance of disorder-induced in-gap states, the system may even
occasionally have an appropriate topological invariant without manifesting
isolated end Majorana zero modes. We discuss our findings in the context of a
recent breakthrough experiment from Microsoft reporting the simultaneous
observations of zero bias conductance peaks in local tunneling and gap opening
in nonlocal transport within small patches of parameter space. Based on our
analysis, we believe that the disorder strength to SC gap ratio must decrease
further for the definitive realization of non-Abelian Majorana zero modes in
SM-SC devices.
