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We consider theoretically the physics of bulk topological superconductivityaccompanied by boundary non-Abelian Majorana zero modes insemiconductor-superconductor (SM-SC) hybrid systems consisting of finite wiresin the presence of correlated disorder arising from random charged impurities.We find the system to manifest a highly complex behavior due to the subtleinterplay between finite wire length and finite disorder, leading to copiouslow-energy in-gap states throughout the wire and considerably complicating theinterpretation of tunneling spectroscopic transport measurements usedextensively to search for Majorana modes. The presence of disorder-inducedlow-energy states may lead to the non-existence of end Majorana zero modes evenwhen tunneling spectroscopy manifests zero bias conductance peaks in localtunneling and signatures of bulk gap closing/reopening in the nonlocaltransport. In short wires within the intermediate disorder regime, apparenttopology may manifest in small ranges ("patches") of parameter values, whichmay 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 evenoccasionally have an appropriate topological invariant without manifestingisolated end Majorana zero modes. We discuss our findings in the context of arecent breakthrough experiment from Microsoft reporting the simultaneousobservations of zero bias conductance peaks in local tunneling and gap openingin nonlocal transport within small patches of parameter space. Based on ouranalysis, we believe that the disorder strength to SC gap ratio must decreasefurther for the definitive realization of non-Abelian Majorana zero modes inSM-SC devices.

Spectral properties, topological patches, and effective phase diagrams of finite disordered Majorana nanowires