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We study the decay and oscillations of Majorana fermion wavefunctions andground state (GS) fermion parity in one-dimensional topological superconductinglattice systems. Using a Majorana transfer matrix method, we find that Majoranawavefunction properties are encoded in the associated Lyapunov exponent, whichin turn is the sum of two independent components: a `superconducting component'which characterizes the gap induced decay, and the `normal component', whichdetermines the oscillations and response to chemical potential configurations.The topological phase transition separating phases with and without Majoranaend modes is seen to be a cancellation of these two components. We show thatMajorana wavefunction oscillations are completely determined by an underlyingnon-superconducting tight-binding model and are solely responsible for GSfermion parity switches in finite-sized systems. These observations enable usto analytically chart out wavefunction oscillations, the resultant GS parityconfiguration as a function of parameter space in uniform wires, and specialparity switch points where degenerate zero energy Majorana modes are restoredin spite of finite size effects. For disordered wires, we find that bandoscillations are completely washed out leading to a second localization lengthfor the Majorana mode and the remnant oscillations are randomized as perAnderson localization physics in normal systems. Our transfer matrix methodfurther allows us to i) reproduce known results on the scaling of mid-gapMajorana states and demonstrate the origin of its log-normal distribution, ii)identify contrasting behavior of disorder-dependent GS parity switches for thecases of even versus odd number of lattice sites, and iii) chart out the GSparity configuration and associated parity switch points as a function ofdisorder strength.

Majorana wavefunction oscillations, fermion parity switches, and disorder in Kitaev chains