Majorana wave-function oscillations, fermion parity switches, and disorder in Kitaev chains

We study the decay and oscillations of Majorana fermion wave functions and ground-state ( GS) fermion parity in one-dimensional topological superconducting lattice systems. Using a Majorana transfer matrix method, we find that Majorana wave-function properties are encoded in the associated Lyapunov exponent, which in turn is the sum of two independent components: a superconducting component, which characterizes the gap induced decay, and the normal component, which determines the oscillations and response to chemical potential configurations. The topological phase transition separating phases with and without Majorana end modes is seen to be a cancellation of these two components. We show that Majorana wave-function oscillations are completely determined by an underlying nonsuperconducting tight-binding model and are solely responsible for GS fermion parity switches in finite-sized systems. These observations enable us to analytically chart out wave-function oscillations, the resultant GS parity configuration as a function of parameter space in uniform wires, and special parity switch points where degenerate zero energy Majorana modes are restored in spite of finite size effects. For disordered wires, we find that band oscillations are completely washed out leading to a second localization length for the Majorana mode and the remnant oscillations are randomized as per Anderson localization physics in normal systems. Our transfer matrix method further allows us to ( i) reproduce known results on the scaling of midgap Majorana states and demonstrate the origin of its log-normal distribution, ( ii) identify contrasting behavior of disorder-dependent GS parity switches for the cases of even versus odd number of lattice sites, and ( iii) chart out the GS parity configuration and associated parity switch points as a function of disorder strength.