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

TL;DR

This paper analyzes Majorana wavefunction decay and oscillations in Kitaev chains, revealing how they relate to topological phase transitions, fermion parity switches, and the effects of disorder using a transfer matrix approach.

Contribution

It introduces a transfer matrix method linking Majorana wavefunction properties to Lyapunov exponents, providing analytical insights into phase transitions, parity switches, and disorder effects in Kitaev chains.

Findings

Majorana wavefunction oscillations are determined by a non-superconducting model.

Topological phase transition occurs when superconducting and normal components cancel.

Disorder washes out band oscillations, leading to a second localization length.

Abstract

We study the decay and oscillations of Majorana fermion wavefunctions and ground state (GS) fermion parity in one-dimensional topological superconducting lattice systems. Using a Majorana transfer matrix method, we find that Majorana wavefunction 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 wavefunction oscillations are completely determined by an underlying non-superconducting tight-binding model and are solely responsible for GS fermion parity switches in finite-sized systems. These observations enable us to analytically chart out wavefunction 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 mid-gap 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.