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.

Figure 1

The topological phase diagram for the uniform Kitaev chain as a function of superconducting gap $\mathrm{\Delta }/w$ and chemical potential $\mu /2w$. The focus here is the circle of oscillations (COO) $[{\mu }^2/4w^2+{\mathrm{\Delta }}^2/w^2=1]$ within each topological phase marking the boundary across, which the nature of Majorana wave functions changes. Within the circle, the wave function has oscillations under the decaying envelope whereas they are absent outside the circle.

Figure 2

The homogeneous Kitaev chain Lyapunov exponent(LE), $\gamma$, measuring Majorana end mode decay as a function of chemical potential for fixed superconducting gap $\left(\mathrm{\Delta }=0.6\right)$. The LE is a sum of a normal and a superconducting component $\gamma ={\gamma }_N+{\gamma }_S$, and is a constant within the circle of oscillations since there ${\gamma }_N=0$ and ${\gamma }_S$ is constant for fixed $\mathrm{\Delta }$. On crossing the circle, ${\gamma }_N$ becomes a nonzero increasing function of $\mu$ and ultimately cancels ${\gamma }_S$, resulting in a zero LE, at the topological phase transition at $\mu =2$.

Figure 3

A schematic of the Majorana wave function in (a) the uniform case within the oscillatory regime and (b) the disorder case. (a) Within the parameter regime containing the circle of oscillations (see text), in addition to a decaying envelope having an associated Lyapunov exponent ${\gamma }_s$, stemming purely from the superconducting order, band oscillations are present. (b) For the disordered case, band oscillations are replaced by random oscillations and a second decaying scale associated with a Lyapunov exponent ${\gamma }_N$, both stemming from the underlying Anderson localization setup of a nonsuperconducting normal wire having the same disorder configuration.

Figure 4

Ground-state parity for a uniform finite-length Kitaev chain within the topological phase in the phase diagram of Fig. 1. Within the circle bounding the region where Majorana bound state wave function exhibit oscillations, alternating parity sectors are demarcated by ellipses. These parity sectors depend on the length of the chain. For chains of odd length $\left(N=11\right)$ (a) the sectors are antisymmetric across $\mu =0$ and are symmetric for chains of even length $\left(N=10\right)$ (b). Outside the circle, the Majorana modes are overdamped with no oscillations.

Figure 5

Comparison between the fermion parity in a uniform Kitaev chain, calculated using Eq. (30), and the matrix element ${\tilde{\mathcal{A}}}_{11}$ of the zero-energy Majorana transfer matrix whose vanishing value reflects the existence of a zero-energy state. The matrix element is calculated both analytically using Eq. (40) and numerically for a uniform chain. One can see that the parity switches coincide exactly with the matrix element going to zero. Here, $N=21\text{and}\mathrm{\Delta }=0.6.$

Figure 6

Fermion parity switches are concurrent in wires of differing superconducting gap $\mathrm{\Delta }$ as a function of the scaled chemical potential ${\mu }^{\prime }=\mu \sqrt{1-{\mathrm{\Delta }}^2}$. This is shown here for the uniform Kitaev chain of length $N=20$, where parity is calculated using Eq. (30). The thick red plot is for $\mathrm{\Delta }=0$. The other plots (scaled away from unity for proper visibility) are for finite superconducting gaps.

Figure 7

Density-of-states plots for a disordered Kitaev chain $\left(\mathrm{\Delta }=0.6\right)$ as a function of distance from the Fermi energy for a single disorder configuration of box disorder (44). (a) For weak disorder $\left(W<8\right)$, there is a well-defined superconducting gap in the density of states. (b) As the disorder strength is increased, the gap is filled due to the proliferation of low-energy bulk states. (c) At a critical disorder strength, there is “singularity” in the density of states. The behavior beyond the critical point resembles that of (b).

Figure 8

(a)Variation of a set of lowest energy levels (first three states) of the Kitaev chain as a function of disorder strength for box disorder (44) and other parameters fixed to $\mathrm{\Delta }=0.6,N=30$. For small disorder width, the Majorana states are well separated from the bulk by a superconducting gap. As the disorder is increased there is a proliferation of the bulk states into the gap. (b) Zoomed in view of the two Majorana states split due to finite size. Their scale is exponentially suppressed compared to the bulk states. These states cross zero energy as the disorder width is varied, inducing a fermion parity switch in the ground state. (c) Griffiths phase: at strong disorder, there is an accumulation of large number of bulk states near zero energy. Level crossings between these states are forbidden due to level statistics of class D (level repulsion). Nearing the critical disorder strength, the magnitude of the energy states due to Majorana splitting become comparable to the bulk states.

Figure 9

Comparison between the fermion parity in a disordered Kitaev chain and the transfer matrix element ${\tilde{\mathcal{A}}}_{11}$ as a function of disorder strength (box disorder). The vanishing of the matrix element reflects the existence of a zero-energy Majorana state and can be seen here to coincide with parity switches as with the uniform case of Fig. 5. Here, $N=40\text{and}\mathrm{\Delta }=0.6$.

Figure 10

(a) Parity switches in a wire of an even number of lattice sites, $N=10$, as a function of disorder width for box disorder. Here, $\mathrm{\Delta }=0.6$. Since parity is an even function of chemical potential for even $N$, the initial disorder window lies within a fixed parity as indicated in the uniform chain phase diagram in (b). As the disorder window is increased beyond a length-dependent value ${\mu }_{\mathrm{switch}}^p$ of Eq. (47) (dotted line) to include opposite parity sectors, parity switches begin to occur.

Figure 11

Possibility of no parity switches in the disordered Kitaev chain. (a) Two possible disorder distributions centered around finite chemical potential are shown in which parity switches are not expected - I. Double box disorder in two disjoint sectors of the same parity and II. Box disorder outside the circle of oscillations. (b) In both cases, the plot of parity as a function of disorder strength indeed shows the absence of parity switches.

Figure 12

Parity switches in a wire of odd length $\left(N=11\text{and}\mathrm{\Delta }=0.6\right)$ as a function of disorder strength for box disorder. (a) Given the antisymmetry in parity sectors across $\mu =0$ for the uniform case, the slightest change in disorder strength is expected to produce a parity switch. This is confirmed in (b), which shows a profusion of random parity switches as a function of disorder strengths starting from the smallest amount of disorder.