Odd-even parity dependent transport in an annular Kitaev chain

TL;DR

This paper addresses how magnetic flux and the odd-even parity of lattice points $N$ govern transport in an annular Kitaev chain, focusing on direct transmission $T_{DT}$, local Andreev reflection $T_{LAR}$, and crossed Andreev reflection $T_{CAR}$. Using non-equilibrium Green's function methods with the Landauer-Büttiker formalism on a tight-binding BdG model with a uniform Peierls phase $\phi$, it derives flux-dependent energy bands and gap openings/closings that depend on $N$'s parity. Key findings reveal parity-dependent patterns: for even $N$ with symmetric leads, $T_{DT}$ shows resonances at $\Phi = N\pi/3$ and $2N\pi/3$ while $T_{LAR}$ and $T_{CAR}$ are suppressed; for odd $N$ the DT peak at $\Phi = N\pi/3$ is weakened and $T_{LAR}$/$T_{CAR}$ peaks dominate at the same flux, with the DT peak at $\Phi = 2N\pi/3$ persisting. These parity-driven effects remain robust to weak disorder, offering a practical parity-resolved diagnostic of topological phases in finite Kitaev rings and motivating experimental realizations in semiconductor–superconductor nanowire networks.

Abstract

We investigate the impact of magnetic flux and the odd-even parity of lattice points $N$ on electron transport in an annular Kitaev chain, with an explanation provided from the energy band perspective. Three transport mechanisms including direct transmission (DT), local Andreev reflection (LAR) and crossed Andreev reflection (CAR) are considered. In particular, the connection configuration of electrodes to different lattice sites is studied, where the case that the two electrodes connected to the sites are aligned along a diameter is called as symmetric connection and otherwise as asymmetric connections. For even $N$ and asymmetric connection, the vanished LAR and CAR in symmetric connection will emerge as peaks. A more prominent observation is that the symmetry of the two resonant peaks due to DT processes located at $Φ= Nπ/3$ and $Φ= 2Nπ/3$ for symmetric connection will be broken, and the peak at $Φ= Nπ/3$ will be largely reduced, where $Φ$ is the magnetic flux. Moreover, the peaks around $Φ= Nπ/3$ due to LAR and CAR processes grows drastically even larger than that from DT. For LAR and CAR processes, there is no peak around $Φ= 2Nπ/3$ and transmission due to these two processes are completely suppressed for $Φ>Nπ/2$. Moreover, it is found that the energy bands vary with $Φ$ in a period of $Nπ$ and $2Nπ$ for even or odd $N$. We finally systematically analyze the influence of weak disorder on transport and demonstrate that these parity-dependent effects are robust in the presence of disorder. These behaviors reflect a complicated competition from DT, LAR and CAR processes and the parity of the lattice number in the Kitaev ring, which will be interested for the quantum device based on Kitaev chain.

Figures (7)

• Figure 1: (a) Schematic plot of the annular Kitaev chain model. The central region represents an annular Kitaev chain. Each ellipse denotes a site on the chain, and there are two MZMs at each site. The dark blue lines in the figure indicate the particle interactions between different sites, while the green lines represent the particle interactions at the same site. Electrodes are connected at the left and right ends of the chain. (b) Schematic illustration of DT. In this case, electrons are directly transmitted through the system, and the charge carriers are electrons. (c) Schematic illustration of local Andreev reflection (LAR). A single incoming electron being reflected as a hole, with a corresponding Cooper pair added to the superconductor. (d) Schematic illustration of crossed Andreev reflection (CAR). A single electron impinges on the superconductor interface, a hole is reflected from the opposite interface, with a Cooper pair enters into the superconductor.
• Figure 2: The transmission coefficients $T_\text{DT}$, $T_\text{LAR}$ and $T_\text{CAR}$ as functions of the magnetic flux phase and different connection configuration of the electrodes (insets in every subfigures). The ring has $N = 6$ lattice points. The on-site potential is $\mu = 1$, the hopping energy is $t = 1$, and the superconducting pairing potential is $\Delta = 1$ with $\theta = 0$. In (a)-(f), the left-hand electrode is fixed to connect to the lattice point $j = 1$ of the ring, while the right-hand electrode is connected to the lattice points $j = 1$, $j = 4$, $j = 2$, $j = 6$, $j = 3$, and $j = 5$ in sequence.
• Figure 3: The transmission coefficients $T_\text{DT}$, $T_\text{LAR}$, and $T_\text{CAR}$ as functions of the magnetic flux phase and different lattice-point configurations. The left-hand electrode is connected to the lattice point $j = 1$, while the right-hand electrode is connected to the lattice point $j = 4$. The on-site potential is set to $\mu = 1$, the hopping energy is $t = 1$, and the superconducting pairing potential is given by $\Delta = 1$ with a phase $\theta = 0$. In (a)-(d), the number of lattice points in the system is set to $N = 6$, $N = 8$, $N = 5$, and $N = 7$ successively.
• Figure 4: The differential conductance $G$ as functions of the magnetic flux phase and different lattice-point configurations. The on-site potential is set to $\mu = 1$, the hopping energy is $t = 1$, and the superconducting pairing potential is given by $\Delta = 1$ with a phase $\theta = 0$. In (a)-(d), the number of lattice points in the system is set to $N = 6$, $N = 8$, $N = 5$, and $N = 7$ successively.
• Figure 5: The plots show the energy ($E$) as a function of the magnetic flux phase ($\Phi$) for different numbers of lattice points, presented as $E - \Phi$ diagrams. In (a)-(d), $N$ in the ring is set to $N = 6$, $N = 8$, $N = 5$, and $N = 7$ respectively.