Josephson Diode Effect for a Kitaev Ladder System

TL;DR

This work demonstrates a geometry-based, field-free Josephson diode effect in a two-leg Kitaev ladder Josephson junction, enabled by breaking inversion symmetry and time-reversal symmetry through a leg-phase difference $\phi$ and a finite interchain coupling $t_\perp$. The authors show that the current splits into an interband channel with a phase shift and an intraband channel without, and that interference between these channels (including a $4\pi$-periodic Majorana channel near topological transitions) yields nonreciprocal CPRs. The diode efficiency $\eta$ grows at intermediate $t_\perp$ and depends sensitively on $\phi$, while vanishing for both small and large $t_\perp$ due to symmetry restoration. The results provide a symmetry- and topology-based route to superconducting rectification in 1D topological systems and suggest practical avenues for ladder- and network-based diode devices without magnetic fields or spin-orbit coupling.

Abstract

We study the Josephson diode effect realized purely by geometry in a Kitaev-ladder Josephson junction composed of two parallel spinless $p$-wave chains coupled by an interleg hopping $t_\perp$. The junction is governed by two phases: the superconducting phase difference across the weak link, $θ$, and the leg-to-leg phase difference, $φ$. For $φ\notin \{0, π\}$ (mod $2π$), time-reversal symmetry is broken, and the absence of leg-exchange symmetry leads to a breakdown of the antisymmetry of the current-phase relation, yielding nonreciprocal Josephson transport without magnetic fields or spin-orbit coupling. By resolving transport into bonding and antibonding channels defined by $t_\perp$, it is shown that the leg phase acts as an effective phase shift for interband ($p_ν/p_{-ν}$) tunneling, whereas the same-band ($p_ν/p_ν$) contribution remains unshifted. These channels arise at different perturbative orders and, together with the $4π$-periodic Majorana channel that emerges near the topological transition, interfere to produce a pronounced diode response. The class-D Pfaffian invariant identifies the parameter regime where the ladder hosts Majorana zero modes. Bogoliubov-de Gennes calculations reveal a dome-like dependence of the diode efficiency $η$ on $t_\perp$: $η\to 0$ for $t_\perp\to 0$ and for large $t_\perp$, with a maximum at intermediate coupling that is tunable by $φ$. The present results establish a field-free, geometry-based route to superconducting rectification in one-dimensional topological systems and specify symmetry and topology conditions for optimizing the effect in ladder and network devices.

Figures (14)

• Figure 1: The point-contact Josephson junction of the Kitaev ladders. $\phi$ is the phase difference between the chains, $\theta$ is the phase difference between the junction. $t_{\perp}$ is the interchain hopping term. $\Gamma$ is the tunneling term between the junction. $\mu_{1/2}$ and $t_{1/2}$ are the chemical potential and the hopping term of the inner chains. Notice that they exchange around the junction.
• Figure 2: The phase diagram of a Kitaev ladder as a function of the chemical potential difference $\Delta \mu=\frac{\mu_2-\mu_1}{2}$ and perpendicular hopping term $t_{\perp}$. For $t_1=t_2=t$ the radii of the two circles are given by $R_1=|2t-\frac{\mu_1+\mu_2}{2}|$ and $R_2=2t+\frac{\mu_1+\mu_2}{2}$. The white region represents the trivial phase while the blue region represents the nontrivial phase. For $t_\perp=0$ and $|\Delta \mu| <R_1$ along the pink line, the phase is determined by the sign of $2t-\frac{\mu_1+\mu_2}{2}$. The phase is nontrivial for a plus sign and trivial for a minus sign.
• Figure 3: The phase diagram of a Kitaev ladder as a function of the chemical potential difference $\Delta \mu$ and perpendicular hopping term $t_{\perp}$. The topological trivial and nontrivial phases are realized at $\sqrt{(\Delta \mu/\mu)^2+(t_{\perp}/\mu)^2}<1$ and $\sqrt{(\Delta \mu/\mu)^2+(t_{\perp}/\mu)^2}>1$, respectively, in the blue and white region. For $t_{\perp}=0$ and $|\Delta \mu/\mu|<1$ along yellow line, the phase is determined by the sign of $\mu$. The system is nontrivial for a plus sign and trivial for a minus sign.
• Figure 4: The energy spectrum of the Kitaev ladder with periodic boundary condition in momentum space for $\mu=\Delta \mu=2\Delta=1.0$. The blue (red) curves represent the $\epsilon_+$ ($\epsilon_-$ band. We use the inter chain hopping matrix elements; (a) $t_{\perp}=0.01.$ and (b) $t_{\perp}=0.5.$
• Figure 5: The topology dependency of the diode effect. The red regions schematically illustrate where one can detect the diode effect.