Josephson Diode Effect for a Kitaev Ladder System

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, $\theta$, and the leg-to-leg phase difference, $\phi$. For $\phi\notin {0, \pi}$ (mod $2\pi$), 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 ($p_\nu/p_{-\nu}$) tunneling, whereas the same-band ($p_\nu/p_\nu$) contribution remains unshifted. These channels arise at different perturbative orders and, together with the $4\pi$-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 $\eta$ on $t_\perp$: $\eta\to 0$ for $t_\perp\to 0$ and for large $t_\perp$, with a maximum at intermediate coupling that is tunable by $\phi$. 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.