Odd-frequency Pairing in Josephson Junctions Coupled by Magnetic Textures

TL;DR

This work addresses how magnetic textures in Josephson junctions induce topological superconductivity and Majorana physics, analyzed via a tight-binding Green function framework that decomposes induced pairing into four fermionic channels. The authors map trivial and topological regimes, showing that Majorana edge states drive spin-polarized odd-frequency triplet correlations with a $1/\omega$ signature when decoupled, while Majorana hybridization introduces resonances at $\omega=\pm\varepsilon$ and linear low-energy behavior. They further reveal that a nonmagnetic barrier creates inner Majorana modes whose coupling depends on barrier width, and that a superconducting phase difference $\phi$ can tune the topological transition and the purity of Majorana states, potentially restoring the $1/\omega$ behavior. Overall, the results position odd-frequency triplet correlations as a sensitive probe and tunable handle for topological superconductivity in magnetically engineered Josephson junctions, with implications for superconducting spintronics and Majorana-based platforms.

Abstract

Josephson junctions coupled through magnetic textures provide a controllable platform for odd-frequency superconductivity and Majorana physics. Within a tight-binding Green function framework, induced pair correlations and spectral properties are analyzed under various magnetic and geometric conditions. When the junction is in the topologically trivial regime, even-frequency singlet pairing is dominant, whereas the topological phase is characterized by the coexistence of Majorana bound states and robust odd-frequency equal-spin triplet pairing at the interface edges. The odd-frequency polarized triplets reveal a divergent $1/ω$ behavior when the Majorana states are decoupled, which is intrinsically connected to their self-conjugation property. The zero-frequency divergence evolves into shifted resonances and linear low-frequency behavior once hybridization occurs. A nonmagnetic interruption in the texture separates the topological superconductor into two topological segments and generates additional inner Majorana modes. When the nonmagnetic barrier is comparable to the inner Majorana states localization length, they hybridize and modify their associated odd-frequency triplet pairing, while the outer edge modes preserve their self-conjugated nature. Tuning the superconducting phase difference further controls the onset of the topological regime and the stability of localized Majorana states. The results highlight the central role of odd-frequency triplet correlations as a probe of topological superconductivity in magnetically engineered Josephson junctions.

Figures (9)

• Figure 1: Josephson junction coupled by a magnetic texture. (a) Superconducting loop closed around a magnetic texture (green). (b) Two superconductors, $L$ and $R$, form the junction. Majorana states (red circles) emerge in the nontrivial regime at the interface edges. (c) Helical magnetic texture along the interface with period $\xi_m$ and amplitude $t_m$.
• Figure 2: Different setup configurations: (a,b) Uninterrupted magnetic texture (green) of length $L_y$ (a) longer or (b) comparable to the Majorana localization length. (c,d) Magnetic texture interrupted by a nonmagnetic barrier of length $L_0$ that is (c) shorter or (d) longer than the inner edge states wavefunction decay. The wavefunction localization of the outer and inner edge states is respectively shown in red and blue.
• Figure 3: Interface density of states and pairing. (a,b) Total as a function of $\omega$ in the (a) trivial ($t_m=0.4t$) and (b) topological ($t_m=0.6t$) phases. (c,d) Local correlators at the junction edge ($L_x, a$) in the (c) trivial and (d) topological phases. In all cases, $\mu=-3.9t$, $t_0=t$, $\phi=0$, $\Delta=0.2t$.
• Figure 4: Wide junction in the topological regime. (a) along the interface as a function of the energy. The Majorana edge states are decoupled as shown in the sketch. (b) Ratio between the local polarized odd-frequency triplet and even-frequency singlet, as a function of energy. (c) as a function of the energy computed at the bottom edge $y=a$ and in the middle of the interface $y=L_y/2$. (d) Ratio $p^\text{OTE}/F^\text{ESE}$ at the same points. In all cases, $\mu=-3.9t$, $t_0=t$, $t_m=0.6t$, $\phi=0$, $\xi_m/L_y=0.1$ and $\Delta=0.2t$.
• Figure 5: Narrow junction in the topological regime. (a) along the interface as a function of the energy. The wave function of the different Majorana edge states overlaps as indicated in the sketch. (b) Ratio $p^\text{OTE}/F^\text{ESE}$ along the interface as a function of $\omega$. (c) as a function of the energy computed at $y=a$ and $y=L_y/2$. (d) Ratio $p^\text{OTE}/F^\text{ESE}$ at the same points as a function of $\omega$. Inset: Log-scale plot of the zero-energy slope $\mathcal{B}$ as a function of $\xi_m/L_y$, with an arrow indicating the value corresponding to this figure. In all cases, $\mu=-3.9t$, $t_0=t$, $t_m=0.6t$, $\phi=0$, $\xi_m/L_y=0.5$ and $\Delta=0.04t$.