Odd-frequency Pairing in Josephson Junctions Coupled by Magnetic Textures

This paper demonstrates that Josephson junctions coupled by magnetic textures serve as a controllable platform for odd-frequency superconductivity, where the emergence of Majorana bound states in the topological phase is intrinsically linked to robust, divergent odd-frequency equal-spin triplet pairing that can be probed and manipulated through magnetic textures, nonmagnetic barriers, and superconducting phase differences.

The Gist

Imagine a superconductor as a superhighway where electrons travel in perfect pairs, never bumping into anything or losing energy. Now, imagine you want to build a special kind of "traffic jam" on this highway that creates a very rare, exotic particle called a Majorana. These particles are like "ghosts" in the quantum world: they are their own mirror images (if you look at them in a mirror, you see the same thing looking back). Scientists hope to use them to build super-powerful, unbreakable quantum computers.

This paper explores how to create and spot these "ghosts" using a specific setup: a Josephson Junction. Think of this as a bridge connecting two superconducting islands. Instead of a normal bridge, this one is covered in a magnetic texture—a pattern of magnetic fields that twist and turn like a spiral staircase or a helix.

Here is the breakdown of what the researchers found, using simple analogies:

1. The "Ghost" Signal: The Odd-Frequency Pair

To find these Majorana ghosts, the scientists didn't just look for them directly; they looked for a specific "fingerprint" they leave behind. This fingerprint is called odd-frequency pairing.

• The Analogy: Imagine a dance between two partners (electrons). Usually, they dance in a rhythm that repeats perfectly every time (even-frequency). But in the presence of these Majorana ghosts, the dance changes. They start dancing in a rhythm that is "odd" in time—like a dance step that only makes sense if you look at it in reverse.
• The Fingerprint: When the Majorana ghosts are perfectly isolated and "pure" (not touching anything else), this odd dance has a very specific, wild behavior: it gets infinitely strong as the energy gets close to zero. Mathematically, this looks like a 1/ω curve (a sharp spike). The paper claims this spike is the ultimate proof that the "ghost" is there and is behaving exactly as a Majorana particle should.

2. The "Crowded Room" Problem: Hybridization

The researchers studied what happens when the bridge (the junction) is too narrow.

• The Analogy: Imagine two Majorana ghosts living at opposite ends of a long hallway. They are far apart and can't see each other. They are pure and stable. But if you shorten the hallway so the ghosts are close together, they start to "talk" to each other. In physics, this is called hybridization.
• The Result: When they talk, they lose their "ghostly" purity. They stop being their own mirror images and become regular particles with a tiny bit of energy.
• The Effect on the Fingerprint: Because they are no longer pure ghosts, that sharp 1/ω spike disappears. Instead, the signal becomes a gentle, straight line (linear) near zero energy. The paper shows that by measuring this change from a "spike" to a "line," you can tell if the ghosts are isolated or if they are interfering with each other.

3. The "Wall" in the Middle: Nonmagnetic Barriers

The team also tested what happens if you put a non-magnetic wall in the middle of the magnetic bridge.

• The Analogy: Imagine the magnetic bridge is a long road. If you build a wall in the middle, you split the road into two separate segments. Suddenly, you don't just have ghosts at the ends of the road; you now have new ghosts appearing at the edges of the wall itself.
• The Interaction: If the wall is wide, the new ghosts are far apart and stay pure (spike signal). If the wall is narrow, the ghosts on either side of the wall get close, talk to each other, and lose their purity (linear signal).

4. The "Volume Knob": Tuning with Phase

The most exciting part of the paper is how they can control this using the superconducting phase difference (think of this as a volume knob or a dial that changes the rhythm of the superconductors).

• The Twist:
  • In a single bridge: Turning the dial usually makes the ghosts at the ends get closer and mess up their purity.
  • In a bridge with a wall: Surprisingly, turning the dial can actually push the ghosts apart. It acts like a force that separates the ghosts living on either side of the wall, making them pure again.
• The Takeaway: By simply adjusting this "dial," scientists can switch the system between having messy, hybridized ghosts and having clean, pure, self-conjugated ghosts. This allows them to "tune" the system to get the perfect 1/ω spike signal they need to confirm they have found a Majorana particle.

Summary

The paper argues that odd-frequency pairing is the best way to "hear" the Majorana ghosts.

• If you see a sharp spike (1/ω), the ghosts are pure and isolated.
• If you see a gentle line, the ghosts are crowded and interacting.
• By using a magnetic texture and tuning a phase dial, you can control whether the ghosts are pure or mixed, and even create new ones by splitting the bridge with a wall.

This provides a new, controllable way to detect these elusive particles, which is a crucial step toward building the quantum computers of the future.

Technical Summary

Technical Summary: Odd-frequency Pairing in Josephson Junctions Coupled by Magnetic Textures

Problem Statement
Topological superconductors host Majorana bound states (MBSs), which are promising candidates for decoherence-free quantum computation due to their non-Abelian statistics and self-conjugation properties. However, experimental identification of MBSs remains challenging because topologically trivial states can mimic the zero-energy signatures of MBSs in standard spectroscopy. The field is shifting toward probing the nonlocal behavior and specific symmetry properties of these states. A key theoretical question is how different magnetic textures affect the emergence, symmetry, and coupling of Majorana modes in Josephson junctions (JJs), and how these effects manifest in measurable quantities like induced pairing correlations. Specifically, the relationship between the self-conjugation property of MBSs and the resulting odd-frequency pairing symmetries requires further elucidation in magnetically engineered systems.

Methodology
The authors employ a microscopic tight-binding Green function formalism to analyze a two-dimensional Josephson junction consisting of two conventional $s$-wave superconductors coupled by a one-dimensional magnetic-textured barrier. The system is modeled on a square lattice with a Hamiltonian comprising the left and right superconductors and a magnetic tunnel barrier. The barrier features a spatially modulated magnetization (helical texture) along the interface, with parameters for amplitude ($t_m$) and period ($\xi_m$).

The study investigates two primary configurations:

1. Uninterrupted Magnetic Texture: A continuous magnetic barrier where the junction width ($L_y$) is varied relative to the Majorana localization length.
2. Interrupted Magnetic Texture: A barrier containing a nonmagnetic segment of width $L_0$, which divides the topological superconductor into two segments, creating inner and outer edges.

The analysis focuses on the retarded Green function to compute the local density of states (LDoS) and the anomalous Green function components. These are decomposed into fermionic pairing channels: even-frequency singlet (ESE), odd-frequency singlet (OSO), even-frequency triplet (ETO), and odd-frequency triplet (OTE). Special attention is given to the polarized odd-frequency triplet components ($F^{OTE,\pm}$), which are linked to equal-spin pairing. The study systematically varies the magnetic texture amplitude, the junction width, and the superconducting phase difference ($\phi$) to map the topological phase diagram and the behavior of pairing correlations.

Key Contributions and Results

• Odd-Frequency Pairing as a Topological Fingerprint: In the topologically trivial regime, even-frequency singlet pairing dominates. In contrast, the topological phase is characterized by the coexistence of MBSs and robust odd-frequency equal-spin triplet pairing at the interface edges. The authors demonstrate that the odd-frequency polarized triplets exhibit a divergent $1/\omega$ behavior at zero frequency when MBSs are decoupled. This divergence is intrinsically connected to the self-conjugation property of the Majorana operators.

• Impact of Hybridization on Pairing Symmetry: When the junction is narrow, the wavefunctions of the edge MBSs overlap, leading to hybridization and a finite energy splitting ($\varepsilon > 0$). This hybridization breaks the pure self-conjugation property. Consequently, the odd-frequency triplet pairing transitions from a $1/\omega$ divergence to a linear low-frequency behavior ($\sim \omega$) and develops resonances at $\omega = \pm \varepsilon$. The slope of this linear behavior serves as a signature of the hybridization strength.

• Effect of Nonmagnetic Barriers: Introducing a nonmagnetic interruption in the magnetic texture creates two new topological segments, generating additional "inner" MBSs at the boundaries of the nonmagnetic region.

  • If the nonmagnetic barrier is wide ($L_0 \gg \xi_S$), the inner MBSs remain decoupled, preserving their self-conjugation and the associated $1/\omega$ odd-frequency triplet divergence.
  • If the barrier is narrow ($L_0 \sim \xi_S$), the inner MBSs hybridize, acquiring finite energy and exhibiting linear low-frequency triplet behavior, similar to the narrow junction case. The outer edge modes, however, remain decoupled and retain their self-conjugated nature.

• Phase Control of Topological Regimes: The superconducting phase difference ($\phi$) acts as a tunable parameter.

  • In a single topological segment (uninterrupted barrier), increasing $\phi$ can induce a topological phase transition but often enhances the hybridization of MBSs in narrow junctions, further suppressing the $1/\omega$ behavior.
  • In junctions with a nonmagnetic barrier, the phase difference has a contrasting effect: it can reduce the hybridization energy of the inner MBSs by shortening their wavefunction tails within the nonmagnetic region. This allows the system to recover the self-conjugation property and the characteristic $1/\omega$ odd-frequency triplet resonance for the inner modes, effectively stabilizing the Majorana states.

Significance
The paper establishes odd-frequency equal-spin triplet correlations as a central probe for topological superconductivity in magnetically engineered Josephson junctions. It provides a theoretical framework distinguishing between trivial Andreev bound states and topological Majorana modes based on the frequency dependence of induced pairing. The work highlights that the $1/\omega$ divergence of odd-frequency triplets is a direct consequence of the self-conjugation of Majorana states. Furthermore, it demonstrates that while hybridization destroys this specific signature, the superconducting phase difference can be used as an experimental knob to decouple hybridized states in specific geometries (those with nonmagnetic barriers), thereby recovering the topological signature. This offers a pathway to verify the nonlocal nature and self-conjugation of Majorana modes beyond zero-energy spectroscopy.