Anisotropic Josephson coupling of $d$ vectors in triplet superconductors arising from frustrated spin textures

This paper demonstrates that coupling itinerant electrons to noncollinear, frustrated spin textures induces anisotropic Josephson couplings between triplet superconducting $d$ vectors, leading to spatially varying pairing orders, anomalous vortices, and a Josephson diode effect.

The Gist

Imagine you have a group of dancers (the electrons) trying to hold hands and move in perfect unison across a dance floor. In a standard superconductor, they all hold hands in the same way, forming a smooth, rigid line that flows without friction. This is like a "superfluid stiffness"—it wants everything to be straight, uniform, and orderly.

Now, imagine the dance floor itself is covered in a tricky, twisted pattern of invisible magnets (frustrated spin textures). These magnets don't just sit still; they are arranged in a way that creates a "tug-of-war" or a puzzle that can't be solved by everyone pointing in the same direction. This is what physicists call a "frustrated magnetic texture."

This paper explores what happens when those dancing electrons try to hold hands while navigating this tricky, twisted magnetic floor. Here is the breakdown of their discovery:

1. The "Hand-Holding" Gets Twisted

In these special materials, the electrons don't just hold hands normally; they form "triplet pairs," which is like a dance move where the partners have a specific orientation or "pose" (represented by a vector called the d-vector).

Usually, if two groups of dancers (superconducting grains) meet, they want to align their poses perfectly to keep the dance smooth. However, the authors found that the twisted magnetic floor acts like a mischievous director. It forces the dancers to change their poses slightly as they move from one spot to another.

Instead of a rigid, straight line, the dance formation becomes "pliable" or flexible. The magnetic floor introduces a new kind of force that competes with the natural desire to stay straight. It's like if the floor itself whispered to the dancers, "Hey, tilt your heads a little bit to the left here, and a little bit to the right there."

2. The "Anisotropic" Connection

The paper describes this new force as an "anisotropic Josephson coupling." In simple terms, "anisotropic" means the rules change depending on the direction.

Think of it like a hinge on a door. A normal hinge lets the door swing open easily in one direction but locks it in another. The magnetic texture creates a similar effect for the electron pairs. It allows them to connect, but it makes them "wobble" or rotate their orientation as they pass from one grain to the next. This is compared to famous magnetic interactions (Dzyaloshinskii-Moriya and $\Gamma$-type), but applied to superconductors instead of magnets.

3. Spontaneous Swirls (Vortices)

Because the dancers are being forced to twist and turn by the magnetic floor, they can't stay in a straight line. This creates spontaneous swirls or spirals in the dance formation, even if there is no external wind (magnetic field) blowing on them.

The authors predict that this can create "anomalous vortices." Imagine a whirlpool forming in a river just because the riverbed has a specific rocky pattern, not because of a dam or a storm. In these materials, the "whirlpools" are twists in the electron pairing that happen naturally due to the frustrated magnetic texture underneath.

4. The One-Way Street (Josephson Diode Effect)

Perhaps the most practical-sounding discovery is the "Josephson diode effect."

Think of a diode as a one-way street for electricity. Usually, electricity flows the same way forward and backward. But in these materials, the twisted magnetic texture acts like a traffic cop who lets cars drive fast in one direction but slows them down in the other.

The paper claims that the "efficiency" of this one-way street depends on the "chirality" (or "handedness") of the magnetic texture. If the magnetic spins are arranged in a left-handed spiral, the electricity might flow easily one way but struggle the other. If you flip the magnetic arrangement to a right-handed spiral, the easy direction flips too. This happens without needing any external magnets to be turned on; the material's own internal "twisted" nature does the work.

Real-World Examples Mentioned

The authors point to two specific materials where this "dance" is happening:

• Mn3Ge: A material with a triangular magnetic pattern that creates these twisted effects.
• 4Hb-TaS2: A layered material that acts like a sandwich, where one layer is a "spin liquid" (a very jiggly, frustrated magnetic state) and the other is a superconductor. The "jiggly" layer influences the "smooth" layer to create these twisted patterns.

Summary

In short, this paper shows that if you put superconducting electrons on a floor with a "frustrated" (twisted and conflicting) magnetic pattern, the electrons won't just flow in a straight line. They will be forced to twist, turn, and swirl. This creates a flexible, wobbly superconducting state that can flow electricity more easily in one direction than the other, all driven by the hidden, twisted geometry of the magnetic atoms underneath.

Technical Summary

Technical Summary: Anisotropic Josephson coupling of d vectors in triplet superconductors arising from frustrated spin textures

Problem Statement
Recent experimental observations in materials such as the chiral antiferromagnetic Weyl semimetal Mn$_3$Ge and the van der Waals heterostructure 4Hb-TaS$_2$ have revealed a complex interplay between unconventional superconductivity and frustrated magnetic textures. These systems exhibit phenomena including long-range coherent Josephson supercurrents, anomalous Hall effects, spontaneous vortices, and magnetic memory effects. While frustrated spin textures are widely believed to underlie these behaviors, the microscopic mechanism linking noncollinear magnetic order to the spatial texture of spin-triplet superconducting pairing remains underexplored. Existing theoretical frameworks, such as the Bogoliubov-de Gennes (BdG) equations, are computationally prohibitive for complex real-space spin textures, while quasiclassical methods often assume weak gradients and smooth textures, potentially missing critical anisotropic effects. This work addresses the gap in understanding how frustrated, noncollinear local exchange fields modify the Josephson coupling between superconducting grains, specifically focusing on the orientation of the triplet pairing $d$ vector.

Methodology
The authors employ a multi-scale theoretical approach combining phenomenological free energy analysis with a microscopic perturbative framework:

1. Phenomenological Free Energy: The study begins by generalizing the free energy contribution from spatially varying $d$-vector order parameters. The authors propose that in the presence of a local exchange field, the Josephson coupling between adjacent grains ($n$ and $m$) is not limited to a simple Heisenberg-like term but includes anisotropic contributions analogous to Dzyaloshinskii-Moriya (DM) and $\Gamma$-type interactions found in magnetism.
2. Microscopic Model (s-d Model): To derive these couplings, the authors utilize an $s$-$d$ exchange model on a geometrically frustrated lattice (specifically triangular and kagome lattices). Itinerant $s$-electrons are coupled to a classical, noncollinear local exchange field generated by localized $d$-electron spins.
3. Perturbative Expansion (T-matrix): Using Green's function methods based on the T-matrix formalism, the authors perform a controlled perturbative expansion of the $s$-$d$ coupling ($J_{sd}$) up to third order. This expansion allows them to derive an effective tunneling matrix ($T_{ij}$) between neighboring sites that incorporates the effects of the local spin texture.
4. Derivation of Couplings: The effective tunneling matrix is substituted into the Ambegaokar-Baratoff formalism to calculate the Josephson form factor. This yields explicit expressions for the anisotropic Josephson coupling coefficients ($J_{nm}$, $D_{nm}$, and $\Gamma_{nm}$) in terms of spin correlation functions (scalar products, cross products, and scalar spin chirality) of the underlying magnetic texture.

Key Contributions and Results

• Anisotropic Josephson Coupling: The primary result is the demonstration that coupling itinerant electrons to a noncollinear exchange field induces anisotropic Josephson couplings between $d$ vectors. Specifically, the effective tunneling generates:

  • A symmetric Heisenberg-like term ($J_{nm}$) favoring collinear alignment.
  • An antisymmetric DM-like term ($D_{nm} \propto \mathbf{s}_i \times \mathbf{s}_j$) favoring noncollinear alignment.
  • A symmetric traceless $\Gamma$-type term ($\Gamma_{nm}$).
    Crucially, the DM-like coupling arises even in coplanar spin structures with vanishing scalar spin chirality, provided the spin texture is noncollinear (e.g., 120$^\circ$ ordered antiferromagnetism).

• "Pliability" of the Pairing Order: The competition between the stiffness term ($J_{nm}$) and the anisotropic DM-like term ($D_{nm}$) endows the pairing order with "pliability." This competition can overcome the superfluid stiffness, energetically favoring spatially inhomogeneous $d$-vector textures over uniform ones. The authors estimate characteristic length scales for these textures (e.g., $\lambda \sim 10$ microns for 4Hb-TaS$_2$ parameters), suggesting the formation of large-scale vortex or spiral textures.

• Anomalous Vortices: The spatial variation of the $d$ vector contributes to the superfluid velocity via a gauge connection term ($i\hat{d}^* \cdot \nabla \hat{d}$). For nonunitary pairing orders, this leads to nontrivial circulation of the superfluid velocity even in the absence of an external magnetic field. This mechanism predicts the existence of coreless vortices (analogous to Mermin-Ho vortices in superfluid $^3$He-A) and skyrmion-like defects in the $d$-vector field.

• Josephson Diode Effect: The authors predict a Josephson diode effect in superconductor-insulator-superconductor (SIS) junctions where the barrier possesses nonvanishing spin chirality. The critical current becomes direction-dependent ($I_c^+ \neq I_c^-$), with the efficiency proportional to the spin chirality ($\chi_{ijk}$) and the breaking of time-reversal and parity symmetries at the interface.

• Robustness in Strong Coupling: The study extends to the strong coupling limit ($J_{sd} \gg t_0$), showing that anisotropic couplings persist and can be comparable in magnitude to the Heisenberg term, further promoting inhomogeneous textures. This regime is relevant for materials like Mn$_3$Ge where $s$-$d$ coupling is strong.

Significance and Claims
The paper establishes a direct theoretical link between frustrated magnetism and the spatial textures of triplet superconducting pairing. By deriving anisotropic Josephson couplings from first principles, the work provides a mechanism for the "pliability" of the superconducting order parameter, explaining how magnetic frustration can drive spatially varying superconducting states without external fields.

The authors claim these results offer a unified explanation for several experimental anomalies in Mn$_3$Ge and 4Hb-TaS$_2$, including:

• The observation of long-range supercurrents and hysteretic behavior in chiral antiferromagnets.
• The emergence of spontaneous vortices and magnetic memory effects in 4Hb-TaS$_2$.
• The potential for realizing a field-free Josephson diode effect in these materials by tuning the magnetic texture (e.g., via magnetic fields to induce canting and spin chirality).

The work suggests that the interplay between frustrated spin textures and unconventional superconductivity is a fertile ground for generating topological superconducting states and novel transport phenomena, providing a framework for interpreting current experimental data and guiding future investigations into materials where superconductivity is proximity-induced or intrinsic.