Unconventional mixed state in the nematic superconductor LiFeAs

Using transverse muon-spin spectroscopy, researchers have discovered an unconventional mixed state in LiFeAs single crystals consisting of stripes of "coreless" vortices formed by bound pairs of half-quantum vortices.

The Gist

The Mystery of the "Ghostly" Magnets: A Story of LiFeAs

Imagine you are looking at a perfectly organized marching band. Every musician is standing in a precise, predictable grid, spaced exactly five feet apart. This is how scientists usually expect to see magnetism inside a "superconductor" (a special material that allows electricity to flow with zero resistance). In these materials, magnetism enters in little "packets" called vortices, which act like tiny, organized soldiers standing in a perfect square lattice.

But a team of researchers just discovered that in a specific material called LiFeAs, the marching band has gone rogue. Instead of a neat grid, the magnetism has transformed into something much more exotic and "ghostly."

Here is the breakdown of their discovery using a few simple analogies.

---

1. The "Coreless" Vortex: The Ghost in the Machine

In a normal superconductor, a magnetic vortex is like a whirlpool in a bathtub. At the very center of the whirlpool, there is a "core" where the water is spinning so violently that the structure of the water breaks down. In physics terms, the superconductivity "dies" at the center of the vortex.

However, the researchers found that in LiFeAs, these vortices are "coreless."

The Analogy: Imagine a whirlpool where, instead of the water disappearing at the center, the water simply changes its color or direction as you pass through the middle. The "superconducting" magic never actually stops; it just twists and turns into a different shape. It’s like a ghost that can walk through a wall without ever actually breaking the wall's structure. These are called Skyrmions.

2. The "Split Personality" Vortex

The paper explains that these coreless vortices are actually made of two smaller, "half-quantum" vortices that are stuck together.

The Analogy: Think of a standard vortex as a single, solid marble. Now, imagine that this marble is actually two tiny, magnetized droplets held together by a very thin, invisible rubber band. They are so close they look like one marble, but they are actually two separate entities dancing around each other. Because they are "half-sized," they don't follow the standard rules of the marching band.

3. The "Striped" Formation: From Grids to Lanes

Because these "double-droplet" vortices have a strange, directional personality (called nematicity), they don't like to stand in a square grid. Instead, they prefer to line up in rows.

The Analogy: Instead of the marching band standing in a wide, open field in a square pattern, they have decided to form tightly packed lanes, like cars on a highway.

The researchers used a technique called Muon Spin Spectroscopy (which is essentially like using microscopic "detectives" to probe the magnetic field) to see this. They noticed that the magnetic field didn't look like one single, smooth hum; it looked like two different notes being played at once. This "double note" is the smoking gun that proves the vortices are arranged in these unique, striped chains.

4. Why does this matter?

You might ask, "Who cares about tiny magnetic stripes in a crystal?"

The reason is that this discovery changes our "rulebook" for physics. For decades, we have used a standard mathematical model (the Brandt model) to understand how much electricity these materials can carry. This paper proves that for these "unconventional" materials, the old rulebook is wrong.

If we want to build the super-fast, ultra-efficient quantum computers or power grids of the future, we need to understand these "ghostly" magnetic textures. We can't use the old maps to navigate a new world.

---

In short: The researchers found that in LiFeAs, magnetism doesn't just "sit" in the material; it performs a complex, swirling, striped dance that defies our traditional understanding of how superconductors work.

Technical Summary

Technical Summary: Unconventional Mixed State in the Nematic Superconductor $\text{LiFeAs}$

1. Problem Statement

In conventional type-II superconductors, the mixed state is characterized by the Abrikosov Vortex Lattice (AVL), where magnetic flux penetrates the material in quantized vortices, each carrying a single flux quantum $\Phi_0 = h/2e$. This structure is typical of even-parity (spin-singlet) superconductors.

However, in odd-parity (spin-triplet) superconductors, the order parameter (OP) is a vector ($\mathbf{d}$-vector) rather than a scalar. This internal degree of freedom allows for more complex topological excitations, such as:

• Fractional Vortices (FV): Vortices carrying a fraction of $\Phi_0$.
• Coreless Vortices (CLV) / Skyrmions: Configurations where the OP does not vanish at the core but rotates in space, effectively acting as a bound state of two spatially separated half-quantum vortices (HQVs).

While theoretically predicted, experimental evidence for CLVs in the bulk of a superconductor has been extremely scarce, with most observations limited to mesoscopic samples or surfaces. The authors aim to provide the first unambiguous bulk evidence of such an unconventional mixed state in the nematic superconductor $\text{LiFeAs}$.

2. Methodology

The study employs a combination of advanced experimental spectroscopy and Ginzburg-Landau (GL) theoretical modeling:

• Angle-Resolved Transverse-Field Muon-Spin Spectroscopy (TF-$\mu$SR):
  • The researchers applied magnetic fields in different orientations relative to the crystal axes: in-plane ($B \parallel a$ and $B$ at $45^\circ$ to $a$) and out-of-plane ($B \parallel c$).
  • By analyzing the Fourier Transform (FT) of the muon polarization, they extracted the local magnetic field probability distribution $p(B)$.
  • They used a skewed Gaussian function (skG) to model the asymmetric field distributions resulting from unconventional vortex structures.
• Zero-Field $\mu$SR (ZF-$\mu$SR): Used to check for Time-Reversal Symmetry (TRS) breaking.
• Theoretical Ginzburg-Landau Modeling:
  • A two-component GL model was developed to simulate a nematic superconductor with an in-plane $\mathbf{d}$-vector.
  • The model accounts for the coupling between the two superconducting components and the effects of nematicity.
  • Numerical minimization of the Gibbs free energy was used to find periodic lattice solutions and predict the evolution of $p(B)$ with temperature.

3. Key Results

• Observation of Peak Splitting: The most significant finding is the splitting of the superconducting peak in the $p(B)$ distribution when the magnetic field is applied parallel to the $c$-axis. This splitting is observed between $2\text{ K}$ and $6\text{ K}$.
• Skyrmionic Vortex Lattice: The double-peak structure in $p(B)$ is a signature of a skyrmion (coreless) vortex lattice. In this state, the vortices arrange themselves into stripes (chains). The two peaks in the field distribution correspond to the two distinct length scales: the separation between the stripes and the separation of skyrmions within a single stripe.
• Temperature Evolution: The splitting is not constant; it appears at $2\text{ K}$, reaches a maximum, and then vanishes above $6\text{ K}$. This is interpreted as the melting of the CLV lattice into a conventional AVL as thermal fluctuations increase.
• Nematicity and Symmetry: The data confirms the nematic nature of the superconducting state. The in-plane measurements showed rotational symmetry breaking (RSB), consistent with the spontaneous selection of a preferred orientation in the $\mathbf{d}$-vector.
• Absence of TRS Breaking: ZF-$\mu$SR results showed no additional relaxation below $T_c$, ruling out chiral or non-unitary spin-triplet states that would break time-reversal symmetry.

4. Significance

• First Bulk Evidence: This work provides the first clear, unambiguous experimental evidence of coreless (skyrmionic) vortices in the bulk of a superconducting material.
• Validation of Multicomponent Theory: The results provide strong support for the existence of multicomponent, odd-parity superconducting states in iron-based superconductors, specifically driven by strong spin-orbit coupling and orbital hybridization.
• Impact on Material Characterization: The authors demonstrate that the standard Brandt model, widely used to extract magnetic penetration depth ($\lambda$) from $\mu$SR data, is unsuitable for unconventional vortex lattices. This necessitates the development of new scaling laws for characterizing unconventional superconductors.
• Topological Physics: The discovery links superconductivity with topological textures (skyrmions), opening new avenues for studying topological matter in bulk crystalline systems.