Fractional $1/3$ quantum vortices in chiral $d+id$ kagome superconductors

Through self-consistent microscopic calculations, this study reveals that chiral $d+id$ superconductors on the kagome lattice host fractional vortices carrying one-third of the magnetic flux quantum, each uniquely associated with the lattice's three sublattice degrees of freedom, offering a theoretical explanation for recent experimental observations of time-reversal symmetry breaking in kagome metals.

The Gist

Imagine a superconductor not as a solid block of metal, but as a bustling city made of a very specific, repeating pattern of streets and buildings. This pattern is called a kagome lattice (named after a traditional Japanese basket weave). In this city, the "residents" are electrons, and when the city gets cold enough, they all agree to dance together in a perfect, synchronized rhythm. This is superconductivity: electricity flows with zero resistance because everyone is moving in unison.

Usually, if you poke a hole in this synchronized dance (by applying a magnetic field), you create a vortex. Think of a vortex like a whirlpool in a river. In a normal superconductor, this whirlpool is a single, indivisible unit of chaos. It carries a specific amount of "magnetic water" (called a flux quantum) that cannot be split. It's like a whole apple; you can't have half an apple in this context.

The Big Discovery: The "One-Third" Apple

This paper reports a surprising discovery: in this specific kagome city, the whirlpools can be split. Not just in half, but into three equal pieces.

The researchers found that instead of one big whirlpool, the magnetic field creates six tiny, fractional whirlpools that group together to form a larger structure. Each of these tiny whirlpools carries only one-third of the usual magnetic "apple."

The Secret Ingredient: The Three Neighborhoods

Why does this happen? The kagome city is unique because it has three distinct neighborhoods (called sublattices A, B, and C). Imagine the city is divided into three zones, each painted a different color: Blue, Orange, and Green.

In most materials, these zones blend together seamlessly. But in this kagome city, the electrons in the Blue zone, Orange zone, and Green zone behave very differently. They are like three different dance troupes that usually perform together but have their own unique styles.

The researchers discovered that the "fractional whirlpools" are actually specialized.

• One tiny whirlpool belongs only to the Blue neighborhood.
• Another belongs only to the Orange neighborhood.
• The third belongs only to the Green neighborhood.

Because the three neighborhoods are so distinct, the magnetic field can't force them to share the burden equally. Instead, the field splits up, creating three separate, smaller vortices, each taking care of just one neighborhood.

The "Ghost" Whirlpool

Here is the most magical part. Usually, a whirlpool has a "core" where the dancing stops completely (the center of the storm is calm). But in this case, the researchers found coreless vortices.

Imagine a storm where the wind is swirling violently in six different spots, but the very center of the storm is still calm and the dancers are still moving! The "whirlpool" is actually a ring of six tiny fractional vortices holding hands, surrounding a calm center. The "storm" never fully stops; it just rearranges itself into a hexagonal shape.

Why Does This Matter?

1. It's a New Kind of Physics: For a long time, physicists thought magnetic flux could only be split in half (like in some exotic liquid helium). Finding a "one-third" split is like discovering a new fundamental rule of nature. It proves that the microscopic details of the material (the three neighborhoods) matter just as much as the big picture.
2. Solving a Mystery: Recently, scientists found new superconducting metals (like Vanadium-based kagome metals) that seem to break the rules of time symmetry (meaning the physics looks different if you play the movie backward). This paper suggests that if you look closely at these materials with a magnetic field, you might see these "one-third" whirlpools. Finding them would be the "smoking gun" proof that these materials are indeed in this exotic, chiral state.
3. The "Map" of the Material: The paper shows that if you change the direction of the magnetic field (pointing it up or down), the pattern of these whirlpools changes completely. It's like turning a kaleidoscope; the same pieces rearrange into a totally different, beautiful pattern depending on how you look at it.

The Takeaway

Think of this paper as a map to a hidden world inside a superconductor. The researchers used powerful computer simulations to show that inside these kagome metals, magnetic fields don't just make one big hole. Instead, they shatter into tiny, fractional pieces, each tied to a specific part of the crystal's structure. It's a beautiful example of how the tiny, invisible details of a material can create a completely new and exotic way for the universe to behave.

Technical Summary

1. Problem Statement and Motivation

The paper investigates the nature of vortex states in chiral $d+id$ superconductors on a kagome lattice. This research is motivated by recent experimental discoveries of superconductivity in vanadium-based kagome metals ($AV_3Sb_5$, where $A = K, Rb, Cs$). These materials exhibit evidence of Time-Reversal Symmetry Breaking (TRSB) in the superconducting state, suggesting a chiral order parameter.

While multicomponent superconductors are theoretically known to host fractional vortices (carrying a fraction of the magnetic flux quantum $\Phi_0 = h/2e$), the specific nature of these vortices depends heavily on the microscopic details of the lattice and band structure. Standard effective field theories (like Ginzburg-Landau) often predict half-quantum vortices ($\Phi_0/2$) for two-component order parameters. However, the authors hypothesize that the unique sublattice interference and microscopic degrees of freedom inherent to the kagome lattice may lead to exotic fractional vortices with different flux quantization, specifically $\Phi_0/3$.

2. Methodology

The authors employ a fully self-consistent microscopic Bogoliubov-de Gennes (BdG) formalism to model the system. Key methodological components include:

• Lattice Model: A tight-binding model of the kagome lattice with three sublattices ($A, B, C$) per unit cell. The Hamiltonian includes nearest-neighbor hopping and chemical potential ($\mu$).
• Pairing Mechanism: The study focuses on next-next-nearest-neighbor (NNNN) attractive interactions. Symmetry analysis shows that NNNN pairing favors the $E_{2g}$ irreducible representation, leading to a chiral $d+id$ state. This is crucial because shorter-range interactions (on-site or nearest-neighbor) do not stabilize this specific chiral state near the van Hove singularities.
• Self-Consistency: The gap matrix $\Delta$ is solved iteratively in real space to handle inhomogeneous states (vortices) under an external magnetic field. The system uses Peierls substitution to incorporate the magnetic vector potential, ensuring gauge invariance.
• Boundary Conditions: Periodic boundary conditions are applied to a magnetic unit cell containing two superconducting flux quanta ($\pm 2\Phi_0$). This setup allows the formation of stable vortex structures without edge effects.
• Fermi Level Tuning: Calculations are performed near the Upper van Hove Singularity (UvHS) and the Lower van Hove Singularity (LvHS). These points are chosen because the density of states is high, and the sublattice interference effects are most pronounced.

3. Key Contributions and Results

A. Discovery of $\Phi_0/3$ Fractional Vortices

The central finding is the existence of fractional vortices carrying exactly one-third of the flux quantum ($\Phi_0/3 = h/6e$).

• Structure: When two flux quanta are introduced, the system does not form two conventional Abrikosov vortices. Instead, it forms a "coreless vortex" structure composed of six fractional vortices arranged in a hexagonal pattern (for one field direction) or two triangular clusters of three (for the opposite direction).
• Sublattice Polarization: Each fractional vortex is uniquely associated with one of the three sublattice degrees of freedom ($A, B,$ or $C$).
  • At the UvHS, the Local Density of States (LDOS) peaks at the vortex cores are strictly confined to a single sublattice. For example, one vortex is purely on sublattice A, another on B, etc.
  • This demonstrates that the sublattice interference effectively decouples the sublattices locally, allowing them to host independent fractional flux.

B. Coreless Vortex and Domain Wall Formation

The fractional vortices are not isolated; they are bound together by a closed domain wall.

• The bulk superconducting state is chiral ($\Delta_+ = d+id$). Inside the region enclosed by the fractional vortices, the order parameter switches to the opposite chirality ($\Delta_- = d-id$).
• The boundary between $\Delta_+$ and $\Delta_-$ forms a domain wall. The fractional vortices sit on this wall.
• The total superconducting gap magnitude $|\Delta|$ never vanishes (hence "coreless"), distinguishing these from conventional Abrikosov vortices where the gap goes to zero at the core.

C. Dependence on Magnetic Field Direction and Singularity

The vortex structure is highly sensitive to the direction of the external magnetic field ($B$) and the specific van Hove singularity:

• Field Direction ($B \odot$ vs. $B \otimes$):
  • For $B$ out-of-plane ($\odot$), the six fractional vortices form a hexagonal ring.
  • For $B$ into-plane ($\otimes$), the vortices rearrange into two triangular clusters.
  • This asymmetry arises because the bulk chiral state breaks time-reversal symmetry, making the response to $+B$ and $-B$ non-degenerate.
• UvHS vs. LvHS:
  • UvHS: Exhibits strong sublattice polarization. The six vortices are distinct and carry $\Phi_0/3$ each, strictly localized on specific sublattices.
  • LvHS: The sublattice interference is inverted compared to the UvHS. While a hexagonal LDOS pattern still appears, the vortices are less distinct; the LDOS peaks involve multiple sublattices, and the topological charge density suggests a less clear separation of fractional vortices compared to the UvHS case.

D. Topological Characterization

The authors compute the topological charge density using a basis-independent $CP^{N-1}$ invariant (specifically related to the Bargmann invariant).

• The total topological charge matches the magnetic flux ($Q = \pm 2$).
• The local charge density peaks align perfectly with the fractional vortex locations, confirming their topological nature.
• This confirms that the fractionalization is a robust topological feature of the multicomponent order parameter on the kagome lattice, not an artifact of the basis choice.

4. Significance and Implications

• Beyond Effective Theories: The results challenge the standard Landau paradigm of effective two-component Ginzburg-Landau theories, which typically predict $\Phi_0/2$ vortices. The paper demonstrates that microscopic details (specifically the kagome sublattice structure and NNNN pairing) are essential for predicting the correct flux quantization ($\Phi_0/3$).
• Experimental Relevance: The findings provide a specific theoretical signature for the chiral $d+id$ state in $AV_3Sb_5$ materials.
  • The asymmetry in LDOS patterns between opposite magnetic field directions offers a testable prediction for Scanning Tunneling Microscopy (STM) and Scanning SQUID microscopy.
  • The observation of sublattice-polarized fractional vortices would be strong evidence for a chiral multicomponent ground state and the specific role of sublattice interference in kagome superconductors.
• New State of Matter: The identification of $\Phi_0/3$ vortices expands the known landscape of topological excitations in superconductors, linking concepts from multiband superconductivity, skyrmions, and fractional quantum Hall physics.

In conclusion, the paper establishes that chiral $d+id$ superconductivity on the kagome lattice hosts a unique "coreless" vortex state composed of six $\Phi_0/3$ fractional vortices, driven by the interplay between sublattice degrees of freedom and time-reversal symmetry breaking.