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

We perform a theoretical investigation of the nature of vortices in chiral $d+id$ superconductors on the kagome lattice. The study is motivated by recent experimental developments reporting evidence of time-reversal symmetry breaking in the superconducting state of kagome metals. Using self-consistent microscopic calculations that incorporate the characteristics of the band structure of the kagome lattice, we find that fractional vortices permeate the ground state condensate in the presence of an external field. Each fractional vortex carries one third of the superconducting flux quantum and exhibits a characteristic signature related to one of the three sublattice degrees of freedom of the kagome lattice. We discuss the relevance of these results to recent experimental studies of kagome superconductors in the presence of an external magnetic field.

💡 Research Summary

This paper presents a comprehensive theoretical study of vortex structures in chiral d + id superconductors on the kagome lattice, motivated by recent experimental evidence of time‑reversal‑symmetry breaking (TRSB) in kagome metals such as AV₃Sb₅ (A = K, Rb, Cs). Using a tight‑binding description of the kagome electronic structure that includes only nearest‑neighbor hopping, the authors first map out the three‑band dispersion, highlighting the presence of two Van Hove singularities (upper and lower) and a Dirac point at K. Near the upper Van Hove singularity the Bloch wavefunctions become strongly sublattice‑polarized: the weight on one of the three sublattices (A, B, C) dominates while the other two are essentially zero. This “sublattice interference” is a key ingredient for the pairing mechanism explored later.

The superconducting interaction is introduced as an attractive density‑density term V(r − r′) that is taken to be translation‑invariant. By performing a mean‑field decoupling in the Cooper channel the Bogoliubov‑de Gennes (BdG) Hamiltonian is obtained. Symmetry analysis based on the D₆h point group shows that the two‑dimensional irreducible representation E₂g (d‑wave) is the natural candidate for a chiral order parameter. However, on‑site and nearest‑neighbor attractions do not stabilize E₂g near the Van Hove points; instead the authors focus on next‑next‑nearest‑neighbor (NNNN) pairing, which is the shortest range interaction capable of generating intra‑sublattice Cooper pairs.

A detailed group‑theoretical decomposition of the NNNN bond amplitudes reveals three distinct sets of basis functions: one associated with bonds inside the elementary hexagon (transforming as E₂g) and two associated with bonds outside the hexagon (transforming as E′₂g and E″₂g). The resulting gap matrix Δ(k) is a linear combination of these basis functions with complex coefficients that differ by a ±π/2 phase, giving rise to two chiral states Δ⁺(k) and Δ⁻(k). Self‑consistent solutions of the homogeneous BdG equations confirm that, for chemical potentials tuned close to either the upper or lower Van Hove singularities, the chiral d ± id state is energetically favored.

To address vortex physics, the authors introduce a uniform external magnetic field via Peierls phase factors on the hopping terms and solve the full real‑space BdG equations self‑consistently on a large lattice. The iterative procedure converges when the relative change in the gap matrix falls below 10⁻⁸. By varying the field direction relative to the kagome symmetry axes, they explore how the vortex lattice adapts to the underlying three‑sublattice structure.

The central finding is that, instead of forming conventional Abrikosov vortices each carrying a full flux quantum Φ₀ = h/2e, the system prefers a composite defect consisting of a closed domain wall that separates an inner d − id region from the outer d + id bulk. This domain wall encloses two flux quanta (2 Φ₀) but hosts six localized fractional vortices positioned at the corners of a hexagon. Each fractional vortex is tied to a specific sublattice (A, B, or C) and carries exactly one third of a flux quantum, i.e. Φ₀/3 = h/6e. The current loops around these fractional vortices are sublattice‑selective, reflecting the underlying sublattice interference of the electronic states.

Energy analysis shows that the domain‑wall plus fractional‑vortex configuration reduces both the kinetic energy of the quasiparticles and the magnetic energy compared with a regular Abrikosov lattice at the same field strength. The reduction is most pronounced when the magnetic field aligns with a kagome high‑symmetry direction, allowing the three sublattice vortices to arrange symmetrically. Moreover, the character of the fractional vortices changes subtly when the chemical potential is tuned from the upper to the lower Van Hove singularity: the core size, the suppression of the order parameter, and the local density of states differ, offering distinct experimental signatures.

The authors connect these theoretical predictions to several experimental observations in AV₃Sb₅. Quasiparticle interference patterns that depend on the direction of an applied field, the appearance of a Hebel‑Slichter peak in NMR spin‑lattice relaxation, and anomalously weak suppression of Tc by disorder are all consistent with a chiral d + id state hosting sublattice‑specific fractional vortices. The paper also discusses how scanning tunneling microscopy could directly visualize the hexagonal arrangement of 1/3‑flux vortices and how muon‑spin rotation (μSR) or small‑angle neutron scattering could detect the associated magnetic field distribution.

In summary, the work establishes that chiral d + id superconductivity on the kagome lattice naturally gives rise to 1/3‑flux fractional vortices tied to the three sublattices, mediated by sublattice interference near Van Hove singularities. These vortices form a domain‑wall network that is energetically favorable under magnetic fields and provides a concrete, experimentally testable hallmark of TRSB kagome superconductivity. The study broadens our understanding of topological defects in multicomponent superconductors and points to kagome materials as a fertile platform for realizing exotic fractional flux excitations.