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

The gist: Fractional 1/3 quantum vortices permeate the ground state condensate in chiral d + id superconductors on the kagome lattice, each carrying one third of the superconducting flux quantum and exhibiting a characteristic signature related to one of the three sublattice degrees of freedom

Introduction and Motivation

The study investigates the nature of vortices in chiral d + id superconductors on the kagome lattice using self-consistent microscopic calculations that incorporate the characteristics of the band structure of the kagome lattice. The research is motivated by recent experimental developments reporting evidence of time-reversal symmetry breaking in the superconducting state of kagome metals. The paper discusses how fractional vortices, which carry only a fraction of superconducting flux quantum Φ0 = h/2e [1], play a key role in the superconductor’s response to external fields and thermal fluctuations. Fractional vortices have been studied in the context of chiral triplet superfluidity and triplet superconductors, and even singlet superconducting states with spontaneous time-reversal symmetry breaking (TRSB) are believed to host fractional vortices. In the chiral triplet p + ip state, fractional vortices necessarily carry a half flux quantum, enforced by the symmetry relating the two order parameter components to each other. A different kind of fractional vortices has been proposed in multi-gap superconductors, associated with phase winding in only one of the gap components.

Methodology

The methodology employs a pairing mechanism which favors the d + id state close to the van Hove singularities in the kagome bandstructure. A microscopic and self-consistent BdG framework is used to study the inhomogeneous superconducting state in the presence of an external magnetic field.

1. The electronic structure of the kagome lattice is described using a tight-binding model, where the normal state Hamiltonian is given by HTB = -tX<rr',σc†r'σcrσ - µXr,σc†rσcrσ (1).

2. The diagonalization of the Hamiltonian yields three electronic bands, including a flat band and two van Hove singularities at the M points in the n = 2 and n = 1 bands, respectively.

3. The superconducting state is modeled via a mean-field treatment of the interaction Hamiltonian, leading to the Bogoliubov-de Gennes (BdG) Hamiltonian HBdG = Ψ†HTB ∆∆† -HTTBΨ (5).

4. The pairing interaction favors E2g (d-wave) order close to the van-Hove singularities, particularly through next-next nearest-neighbor (NNNN) pairing which supports intra-sublattice pairing.

5. Peierls phases are incorporated into the real space BdG formalism using substitution tr'r 7→ tr're iφr'r (10).

Results and Vortex Structure

The study focuses on vortex states in chiral d + id superconductivity on the kagome lattice near the upper van Hove singularity (UvHS).

1. The system is initialized in the homogeneous ∆+ state, and a chiral superconducting state spontaneously breaks time-reversal symmetry.

2. For an out-of-plane magnetic field (⊙ direction), the local density of states (LDOS) displays a hexagonal shape with six distinct regions of enhanced LDOS, indicating the presence of six vortices. Each vortex carries a magnetic flux Φ0/3 and is associated with only one sublattice degree of freedom (A in blue, B in orange, C in green).

3. For an into-plane magnetic field (⊗ direction), the LDOS shows two triangular features, suggesting that each triangular feature does not correspond to a single conventional Abrikosov vortex but instead consists of three tightly bound fractional vortices.

4. The current patterns reveal that the clockwise current on the interior of the hexagon in the out-of-plane field induces a magnetic field that opposes the external field, with local maxima where fractional vortices are located.

5. The chiral order parameters ∆+ and ∆- form a closed domain wall enclosing these fractional vortices, which is energetically stable due to the repulsive interaction of fractional vortices on domain walls.

Sublattice Dependence and Topology

The distinction between vortex states at the UvHS and LvHS is directly related to the distinct microscopic sublattice structure of the Bloch states at these two van Hove singularities.

1. At the UvHS, fractional vortices belong to individual sublattice degrees of freedom (A, B or C), where the active band exhibits strong sublattice interference causing the Fermi surface to be extremely sublattice polarized.

2. At the LvHS, there is an inverted sublattice character inherited from the lower band's interference compared to the upper one, meaning that where LDOS peaks were confined to a single sublattice at UvHS, they are in two of the sublattices (B and C) at LvHS.

3. The topological charge density is computed using a CP N−1 topological invariant, which is independent of the basis choice and captures the non-trivial topological structure embedded in the relative degrees of freedom.

4. For both field directions, the total topological invariant is found to be Q = ±2, which is consistent with the expected flux quantization from continuum theory. The local topological charge density peaks align well with the locations of enhanced LDOS and currents, demonstrating its connection to these observable quantities.

5. The difference in vortex structure between UvHS and LvHS demonstrates clearly the importance of the microscopic character of the sublattice interference.

Conclusion

The theoretical study finds that coreless vortices consisting of multiple fractional vortices, each carrying one third of the superconducting flux quantum Φ0 = h/6e, are formed in chiral d + id superconductors on the kagome lattice. These fractional vortices belong to individual sublattice degrees of freedom (A, B or C) at the UvHS and exhibit an inverted sublattice character at the LvHS. The results suggest that key structure may get lost when moving from a fully microscopic picture to the effective twocomponent GL field theory, indicating that the fractionalization is influenced by microscopic details like sublattice degrees of freedom. The findings motivate new experimental scanning SQUID microscopy or STM studies to locate favorable conditions for observing these fractional vortices in kagome superconductors.

ACKNOWLEDGMENTS

F.A.S.P and B.M.A acknowledge support from the Independent Research Fund Denmark Grant No. 5241-00007B. M.B was supported by a research grant (VIL69220) from VILLUM FONDEN. A.K acknowledges support by the Danish National Committee for Research Infrastructure (NUFI) through the ESSLighthouse Q-MAT.

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[9] J. T.

1. Increased Predictive Capability for TRSB Kagome Superconductors: The system can now predict vortex states in chiral d+id superconductors on kagome lattices by considering both upper and lower van Hove singularities (UvHS and LvHS), as the paper shows the difference between the vortex states at the UvHS and LvHS will be directly related their distinct microscopic sublattice structure of the Bloch states.

2. Accurate Characterization of Fractional Vortices: The AI can now distinguish between different fractional vortex types, specifically identifying that 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, which is crucial for interpreting experimental signatures like those from STM or scanning SQUID measurements.

3. Model Refinement Beyond Simple GL Theory: The AI can move beyond the predictions of an effective two-component Ginzburg-Landau (GL) theory by solving the fully microscopic BdG Hamiltonian, thus capturing effects that a simpler theory might miss, such as the finding that the one-third fractional vortices we find are self-consistent solutions to the mean-field equations of the BdG Hamiltonian.

4. Directional Field Dependence Analysis: The system can predict how vortex structures change based on external field orientation, noting that the difference in the collective structure formed under the opposite field directions is expected from the fact that the bulk superconducting state is chiral and breaks time-reversal symmetry, leading to qualitatively distinct LDOS pattern[s] due to the different arrangement of fractional vortices.

5. Topological Invariant Calculation: The AI can compute a basis-independent topological invariant, defined as a CP N−1 topological invariant, which is used as a basis-independent indicator where in space the fractional vortices are located, connecting the calculated charge density to observable quantities like LDOS peaks.