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

Abstract

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.

Figures (12)

• Figure 1: (a) Illustration of vortex state in a $d+id$ superconductor on the kagome lattice: Due to the interplay of the sublattice degrees of freedom (A,B,C), a phase boundary between the $d+id$ phase ($\Delta^+$) and a $d-id$ phase ($\Delta^-$) forms. At the boundary, a hexagonal structure with fractionalized vortices and suppressed order parameter on the three sublattices (blue, orange, green dots) form, exhibiting individual current loops around each of them. (b) Kagome lattice with the three sublattices A, B and C. The color and orientation of the diamonds are unique for each sublattice degree of freedom, and will be used throughout this work.
• Figure 2: (a) Band structure of the kagome lattice along the high-symmetry path (see inset) for the three bands. The upper (lower) van Hove singularities are labeled as UvHS (LvHS). (b) Weight of the Bloch states in the second band ($n=2$) on the A, B and C sublattices displayed in the first Brillouin zone. The Fermi surface for $\mu=0$ is indicated in black.
• Figure 3: Visualization of the real-space representations (reps) describing the allowed singlet pairing states assuming only NNNN interactions. The reps are denoted by the irreducible reps (irrep) which they transform as. We distinguish between reps transforming as the same irrep $\Gamma$ as $\Gamma'$ and $\Gamma"$. For the two-dimensional reps ($E_{2g}$, $E_{2g}'$ and $E_{2g}"$) the outer index enumerates the components of the rep. The color of the line indicates the sign of $\Delta_{\textbf{r}\textbf{r}'}$: red positive and blue negative. The line thickness indicates the magnitude of $\Delta_{\textbf{r}\textbf{r}'}$: no line $0$, thin line $1$, and line thick $2$.
• Figure 4: LDOS at the Fermi level for the vortex states found at the upper van Hove singularity. The magnetic field in (a) is in the out-of-plane ($\odot$ direction), and in (b) it is in the into-plane ($\otimes$ direction). In both cases the bulk chirality is $\Delta^+$ and the magnetic flux through the system is $\pm 2 \Phi_0$. The orientation of each diamond along with the three different colormaps tells which sublattice degree of freedom each site belongs to (see Fig \ref{['fig:sketch']}(b) for reference). Although the structure is different for the two field directions, we clearly observe six regions of enhanced LDOS indicating the presence of six vortices. Each vortex carries a magnetic flux $\Phi_0 / 3$ and is associated with only one sublattice degree of freedom (A in blue, B in orange, C in green).
• Figure 5: LDOS at the Fermi level in FIG. \ref{['fig:LDOS_UvH']} shown separately for each of the sublattices. The top (bottom) row correspond to magnetic field in the $\odot$ ($\otimes$) direction, while each column (left to right) correspond to the A, B and C sublattices.