Nematic Skyrmions in Odd-Parity Superconductors

We study topological excitations in two-component nematic superconductors, with a particular focus on ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ as a candidate material. We find that the lowest-energy topological excitations are coreless vortices: a bound state of two spatially separated half-quantum vortices. These objects are nematic Skyrmions, since they are characterized by an additional topological charge. The inter-Skyrmion forces are dipolar in this model, i.e., attractive for certain relative orientations of the Skyrmions, hence forming multi-Skyrmion bound states.

Figure 1

Close-up view of the vortex core structure in the nematic phase in which we set $v=v_z$, ${\beta }_z=0$, and for $\kappa =3$. On the first line, the first panel shows the distribution of the magnetic field, while the second displays the relative phase between the two components of the order parameter. The second line shows the densities of the two components of the order parameter. Clearly, the cores in both components do not superimpose, thus implying that vortices in the nematic phase are coreless defects. Since cores do not overlap, the relative phase has $\pm 2\pi$ winding around each core. Furthermore, the relative phase exhibits a dipolar mode that is long-ranged.

Figure 2

A close-up view of a bound state of two coreless vortices carrying one flux quantum each. Each coreless vortex is a well-localized bound state of two HQVs. The dipolelike forces in the relative phase yields a long-range attraction that binds the single-quantum vortices together. Note that the dipoles are antialigned in the bound state as dictated by relative phase interaction. Displayed quantities are the same as in Fig. 1.

Figure 3

Texture of the unit vector $\mathbf{n}$ defined as the projection of the superconducting d.o.f. onto the spin-$1/2$ Pauli matrices. The left panel shows a single-quanta solution corresponding to Fig. 1, while the right panel corresponds to the bound state displayed in Fig. 2. Inspection of the single-quanta solution show how $\mathbf{n}$ wraps the target two-sphere. The north (respectively, south) pole signals zero of ${\mathrm{\Delta }}_{+}$ (respectively, ${\mathrm{\Delta }}_{-}$) and thus the position of respective HQVs. From this one, can see structural difference of nematic Skyrmions, compared to Skyrmions in chiral superconductors [31]. Here the solution has clearly the form of weakly interacting well-separated Skyrmions with unit topological charge each. By contrast, in the chiral case the Skyrmionic topological charge tends to be relatively uniformly spread along closed domain walls.