Magnetic Response of Nematic Superconductors: Skyrmion Stripes and Their Signatures in Muon Spin Relaxation Experiments

We investigate the magnetic response of nematic superconductors, presenting a new approach to find vortex and skyrmion structures beyond symmetry-constraining Ansätze. Using this approach we show that nematic superconductors form distinctive skyrmion stripes. Our approach lends itself to accurate determination of the field distribution for muon spin rotation probes. We use this to show that the skyrmion structure manifests as a double peak in the field distribution, markedly different from the signal of standard vortex lattices.

Figure 1

Diagram of the geometry of a general unit cell for a vortex lattice, defined by the two vectors $V_1$ and $V_2$ with angle $\alpha$.

Figure 2

Contour plots of the lattice solutions (skyrmion chains) for the model Eq. (1) in the basal plane, with $\eta =3$ and increasing external field $H$. The fields plotted are the order parameter magnitudes $|{\psi }_1|$ and $|{\psi }_2|$, the phase difference between the two complex order parameters ${\theta }_{12}={\phi }_1-{\phi }_2$ and the local magnetic field strength $B$. The lattice consists of skyrmion chains, formed by rectangular unit cells (marked with black lines) containing four half-quantum vortices or winding two. As the external field increases the skyrmion chains squash together. Then near $H_{c_2}$ a transition occurs and the vortices are forced into the conventional triangular lattice with unit cells of winding one. Note that there is no simple symmetry between $|{\psi }_1|$ and $|{\psi }_2|$ as the symmetry is broken to a mixed symmetry with spatial rotations [33].

Figure 3

Magnetic field distribution for a skyrmion state in nematic superconductor (top) and vortex lattice in single component superconductor (bottom) for external fields closer to $H_{c1}$ than $H_{c2}$. The nematic superconductor is described in Eq. (1) with $\eta =3$. The structure of the field distribution is markedly different for the skyrmion state in the nematic superconductor with a double peak, rather than the familiar single peak for the standard GL model.

Figure 4

Comparing the field distribution $p(B)$ of a nematic superconductor in the lower plot with the areas of the local magnetic field configuration $B(x)$ each peak corresponds to for external field strength $H=2$.

Figure 5

Magnetic field distribution for a nematic superconductor (top) and single component superconductor (bottom) for external fields $H_{c2}\approx H\gg H_{c1}$. The nematic superconductor is described in Eq. (1) with $\eta =3$. As one approaches $H_{c2}$, the structure of the field distribution for the nematic superconductor becomes similar to the single peak structure of the familiar single peak for the standard GL model. This corresponds with the lattice becoming a triangular lattice for external fields near $H_{c2}$.