Skyrmion-like textures in superconductors with competing orders

It is well known that competing phases can manifest within the core regions of superconducting vortices. In strong magnetic fields, neighboring vortices can overlap to give rise to long-ranged correlations in the competing phase. We show that the resulting textures can have topological character with similarities to skyrmion crystals in magnets. We illustrate this using the SO(3) theory of competing orders wherein a scalar charge-density-wave (CDW) order competes with superconductivity. We study this theory on a spherical surface enclosing a magnetic monopole, giving rise to two vortices on the surface. The lowest-energy solution has both vortex cores ordered in the same sense, with coherent CDW order developing over the entire sphere. This changes dramatically when a perturbation that breaks inversion symmetry is introduced. The CDW order becomes anticorrelated between vortices, resulting in a topologically stable configuration that is analogous to a skyrmion in a ferromagnet. We extend this idea to vortex lattices in the two-dimensional plane, modeled by the attractive Hubbard model with an orbital field. Upon introducing Rashba spin-orbit coupling (RSOC), we find spatially modulated textures with CDW order developing around vortex cores in a stripe-like anticorrelated fashion. In a wide parameter regime, RSOC lowers the energy cost for domain walls separating vortices with opposite CDW orders. As a consequence, the competing CDW order loses its rigidity and becomes short-ranged. We discuss implications for the underdoped cuprates where charge order develops without a sharp diffraction peak.

Figure 1

Magnetic field of a monopole. We take the Dirac string to be flattened into a sheet in the equatorial plane.

Figure 2

Parallel solution. Left: A cross section of the sphere in the XZ plane showing the pseudospin orientation vs $\theta$, for $\xi =0.9$. The $z$ component of the pseudospin has the same sign everywhere, with the strongest magnitude at the north and south poles. This corresponds to CDW order developing in the same sense at both vortex cores. Center: Superconducting amplitude vs $\theta$ for different $\xi$ values. Right: CDW order parameter vs $\theta$ for different $\xi$ values. Note that CDW order is nonzero at the equator for $\xi \gtrsim 0.4$.

Figure 3

Antiparallel solution. Left: A cross section of the sphere in the XZ plane showing the pseudospin orientation vs $\theta$, for $\xi =0.9$. The $z$ component of the pseudospin has opposite signs in the two hemispheres, with the highest amplitude at the poles. This corresponds to CDW order at the two vortex cores in opposite senses. Center: Superconducting amplitude vs $\theta$ for different $\xi$ values. Right: CDW order parameter vs $\theta$ for different $\xi$ values. Note that CDW order vanishes at the equator.

Figure 4

Top: Free energies of the three saddle-point solutions vs $\xi$ within SO(3) theory. Note that the parallel solution is always the ground state. Bottom: Free energies of the three solutions in the presence of SOC. The antiparallel solution becomes the ground state over a range of $\xi$ values. The data are for an SOC strength given by $\lambda =-0.8$.

Figure 5

Rashba spin-orbit coupling arising from a perpendicular electric field. Hopping on each bond develops a spin-orbit term determined by the vector ${\vec{\lambda }}_{ij}=\vec{E}\times {\vec{r}}_{ij}$. This term is proportional to the amplitude of the electric field; we fix its strength to be $t_{\lambda }$, a tuning parameter. We only consider this term on nearest-neighbor bonds.

Figure 6

Skyrmion-like stripe in the Hubbard model for $U=2t$, $t_{\lambda }=0.5t$, and a field corresponding to 16 vortices. The superconducting amplitude as a function of the position (left) shows clear vortices. The CDW order parameter (right) has its maximum amplitude at vortex cores but alternates in sign to form stripes.

Figure 7

Two solutions obtained for $U=4t$, $t_{\lambda }=0.5t$ at a flux corresponding to 12 vortices. A regular supersolid solution is shown on the left: top left, superconducting amplitude; bottom left, CDW order parameter. A domain wall solution is shown in the corresponding panels on the right. We have coherent superconducting order, but CDW order is disrupted at a domain wall. These two solutions have the same free energy within our simulation resolution.

Figure 8

A circular domain wall solution. The parameters used are $U=4t$, $t_{\lambda }=0.5t$, with a large flux, corresponding to 20 vortices.

Figure 9

Mean-field solutions for $U=4t$, $t_{\lambda }=0.5t$ at a flux corresponding to 16 vortices. Left: A regular supersolid solution. Center: Two solutions, both containing domain walls in CDW order. Right: A circular domain wall. All four solutions have comparable energies, with the differences being smaller than our resolution.