Interactions of composite magnetic skyrmion-superconducting vortex pairs in ferromagnetic superconductors

TL;DR

The paper develops a self-consistent Ginzburg–Landau framework for ferromagnetic superconductors to study composite magnetic skyrmion–superconducting vortex pairs (SVPs). By linearizing around the ground state, it identifies three distinct decay lengths and derives explicit far-field profiles, enabling a quantitative analysis of SVP–SVP interactions. It shows that short-range repulsion and long-range attraction coexist, with the attractive channel favored at relative iso-rotation chi = pi, leading to stable SVP bound states and possible clustering in a type-1.5–like regime. The work provides a field-theoretical basis for engineering and controlling hybrid topological matter in bulk ferromagnetic superconductors and highlights the role of back-reaction between superconducting and magnetic sectors. Key results include explicit interaction energies and the dependence on orientation, which together explain bound-state formation and potential manipulation of SVPs.

Abstract

We study composite topological excitations in ferromagnetic superconductors consisting of bound states of magnetic spin textures (skyrmions) and superconducting vortices. Using a Ginzburg--Landau framework with Zeeman coupling between the magnetization and superconducting magnetic field, we demonstrate that skyrmion-vortex pairs (SVPs) form energetically stable bound states. By analyzing their asymptotic interactions, we identify regimes in which SVPs exhibit both short-range repulsion and long-range attraction, leading to clustering phenomena. Our results provide a field-theoretical basis for understanding suggest pathways for controlling hybrid topological matter through long-range interactions.

Figures (5)

• Figure 1: A composite skyrmion-vortex pair, consisting of a $N=1$ superconducting vortex and a $n=-1$ magnetic Bloch skyrmion in the mixed superconducting/ferromagnetic phase. They are obtained by numerically solving the full field theory equations of motions \ref{['eq: SC EL']}, associated to the normalized energy \ref{['eq: Normalized SC energy']}, with the arrested Newton flow minimization method. The parameters used here are $a=-1, b=1/4$ and $\beta=-\alpha=1$, which gives the vacuum values $m_0^2=1$ and $u^2=4$. The associated decay length for the superconducting order parameter is $\xi_s=1/\sqrt{2}$, the magnetization decay length is $l_m =1/\sqrt{3}$ and the magnetic penetration depth is $\lambda=1/2$.
• Figure 2: A bound state of composite SVPs, where each SVP experiences short-range repulsion and long-range attraction. The parameters used here are $a=-4, b=1$ and $\beta=-\alpha=1$, which gives the vacuum values $m_0^2=1$ and $u^2=4$. The associated decay length for the superconducting order parameter is $\xi_s=1/\sqrt{8}$, the magnetization decay length is $l_m =1/\sqrt{3}$ and the magnetic penetration depth is $\lambda=1/2$. Therefore we are in the bound state regime since $\xi_s<\lambda< l_m$ and the binding energy is negative, $E_{\textup{SVP-SVP}}-2E_{\textup{SVP}}<0$.
• Figure 3: Depiction of the bulk magnetic spin texture and superconducting order parameter density for the SVP-SVP bound state. The Bloch configuration for the magnetization is the skyrmion type that yields the lowest energy SVP, in comparison to Néel and anti-skyrmions.
• Figure 4: Setup for computing the long-range interaction between two well-separated composite skyrmion-vortex pairs. One pair is located at $\vec{x}_1$ and the other pair at $\vec{x}_2$. The distance between the pair is $R=|\vec{x}_2-\vec{x}_1|$, with the red dots indicating the zeroes of the order parameter $\psi$, i.e. the vortex locations. There is a relative phase orientation between the two skyrmions, characterized by the rotation angle $\chi\in[0,2\pi)$ that acts on the in-plane magnetization $(m_x,m_y)$-components, via a rotation matrix $R_z(\chi)\in\textup{SO}(2)$.
• Figure 5: Comparison of composite skyrmion-mulitvortex configurations in the different regimes. Shown here is a 6-vortex configuration coinciding with a single Block skyrmion. The parameters for the magnetization are fixed with $\beta=-\alpha=1$, such that the ground state magnetization has value $m_0^2=1$. For the superconducting order parameter, we fix the ground state value $u^2=-a/b=4$ and consider two cases. The first is (a) the type-I regime (corresponding to $b<\tfrac{1}{2}q^2$) where vortex cores coincide to form one big axially symmetric state, with $\lambda<\xi_s$. The second is (b) the bound state regime (corresponding to $b>\tfrac{1}{2}q^2$) where vortex clustering occurs, with $\xi_s<\lambda< l_m$.