Modelling the elliptical instability of magnetic skyrmions

TL;DR

This work advances analytical understanding of elliptical instabilities in chiral magnets by applying two heuristic methods—the zero-energy domain-wall approach and the diverging tail lengthscale approach—to new parameter regimes: the symmetry-breaking phase and the general tilted-field with uniaxial anisotropy. The domain-wall method yields exact instability boundaries in the symmetry-retaining case, while the diverging-lengthscale method provides a robust predictive boundary that aligns with known numerics, notably predicting $h_a\ge-2k^2$ in the symmetry-breaking regime. In the tilted-field case with anisotropy, both methods generate complementary phase-boundaries, revealing how easy-axis and easy-plane anisotropies differently shape the onset of instability and how a critical point marks the convergence of predictions; the framework also suggests extensions to non-axisymmetric DMI and three-dimensional solitons. Overall, the results offer simple, explicit criteria to delineate regions of elliptical instability and guide future numerical and experimental explorations in chiral magnets.

Abstract

Two recently developed methods of modelling chiral magnetic soliton elliptical instability are applied in two novel scenarios, the tilted ferromagnetic phase of chiral magnets dominated by easy-plane anisotropy and the general case of the chiral magnet with tilted applied field and arbitrary uniaxial anisotropy. In the former case, the analytical predictions are found to exactly match previous numerical results. In the latter case, instability of isolated chiral skyrmions has not yet been studied, although the predictions correspond interestingly to previous numerical investigation of the phase diagram.

Figures (5)

• Figure 1: Phase diagram of the vacua of the general potential with easy-axis ($h_a>0$) or easy-plane ($h_a<0$) anisotropy and Zeeman interaction of strength $h_z$ tilted at $\theta_h$ to the normal of the plane, leading to an in-plane component of the magnetisation $h_z\sin\theta_h$ and an out-of-plane component $h_z\cos\theta_h$. The ambient space around the blue and red surfaces represents potentials with a unique minimum, which varies continuously as a function of $(h_z,h_a,\theta_h)$. Crossing through either surface leads to the vacuum changing discontinuously. The blue surface represents potentials with a $U(1)$ symmetry that is spontaneously broken, resulting in a tilted ferromagnetic phase. The red surface represents potentials with a $\mathbb{Z}_2$ symmetry $n_3\to-n_3$ that is again spontaneously broken.
• Figure 2: Comparison of the different analytical methods of estimating soliton elliptical instability within the axisymmetric chiral magnet, including both symmetry-retaining and symmetry-breaking phase. Within the symmetry-breaking phase there are two possible domain walls, short and long, whose energy per unit length is calculated in Ross20. This gives rise to three boundaries as the sign of different domain wall energies change. Also within the symmetry-breaking phase, our diverging lengthscale method predicts elliptical instability for $h_a\geq-2k^2$, and this is borne out by numerical simulation Leonov17. The critical coupling point Schroers18 is marked with a purple dot.
• Figure 3: Skyrmion region of elliptical instability according to the domain wall method (blue, dashed border) compared to skyrmion region of instability according to divergence of decay lengthscale (grey, solid border), and first-order transition of the vacuum (red) in the case of easy-axis anisotropy.
• Figure 4: Skyrmion region of elliptical instability according to the domain wall method (blue, dashed border) compared to skyrmion region of instability according to divergence of decay lengthscale (grey, solid border), and first-order transition of the vacuum (red) in the case of easy-plane anisotropy. The critical coupling discussed in Schroers18 is seen in the middle plot at the intersection of the three curves, marked by a purple dot.
• Figure 5: The equilibrium angle $\Theta_0$ of magnetisation as a function of applied magnetic field tilt $\theta_h$ and the ratio of uniaxial anisotropy to Zeeman interaction strength $\rho$. The point $\theta_h=0$, $\rho=-\frac{1}{2}$ corresponds to the second-order phase boundary $h_z+2h_a=0$, and the line $\theta_h=0$, $\rho<-\frac{1}{2}$ corresponds to the symmetry-breaking tilted phase.