Network of localized magnetic textures revealed using a saddle-point search framework

TL;DR

The study addresses how metastable localized magnetic textures transition via energy barriers in 2D chiral magnets. It introduces a modular saddle-point search framework (SPSF) with four stages—preprocessing, subsystem-based escape, geodesic minimum mode following convergence, and postprocessing—to build a state network of metastable textures connected by first-order saddle points. The authors identify five SP categories corresponding to distinct transformation mechanisms and show a near-universal hierarchy of energy barriers that persists toward the continuum limit, while demonstrating that homotopies do not always correspond to minimum energy paths. This framework enables systematic exploration of long-timescale dynamics, offers insights into the role of topology in texture transformations, and provides a foundation for kinetic simulations and global optimization in topological magnetism.

Abstract

A computational framework is presented for the sampling of the energy surface of magnetic systems via the systematic identification of first-order saddle points that determine connectivity of metastable states and define the mechanisms of transitions between them. The framework combines four stages: first, the symmetry of a given minimum-energy configuration is identified and used to define subsystems whose eigenmodes provide relevant deformation directions; the subsystem eigenmodes are then used to guide the system toward the vicinity of different saddle points surrounding the energy minimum; next, the geodesic minimum mode following method is employed to efficiently converge onto the saddle points; and finally, the identified saddle points are embedded into the state network. Applied to metastable textures in two-dimensional chiral magnets described by a lattice Hamiltonian, the method reveals a hierarchy of transition mechanisms governing the nucleation, annihilation, and rearrangement of the fundamental components of localized magnetic textures. The identified saddle points enable the construction of the network of metastable states, where saddle points define the connectivity between them, providing a comprehensive map of accessible transitions and their associated energy barriers. Transitions corresponding to both homotopies that preserve the topological charge and transformations that change it are identified. By scaling the interaction parameters, the distinct behavior of these two classes is obtained as the continuum limit is approached. Finally, it is shown that textures with the same topological charge are not always connected by a homotopy corresponding to a minimum-energy path: in specific parameter regimes, the total topological charge necessarily increases and then decreases (or vice versa) during the transition, returning to its initial value at the final state.

Figures (11)

• Figure 1: Examples of magnetic textures with different topological charges $Q$ and symmetries, described by spin point groups $\mathrm{C}_n$ and $\mathrm{D}_n$, for a magnetic field of $h=0.623$ and $L_D=40$. The contours ($m_z=0$) are shown in white. The inset explains the hue-saturation-lightness color scheme encoding the azimuthal angle $\Phi$ of the magnetic moments by the hue, while the lightness represents the polar angle $\Theta$ for a fixed saturation of $1.0$. a: Skyrmion with axial symmetry ($\mathrm{D}_\infty$). b: Texture featuring a chiral kink, a tail and two nested closed contours $\gamma_1$ and $\gamma_2$. The magnetic moments along the outer contour are shown by white arrows. c-f: Chiral droplet (c), skyrmion bag with two inner contours (d), two-tailed texture with two chiral kinks (e) and three-tailed state (f) with mirror axes depicted in cyan. The orientation of the mirror axes is denoted $\vec{p}\,_j$. The fundamental domain is highlighted while the remaining part of the texture is grayed out.
• Figure 2: The SPSF is structured into different stages. During the preprocessing stage, the properties of the local energy minimum configuration are analyzed to define suitable settings for escaping the convex region (escape stage) and then converging onto first order SPs (convergence stage). The tree on the right highlights the fact that the convex region is escaped at multiple points on the high dimensional energy surface resulting into many SP search attempts computed in parallel. Finally, during postprocessing the redundant SPs are filtered, the connectivity of the identified SPs to the initial minimum is tested and the adjacent energy minimum state is revealed.
• Figure 3: Flowchart describing the preprocessing of the SPSF for an exemplary magnetic state (a) in a chiral magnet for a magnetic field of $h=0.65$ and model parameters chosen according to $L_D=64a$ (cf. Tab. \ref{['tab:modelparameters']}). The DM interaction angle is set to $\beta=30^\circ$ [cf. Eq. \ref{['eq:hamiltonian']}]. b: The black lines represent the $m_z=0$ contours. c: Magnetic textures defined by their outer contour ($\gamma_i$) and separated using a density-based spatial clustering. Each color corresponds to one magnetic texture. Inner contours have been excluded ($\gamma_j$). d: Texture selected from a,b centered at $\vec{c}\,$. A two-fold rotational symmetry was detected. The red surrounded part of the texture marks one of two equivalent sectors. e: Sector transformed using Eqs. \ref{['eq:trafo_m']},\ref{['eq:trafo_r']}. The orientation $\vec{p}\,$ of the mirror axis (cyan) is determined by the connection line between two fix-points $\vec{f}\,$ and $\vec{g}\,$ (see text). For the original texture this yields $n=2$ mirror axes $\vec{p}\,_n$ with angles $\kappa_n$ to the $x$-axis depicted in f, where the fundamental domain is highlighted and the remaining texture is grayed out.
• Figure 4: Illustration of the proposed subsystem-based escape stage of the SPSF. A sampling of elliptical subsystems centered in the fundamental domain is generated (a). A set of low-energy subsystem eigenmodes is defined for each subsystem (see b-e). Taking into account both signs of the respective eigenvectors each escape attempt iteratively displaces the configuration within the subsystem along a subsystem eigenmodes until the escape criterion is satisfied [Eqs. \ref{['eq:escape_crit_fullsys']},\ref{['eq:valley_entrance']},\ref{['eq:escape_crit_single']}]. The displacement is guided by a retraction by a distance $\delta$edelman1998schrautzer2025. In each iteration, after the displacement of the texture, the energy of the rest of the system is minimized while keeping the subsystem fixed. Successful escape attempts mark entry points for the subsequent convergence stage of the SPSF.
• Figure 5: Flowchart of the convergence stage. For each of the final configurations from the escape stage an S-GMMF calculation is initialized while using the same subsystem as in the corresponding escape attempt. First the GMMF force is computed in the subsystem [Eq. \ref{['eq:GMMF_force_full']}]. If none of the exit criteria for S-GMMF is satisfied (see text) the search direction and the displacement parameter $\delta$ is computed using an L-BFGS solver. The subsystem is subsequently retracted along this search direction and after minimizing the energy of the rest of the system the process repeats until either a subsystem-constrained SP has been identified, a convex region has been entered for the subsystem or the number of iterations exceeds $N_\text{max}^s$. In all cases subsequently the full system GMMF algorithm is applied. For details about the GMMF implementation refer to Ref. schrautzer2025. The convergence stage yields a set of SPs that is passed to postprocessing.