Towards a universal phase diagram of planar chiral magnets

TL;DR

This work provides a comprehensive map of ground states in planar chiral magnets with the most general DMI and arbitrary magnetic fields. By reducing the DMI parameters to a single ellipse parameter $b\in[0,1]$ and rotating the field, the authors classify ground states into ferromagnetic, spiral, and skyrmion‑lattice phases and develop both analytic bounds (sine‑Gordon and conical) and robust numerical schemes to locate spiral and skyrmion configurations. The results show skyrmion lattices exist only in a limited regime near $b=1$ and fields close to orthogonal to the DMI plane, while rank‑1 DMI ($b=0$) cannot support skyrmion lattices as ground states, highlighting the delicate balance required for topological textures. The work lays groundwork for a unified analytical understanding of ground states across toroidal and planar geometries and points to directions for rigorous ties to existing stability results in the literature.

Abstract

In planar chiral magnets, the competition of the positive definite Heisenberg exchange and Zeeman energies with the indefinite Dzyaloshinskii-Moriya interaction (DMI) energy allows for the possibility of negative energy ground states, and leads to an intricate dependence of the ground states on the parameters of the theory. In this paper, we consider arbitrary spiralization tensors for the DMI interaction and arbitrary directions for the external magnetic field, and study the nature of the ground states in this parameter space, using a combination of analytical and numerical methods. Classifying ground states by their symmetry into ferromagnetic (invariant under under arbitrary translations in the plane), spiral (invariant under arbitrary translations in one direction) and skyrmion lattice ground states (invariant under a two dimensional lattice group), we give a complete description of the phase diagram of this class of theories.

Figures (13)

• Figure 1: The magnetization field $\hbox{\boldmath{$m$}}$ for a single skyrmion ($Q = -1$), for the model with standard isotropic DMI term and applied magnetic field $\hbox{\bm{$H$}}=(0,0,1)$. Left and right panels depict the same field. The left panel records the orientation of $\hbox{\boldmath{$m$}}$ by assigning a colour to each point on $S^2$, while the right panel shows $\hbox{\boldmath{$m$}}$ using oriented arrows in ${\mathbb{R}}^3$ coloured using the same rule. So black denotes $(0,0,-1)$ and white denotes $(0,0,1)$, for example. We will use the colouring scheme of the left panel throughout the paper.
• Figure 2: DMI terms classified in terms of ellipses with major axis 1, minor axis $b$ and eccentricity $\varepsilon =\sqrt{1-b^2}$
• Figure 3: The nesting of the ellipses implies a nesting property of the spiral domain: $\mathscr{H}_b\subseteq\mathscr{H}_{b'}$ for all $0\leq b\leq b'\leq 1$.
• Figure 4: Sections through the spiral domain $\mathscr{H}_b$ in the coordinate planes $H_2=0$ (top row), $H_1=0$ (middle row) and $H_3=0$ (bottom row) for DMI parameters $b=0$ (the rank 1 case, left column), $b=0.5$ (middle column) and $b=1$ (the standard isotropic case, right column). In each case, $\mathscr{H}_b$ is the region bounded by the black closed curve and is, by definition, the set of applied field values $\hbox{\bm{$H$}}$ for which the model has a negative energy spiral phase. The dashed curves correspond to lower bounds on the radius of the boundary curve, calculated by assuming the SP state is a conical spiral (red) or sine-Gordon state (green).
• Figure 5: Diagram of the geometry of a general unit cell $T^2_\Omega$ for a skyrmion lattice, defined by the two vectors $\hbox{\boldmath{$v$}}_1$ and $\hbox{\boldmath{$v$}}_2$ with angle $\alpha$. Note that $\hbox{\boldmath{$v$}}_1$, $\hbox{\boldmath{$v$}}_2$ need not be oriented and if $\det M < 0$ then $\hbox{\boldmath{$m$}}(x)$ has degree $N = - N_\square$.