Magnetic degeneracy points in interacting two-spin systems: Geometrical patterns, topological charge distributions, and their stability

Spectral degeneracies of quantum magnets are often described as diabolical points or magnetic Weyl points, which carry topological charge. Here, we study a simple, yet experimentally relevant, quantum magnet: two localized interacting electrons subject to spin-orbit coupling. In this setting, the degeneracies are not necessarily isolated points, but can also form a line or a surface. We identify 10 different possible geometrical patterns formed by these degeneracy points, and study their stability under small perturbations of the Hamiltonian. Stable structures are found to depend on the relative sign $\mathcal{S}$ of the determinants of the two $g$ tensors. Both for $\mathcal{S}=+1$ and $-1$, two stable configurations are found, and three out of these four configurations are formed by pairs of Weyl points. These stable regions are separated by a surface of almost stable configurations, with a structure akin to codimension-one bifurcations. These properties of magnetic degeneracy points can be practically important for control and readout of spin-based quantum bits.

Figure 1

Magnetic degeneracy points of two interacting spin-$\frac{1}{2}$ electrons. (a) Exchange interaction between the spins is described by its strength $J$ and a rotation $\widehat{\mathbf{R}}$. External magnetic field $\mathbf{B}$ couples to the spins through the $g$ tensors. (b) Geometry and topological charge distribution of magnetic degeneracy points for $\text{det}{\widehat{\mathbf{g}}}_{\text{L}},\text{det}{\widehat{\mathbf{g}}}_{\text{R}}>0$. Colors indicate topological charge: $+1$ (red), $-1$ (blue), 0 (black). Six-point and two-point patterns are stable, and the generic transition between these patterns is when two oppositely charged pairs meet and their charges annihilate each other (four points).

Figure 2

Magnetic degeneracy points for two interacting spin-$\frac{1}{2}$ electrons, when the relative sign of the $g$ tensors is negative, $\mathcal{S}=-1$. Color of each point indicates its topological charge: $+1$ (red), $-1$ (blue), 0 (black). Four-points and no-points patterns are stable, and the generic transition between them is that the two oppositely charged pairs meet and their charges annihilate each other (two points).