Chern Numbers for Spin Models of Transition Metal Nanomagnets

We argue that ferromagnetic transition metal nanoparticles with fewer than approximately 100 atoms can be described by an effective Hamiltonian with a single giant spin degree of freedom. The total spin $S$ of the effective Hamiltonian is specified by a Berry curvature Chern number that characterizes the topologically nontrivial dependence of a nanoparticle’s many-electron wave function on magnetization orientation. The Berry curvatures and associated Chern numbers have a complex dependence on spin-orbit coupling in the nanoparticle and influence the semiclassical Landau-Liftshitz equations that describe magnetization-orientation dynamics.

Figure 1

Chern numbers $S$ of quasiparticle levels of a 25-atom cobalt nanoparticle, as a function of the spin-orbit coupling strength $\xi$. Here I, II, and III label contiguous orbitals whose energies are ordered in ascending order. Interchanges between and deviations from $S=\pm 1/2$ (the values at $\xi =0$) occur when two levels cross. The sum of the two Chern numbers is conserved at crossing.

Figure 2

Chern numbers $S$ of all the individual quasiparticle levels of a 25-atom nanoparticle for a finite value of $\xi$. Several of these numbers differ strongly from $\pm 1/2$.

Figure 3

Planar projection of the total (negative) Berry curvature $-\mathcal{C}[\widehat{n}]$ for a 25-atom nanoparticle. The strength of the spin-orbit coupling $\xi =165\mathrm{m}\mathrm{e}\mathrm{V}$ is chosen in proximity to a value at which the Fermi level and the first unoccupied level cross. Level crossing occurs for a given $\widehat{n}$ and its opposite $-\widehat{n}$. In correspondence of these two values, $-\mathcal{C}[\widehat{n}]$ has large peaks.