Chirality tunneling and quantum dynamics for domain walls in mesoscopic ferromagnets

We studied the quantum dynamics of ferromagnetic domain walls (topological kink-type solitons) in one-dimensional ferromagnetic spin chains. We show that the tunneling probability does not depend on the number of spins in a domain wall; thus, this probability can be large even for a domain wall containing a large number of spins. We also predict that there is a strong interplay between the tunneling of a wall from one lattice site to another (tunneling of the kink coordinate) and the tunneling of the kink topological charge (so-called chirality). Both of these elementary processes are suppressed for kinks in one-dimensional ferromagnets with half-integer spin. The dispersion law (i.e., the domain wall energy versus momentum) is essentially different for chains with either integer or half-integer spins. The predicted quantum effects could be observed for mesoscopic magnetic structures, e.g., chains of magnetic clusters, large-spin molecules, or nanosize magnetic dots.

Figure 1

(Color online) The geometry of the problem: (a) anisotropy axis for a single magnetic particle and (b) a schematic representation of a chain of particles. Each particle is represented by its spin.

Figure 2

(a) Trajectories on the sphere ${\mathbf{m}}^2=1$ (the direction of motion is indicated by the arrows) corresponding to different kinks; 1, 2, and 3 indicate the anisotropy axes of the magnet. Two of the most favorable kinks with chirality $\chi =\pm 1$ are shown by $B+$ and $B-$; unfavorable kinks with indefinite chirality are labeled by $N_1$ and $N_2$. The crossings between the trajectories and axes 1 or 2 are shown by four gray ovals in (a). The trajectories $N_1$ and $N_2$ divide the sphere in two domains, associated with kinks of different chiralities, as discussed in the text. (b) Spin distribution for favorable kinks with chirality $\chi =\pm 1$.

Figure 3

Shape of the domain wall pinning potential $U\left(X\right)$, normalized by its maximal value $U(a∕2)$, for two values of the anisotropy constant. Symbols denote the numerical data. The fit for the model dependence [Eq. (21)] is shown by the full line.

Figure 4

Coefficients $U_0$ and $U_1$ (in units of $J{S}^2$) in Eq. (21) for some values of anisotropy constant $K_1$.

Figure 5

Dispersion relation of a kink in a ferromagnet with spin $S=1$ subject to a weak pinning potential. The solid line and the dashed line correspond to the states $(\mid +⟩-\mid -⟩)$ and $(\mid +⟩-\mid -⟩)$, antisymmetric and symmetric over chirality, respectively. Here and Figs. 6, 7, 8, the vertical dotted lines show the boundaries of the usual Brillouin zone, $-P_B∕2<P⩽P_B∕2$.

Figure 6

Same as in Fig. 5 for spin value $S=2$. The region near the crossing points, schematically shown by shadowed ellipses on the main figure, is magnified in the upward left inset.

Figure 7

Dispersion relation of a kink in a ferromagnet with spin $S=1∕2$, subject to a weak pinning potential. The solid line and dashed line correspond to the states with chiralities $\chi =-1$ and $\chi =1$, respectively. Rectangles denote the points where the value of the chirality is not determined.

Figure 8

Same as in Fig. 7 for the spin value $S=3∕2$. The gray and the light circles schematically show the areas in the vicinity of the crossing of unperturbed spectra with and without chirality hybridization, respectively.

Figure 9

Dependence on the parameters $a_E$ and ${\overline{a}}_E$, determining the value of the Euclidean action for the tunneling processes, as a function of $U_0∕T_0$ (in logarithmic scale). Two analytic asymptotic dependencies are also indicated in the figure with a dashed and a dot-dashed curves.