Topological dipoles of quantum skyrmions

Abstract

Magnetic skyrmions are spatially localized whirls of spin moments in two dimension, featuring a nontrivial topological charge and a well-defined topological charge density. We demonstrate that the quantum dynamics of magnetic skyrmions is governed by a dipole conservation law associated with the topological charge, akin to that in fracton theories of excitations with constrained mobility. The dipole conservation law enables a natural definition of the collective coordinate to specify the skymion's position, which ultimately leads to a greatly simplified equation of motion in the form of the Thiele equation. In this formulation, the skyrmion mass, whose existence is often debated, actually vanishes. As a result, an isolated skyrmion is intrinsically pinned to be immobile and cannot move at a constant velocity. In a spin-wave theory, we show that such dynamics corresponds to a precise cancellation between a highly nontrivial motion of the quasi-classical skyrmion spin texture and a cloud of quantum fluctuations in the form of spin waves. Given this quenched kinetic energy of quantum skyrmions, we identify close analogies to the bosonic quantum Hall problem. In particular, the topological charge density is shown to obey the Girvin-MacDonald-Platzman algebra that describes neutral modes of the lowest Landau level in the fractional quantum Hall problem. Consequently, the conservation of the topological dipole suggests that magnetic skyrmion materials offer a promising platform for exploring fractonic phenomena with close analogies to fractional quantum Hall states.

Figures (2)

• Figure 1: Example of a topological skyrmion texture with charge $Q_{\rm top} = - 1$. The arrows represent the field $\vec{n}$ and the brightness of the background color indicates the magnitude of the associated topological density $\rho_{\text{top}}$. The density $\rho_{\text{top}}$ is only finite in regions with spatially varying $\vec{n}$.
• Figure 2: Schematic illustration of the collective coordinate $R_i = D_i/Q_{\rm top}$ of a topological spin texture with the topological charge $Q_{\rm top} \neq 0$ and the topological dipole $D_i$, that decomposes into a classical and a magnon component, $R_i = R_{\text{cl},i} + R_{\text{mag},i}$. Whereas the dipole moment is conserved, $\partial_t R_i = 0$, the magnon excitations impose a non-trivial dynamics on the classical coordinate such that the time-evolution of $R_{\text{cl},i}$ is counterbalanced by a magnon cloud with coordinate $R_{\text{mag},i}$.