Magnetic Skyrmions are Quasi-Magnetic Monopoles in Two-Dimensional Magnetic Materials

TL;DR

This paper demonstrates that magnetic skyrmions in two-dimensional magnetic materials can be understood as quasi-magnetic monopoles by relating Wu-Yang potentials to Berry connections through $SU(N)$ gauge theory.

Contribution

It establishes a novel algebraic and geometric connection between magnetic skyrmions, Wu-Yang potentials, and Berry connections using $SU(N)$ gauge transformations.

Findings

Magnetic skyrmions are algebraically similar to Wu-Yang magnetic monopoles.

The Wu-Yang potential of skyrmions is proportional to the Berry connection.

The relation between Wu-Yang potentials and Berry connection is generalized for $SU(N)$ systems.

Abstract

Using $su(N)$ Cartan subalgebra local bases parametrization of density operator $\rho$, we prove that the Wu-Yang potentials of a general $N$-level quantum system are completely expressed by $SU(N)$ gauge transformation. By taking the $SU(2)$ Cartan subalgebra local basis as a local normalized magnetization vector, we find that magnetic skyrmion and Wu-Yang magnetic monopole have the same algebraic structure. Moreover, by taking the adiabatic unitary evolution of magnetization as local gauge transformation, we verify that the Wu-Yang potential of magnetic skyrmions is proportional to the Berry connection, this means that magnetic skyrmion is quasi-magnetic monopole. The exact relation between Wu-Yang potentials and Berry connection is discussed in detail for the general $SU(N)$ case, i.e. the Berry connection for the pure or mixed state is the weighted average of the $(N-1)$ Wu-Yang potentials.