Real-space Berry phases: Skyrmion soccer (invited)

Abstract

Berry phases occur when a system adiabatically evolves along a closed curve in parameter space. This tutoriallike article focuses on Berry phases accumulated in real space. In particular, we consider the situation where an electron traverses a smooth magnetic structure, while its magnetic moment adjusts to the local magnetization direction. Mapping the adiabatic physics to an effective problem in terms of emergent fields reveals that certain magnetic textures -skyrmions -are tailormade to study these Berry phase effects.Over the last decades physicists have realized that a seemingly harmless phase -the Berry phase -leads to a lot of interesting physical consequences like quantum, This paper is structured as follows: First, we review the general concept of Berry phases and introduce the main formulas. In Sec. II, we focus on real-space Berry phase effects. Here we discuss the example of a free electron traversing a spatially or temporally inhomogeneous, smooth magnetic structure. In the adiabatic limit, the magnetic moment of the electron adjusts constantly to the local magnetization direction and thereby the electron picks up a Berry phase. To interpret the physical consequences of these Berry phases, we review the mapping onto a problem, where the electron moves in a uniform Zeeman magnetic field, but instead "feels" an emergent electric field and an emergent orbital magnetic field, leading, for example, to the topological Hall effect 4 . In Sec. III, we first introduce skyrmions in a formal way and then later focussing on skyrmions in magnetic systems, in particular those in a certain material class denoted as B20 materials. We discuss that those magnetic textures are tailormade to observe the emergent electric fields introduced in Sec. II A. In Sec. III C we switch the perspective and discuss the back reaction of the electronic Berry phase effects on the skyrmion lattice. I. BERRY PHASESBerry phases occur in all parts of physics and arise whenever a system adiabatically evolves along a cyclic process C in some parameter space X. A simple classical example is the parallel transport of a vector along a closed curve on a sphere (see Fig. 1). Although the system returns to its initial state it aquires a geometric phase characterized by the enclosed area on the sphere (Gauss-Bonnet theorem).A quantum-mechanical example is a system whose states are non-degenerate and which evolves adiabatically in the parameter space X = X(t). In that case, the solution of the time-dependent Schrödinger equation i ∂ t |ψ(t) = H(X(t)) |ψ(t) is given by 11 |ψ(t) = n a n (t 0 )e iγn(C) e − i t 0 dt n (X(t )) |ψ n (t) , (1) where |ψ n (t) and n (X(t)) are the time-evolved eigenstates and energies of the Hamiltonian H(X(t)), |ψ(t 0 ) = n a n (t 0 ) |ψ n (t 0

Direct observation of the skyrmion Hall effect

“…1 (a)], the translational motion driven by the spin Hall effect can be described by a modified Thiele equation 2,3,[27][28][29] :…”

“…Because of their topologically non-trivial spin textures, chiral magnetic skyrmions enable many intriguing phenomena based on their topology [2][3][4] , such as emergent electrodynamics 10 and effective magnetic monopoles 11 . As compared to most (vortex-like)…”

Insights into the orbital magnetism of noncollinear magnetic systems

“…When electrons travel throught the noncollinear magnetic structure of the skyrmion, they experience emergent electromagnetic fields. [10][11][12] This strong coupling between the electronic and magnetic degrees of freedom leads to very efficient motion of skyrmions with electric currents. 13 It also generates a topological contribution to the Hall effect, a transport signature of a skyrmion-hosting sample, 14,15 and was shown to enable the electrical detection of an isolated skyrmion.…”

“…If the magnetic exchange is much stronger than all the other terms, we can adopt the frequently used adiabatic approximation. 12 The spin projectors that diagonalize the magnetic exchange interaction are (see Appendix A)…”

“…10,12 Consider the non-interacting electron hamiltonian consisting of a kinetic term and a spin exchange coupling to an underlying magnetic structure:…”