Chiral Edge Mode in the Coupled Dynamics of Magnetic Solitons in a Honeycomb Lattice

Abstract

Motivated by a recent experimental demonstration of a chiral edge mode in an array of spinning gyroscopes, we theoretically study the coupled gyration modes of topological magnetic solitons, vortices and magnetic bubbles, arranged as a honeycomb lattice. The soliton lattice under suitable conditions is shown to support a chiral edge mode like its mechanical analogue, the existence of which can be understood by mapping the system to the Haldane model for an electronic system. The direction of the chiral edge mode is associated with the topological charge of the constituent solitons, which can be manipulated by an external field or by an electric-current pulse. The direction can also be controlled by distorting the honeycomb lattice. Our results indicate that the lattices of magnetic solitons can serve as reprogrammable topological metamaterials.Introduction.-The term metamaterials refer to a class of man-made composite materials which can offer functionalities beyond those found in nature via collective dynamics of constituent elements [1] [7] that a honeycomb lattice of spinning gyroscopes can support a chiral edge mode that is protected from small perturbations such as lattice distortions and thus can be identified as a topological mechanical metamaterial. As discussed in Ref. [7], an open challenge for its practical applications is to find a feasible way to keep gyroscopes spinning.Quantum-mechanically, nature has already endowed us a permanent gyroscope: spin of a particle. This intrinsic angular momentum manifests itself macroscopically through the gyrotropic force in the dynamics of magnetic solitons with topologically nontrivial textures such as magnetic bubbles (also known as skyrmions) and vortices [8]. These solitons and their dynamics have attracted much attention of physicists due to their fundamental properties [9] and technological promise [10,11]. In particular, the collective gyration modes of arrays of vortex disks have been studied theoretically [12] and experimentally [13][14][15][16] as reprogrammable metamaterials whose functionalities can be controlled by changing vortices' polarities and chiralities [17].When viewing topological magnetic solitons as gyroscopes, it is natural to expect that a honeycomb lattice of the solitons can support a chiral edge mode as its mechanical analogues [6,7]. In this Letter, we verify the expectation both by numerically solving the equations of motion for the dynamics of coupled solitons and by mapping the system to the Haldane model for an electron in graphene, which is known to exhibit the quantum Hall effect [18]. We also show that the direction of the edge mode can be controlled either by changing the topological charge of the solitons or by distorting the geometry of the honeycomb lattice. We conclude the Letter with

Edge states in a two-dimensional honeycomb lattice of massive magnetic skyrmions

“…In Ref. [40], Kim and Tserkovnyak generalized the situation to 2D: they theoretically studied the coupled oscillations of magnetic vortices and bubbles in a honeycomb lattice. By mapping the massless Thiele's equation into the Haldane model [41], they predicted the chiral edge modes near the gyration frequency of the single soliton.…”

Higher-order topological solitonic insulators

“…1(b)], m is the unit vector of magnetization, w is the thickness of nanodisk, M s is the saturation magnetization, γ is the gyromagnetic ratio, M is the effective mass of the magnetic vortex [50][51][52], and G 3 is the third-order gyroscopic coefficient [53][54][55]. The conservative force can be expressed as F j = −∂W/∂U j where W is the potential energy as a function of the vortex displacement: [49,56,57]. Here K is the spring constant which is determined by the identity ω 0 = K/|G|, ω 0 is the gyrotropic frequency of a single vortex (see Supplementary Note 1), I and I ⊥ are the longitudinal and transverse coupling constants, respectively.…”

“…When embracing the topological properties of vortex states, it paves the way for robust spintronic information processing. Topological chiral edge states are discovered in a two-dimensional honeycomb lattice of magnetic solitons [48,49]. However, all these topological magnonic and solitonic states are first-order in nature, according to the classification of topological insulators mentioned above.…”

Edge solitons in a nonlinear mechanical topological insulator

“…The derivation can be readily generalized to other mechanical TIs with discrete elements [17,52], possibly with dissipation [53] and forcing [30] included. Further extensions may include TIs in continuous media that can exhibit significant nonlinearity, such as recent proposals based on magnetic solitons [54] and water waves [55]. The existence of TPES in both photonic and phononic TIs may have significant impacts on practical applications such as optical and acoustic delay lines [56,57] and robust manipulation of light and sound [58,59] The Supplementary material here is organized into four sections and accompanying films for the bright and dark soliton simulations.…”