Static Hopf Solitons and Knotted Emergent Fields in Solid-State Noncentrosymmetric Magnetic Nanostructures

Abstract

Two-dimensional topological solitons, commonly called Skyrmions, are extensively studied in solid-state magnetic nanostructures and promise many spintronics applications. However, three-dimensional topological solitons dubbed hopfions have not been demonstrated as stable spatially localized structures in solid-state magnetic materials. Here we model the existence of such static solitons with different Hopf index values in noncentrosymmetric solid magnetic nanostructures with a perpendicular interfacial magnetic anisotropy. We show how this surface anisotropy, along with the Dzyaloshinskii-Moriya interactions and the geometry of nanostructures, stabilize hopfions. We demonstrate knots in emergent field lines and computer simulate Lorentz transmission electron microscopy images of such solitonic configurations to guide their experimental discovery in magnetic solids.Topological solitons exist in the effective field theories of many physical systems [1][2][3][4][5][6][7][8][9]. For example, two-dimensional (2D) particlelike Skyrmions are solitonic field configurations classified by elements of the second homotopy group ! " ($ " ) = ℤ and indexed by a topological charge, the Skyrmions number. Such Skyrmions are widely studied in solid magnets due to the wealth of new

Three-dimensional crystals of adaptive knots

“…The Frank elastic constants adopted in the numerical modeling are based on values for the two nematic hosts used in this study (table S1) (19,21,23). The simulations with the saddle-splay elastic constant / 2H = / 22 and / 2H = 0 yielded similar results, consistent with the absence of singular defects (21,31). The 3D spatial discretization is performed on dense 3D square grids, with one helicoidal pitch ' distance equivalent sampled by 24 or 32 grid points.…”

“…Numerical modeling of the energy-minimizing ! (#) structures is performed using a variational-method-based relaxation routine implemented using the common finite differences approach (21)(22)(23)31). [)] ö õ , where the subscript ú denotes spatial coordinates, [)] ö õ denotes the functional derivative of ) with respect to U M , and MSTS is the maximum stable time step in the minimization routine, determined by the values of elastic constants and the spacing of the computational grid (31).…”

“…inclusions (18,19), confinement and boundary conditions (20)(21)(22), as well as could not selforganize into 3D lattices (22,23). We introduce energetically stable micrometer-sized adaptive knots in chiral LCs that, unexpectedly, materialize the knotted vortices and nonsingular solitonic knots at the same time and indeed behave like particles, undergoing 3D Brownian motion and self-assembling into 3D crystals.…”

“…Unlike the atomic, molecular and colloidal crystals, heliknoton crystals exhibit giant electrostriction and dramatic symmetry transformations under <1V voltage changes. We envisage that such solitons can emerge in helical phases of solid-state non-centrosymmetric magnets (13,14,22) and ferromagnetic LCs (23) Figures S1-S6 Table S1 Movies S1-S8…”

Spin, Orbital, Weyl and Other Glasses in Topological Superfluids

“…In these solid-state magnetic materials, the particle-like topological excitations described by the Hopf invariant—hopfions in Fig. 1 (or knots [ 28 ])—are suggested for spintronics applications [ 29 , 30 ].…”