Stability of highly-twisted Skyrmions from contact topology
SStability of highly-twisted Skyrmions from contact topology
Yichen Hu and Thomas Machon The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, UK H.H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK
We describe a topological protection mechanism for highly-twisted two-dimensional Skyrmions in systemswith Dyloshinskii-Moriya (DM) coupling, where non-zero DM energy density (dubbed “twisting energy den-sity”) acts as a kind of band gap in real space, yielding an N invariant for highly twisted Skyrmions. We proveour result through the application of contact topology by extending our system along a fictitious third dimension,and further establish the structural stability of highly-twisted Skyrmions under arbitrary chirality-preserving dis-tortions. Our results apply for all two-dimensional systems hosting Skyrmion excitations including spin-orbitcoupled systems exhibiting quantum Hall ferromagnetism. Introduction -
Skyrmions and their variants(anti-Skyrmions,merons, etc.) are non-trivial spin textures characterized by aSkyrmion charge counting the number of times spin directionscover the sphere. This topological aspect of spin textures inreal-space leads to their stability with physical consequencessuch as the topological Hall effect[1]. Skyrmion-like struc-tures are ubiquitous in condensed matter physics, appearingin magnetic materials[2–7], cold atom systems[8–10], liquidcrystals[11–15] and quantum Hall ferromagnetic systems[16–18]. Together with their solitonic nature, Skyrmions canbe created, annihilated, and manipulated at high speed byextremely low current density[19, 20]. This makes themideal for information storage and processing, particularly asSkyrmions are nanometer-sized for certain magnetic materi-als or thin film heterostructures. Skyrmion-based racetrackmemory[21–24] is predicted to have higher storage densityand lower energy consumption compared to all present-daysolid state devices. Current research efforts[25–32] aim togenerate more robust and enduring Skyrmions in room tem-perature for future applications in spintronic devices.Skyrmions and their variants (see Fig. 1) can be generatedby various mechanisms in magnetic systems such as dipo-lar interactions[33–35], frustrated exchange interactions[36]and four-spin interactions[5]. In this paper, we will focus onhighly-twisted Skyrmions arising from Dyloshinskii-Moriyainteractions[37, 38]. These inversion symmetry breakinginteractions induced by spin-orbit coupling are commonlyfound in magnetic materials with non-centrosymmetric lat-tices. The DM interaction controls several key aspects ofSkyrmions. Its strength determines the size of Skyrmionsand its sign determines the chirality of Skyrmions (twistingdirection of spin rotation). Here we show that surprisingly,DM interactions also constitute a crucial part of a novel topo-logical protection mechanism for stability of highly-twistedSkyrmions. The first hint comes from observation of stable π twisted Skyrmions - “Skyrmioniums” (Fig. 1)[7, 39–41].Unlike usual Skyrmions, the total Skyrmion charge of thesespin textures is zero. Conventional wisdom would suggestthat non-topological Skyrmioniums are fragile as spins withinthem can be smoothly deformed to a uniform spin config-uration without any topological obstruction. However, theextra “twist” from the DM interaction completely changes FIG. 1.
Top a: Isolated π twisted Skyrmion with Skyrmion charge 1in a vertical background field, b: Arbitrary chirality-preserving dis-tortion of the Skyrmion of the type considered here. Bottom c: π twisted Skyrmion with Skyrmion charge 0, d: Arbitrary chirality-preserving distortion of the Skyrmion of the type considered here.Color gives the value of v z . this picture. We claim and prove in this paper that, for re-gions with non-vanishing DM energy density (“twisting en-ergy density”), highly-twisted Skyrmions together with theirSkyrmion charges enclosed remain invariant under arbitrarychirality-preserving distortions. By extending our 2D thinfilm geometry to a fictitious third direction, we turn the non-vanishing twisting energy density condition for the magneti-zation vector field to the contact condition for its dual -form.This enables us to leverage a fundamental theorem in contacttopology – Gray’s theorem[42, 43] – to rigorously establishthe robustness of these highly-twisted Skyrmions character-ized by a new N topological invariant. Physically, the non-vanishing twisting energy density in real-space bears strikingresemblance with the band gap for insulators in momentum-space[44, 45]. Both are quantum effects induced by rela-tivistic spin-orbit interactions. Just as a topological insulator a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 2. Construction of structural stability for chiral Skyrmions. The initial Skyrmion, or similar twisted soliton, is contained in the region S and we artificially extend the system symmetrically in the z direction (bottom left). The non-zero DM density within S is given by α ∧ dα (cid:54) = 0 . Under time evolution that preserves the contact condition within the region S t , but which is otherwise arbitrary, the Skyrmionis contained within the region S t (bottom right). We then construct a diffeomorphism φ t of the domain, so that the pullback of α t under φ t is α . The pullback F ∗ t of α t , gives β t , with the Skyrmion contained in S (center) top. Pulling back by G t then yields α . phase stays as a robust phase of matter provided there is anon-vanishing band gap, a highly-twisted spin texture suchas a Skyrmionium stays unaltered provided there is a non-vanishing twisting energy density. Any transition to an or-dinary insulator phase or trivial spin configuration must passthrough at least one configuration where the band gap or twist-ing energy density goes to zero respectively. More intrigu-ingly, symmetries are fundamental for each case but take op-posite roles. Time-reversal and inversion symmetry[46] areessential for the existence of topological insulators and theirhelical edge modes while these symmetries are explicitly bro-ken by DM interactions. Energy of a Skyrmion -
We start with a region M , which istaken as a simply connected domain in R . We associate toeach point in M a magnetization vector v ( x, y ) : M → S .This is a generic model of a thin film ferromagnet. The totalenergy E has the following components[47]: E = kE ke + E DM + aE f , (1) E ke = (cid:82) M |∇ v | is the kinetic energy of magnetization vec-tor fields and k is the stiffness. The second term E DM = (cid:82) M e DM corresponds to energy from DM interactions. For abulk material in 3D, the local twisting energy density is sim-ply e DM ( x, y, z ) = d v ( x, y, z ) · ∇ × v ( x, y, z ) . (2)This term explicitly breaks time-reversal and inversion sym-metry in real-space. The sign of d and v · ∇ × v combinedtogether determine chirality of twisting direction for v . Asa spin-orbit coupling term, it breaks spin or orbital rotationalsymmetry individually, but is invariant under a simultaneous SO(3) rotation of spin and space. In a thin film geometry, thefull
SO(3) rotation in spin and orbital space gets modified to
SO(2) symmetry within xy -plane of spin and orbital space. Therefore, a generic twisting energy density term appropriatefor a thin film becomes e DM ( x, y ) = d b v ( x, y ) · ∇ (cid:48) × v ( x, y )+ d n [( v ( x, y ) · ∇ (cid:48) ) v z ( x, y ) − v z ( x, y )( ∇ (cid:48) · v ( x, y ))] (3)where ∇ (cid:48) = ( ∂ x , ∂ y , . The first term prefers chiral (twisted)configurations such as Bloch Skyrmions whereas the secondterm prefers achiral N´eel Skyrmions (hedgehogs). Our resultsapply exclusively to twisted configurations so in the followingwe set d n = 0 .The last term E f makes sure that the minimal energyconfiguration of a Skyrmion has finite radius. This canbe achieved by introducing shape or magnetocrystallineanisotropy or an external magnetic field which induces Zee-man energy. Using parameters of (1), the radius of a Skyrmionis R ∼ da [47]. Structural stability of Skyrmions -
We first derive the DM-protected structural stability of a Skrymion-like soliton. Ourargument assumes a single isolated object surrounded by auniform background, however the construction can be easilyextended to an arbitrary collection of Skyrmions, assumingthey do not overlap.The configuration is defined by a time-dependent magne-tization vector v ( x, y ) at each point in M , which we as-sume varies smoothly with x and y . We also assume that ateach time t there is a sub region S t ⊂ M , representing theSkyrmion or similar twisted soliton, with boundary a smoothtopological circle, such that within S t , v · ∇ (cid:48) × v (cid:54) = 0 at alltimes and outside S t , v = e z points in the z -direction[48].We now artificially extend v along the third dimension z to M such that v at a fixed ( x, y ) is constant along the z -direction. We then define a new non-singular unit vector field m ( x, y, z ) ≡ v ( x, y ) for all z ∈ R . The previous twistingenergy term v · ∇ (cid:48) × v now becomes m · ∇ × m , (4)on M × R . In three-dimensional Euclidean space, the condi-tion of a non-vanishing twisting energy term m · ∇ × m (cid:54) = 0 can be reformulated in terms of its dual -form α as the con-tact condition[42] α ∧ dα (cid:54) = 0 , (5)where component-wise ( α ) i = (cid:80) j g ij m j and g ij is the Eu-clidean metric ( i, j ∈ { x, y, z } ). Using the 1-form α ratherthan m is done for technical reasons – the construction belowrequires using the covariant form of m . Let α t denote α attime t . We now adapt Gray’s theorem[42, 49] to show that,under our assumptions, there is a time-dependent coordinatetransformation φ t : M × R → M × R such that φ ∗ t α t = λ t α , (6)where φ ∗ t α t is the pullback of α t under φ t and λ t is a smoothpositive function. This states that provided the interior of theSkyrmion has non-zero twisting energy density, the time evo-lution of the system is equivalent to a coordinate transforma-tion ( φ t ) under which v transforms covariantly. The particularform of φ t is not important, but the fact that it exists impliesthat various topological and geometric properties of v are pre-served.To establish this we first construct an arbitrary diffeomor-phism f t : M → M , equal to the identity on ∂M , such that f t ( S ) = S t . Extending f t to a map F t : M × R → M × R asthe identity in the z direction, the 1-form β t = F ∗ t α t restrictedto the interior of S then defines a contact structure in S × R ,with β = α and β t = a t dz on the boundary of S , for somepositive constant a t . By Gray’s theorem, there is then a dif-feomorphism G t : S × R → S × R , restricting to the identityon ∂S × R , such that G ∗ t β t = µ t β , for some positive func-tion µ t . We can extend the map G t to the entirety of M × R by making it the identity outside of S × R . The diffeomor-phism φ t = F t ◦ G t then satisfies φ ∗ t α t = G ∗ t F ∗ t α t = λ t α ,establishing (6).This tells us that deformations of a Skyrmion, or similartwisted soliton, that preserve a non-zero twisting energy den-sity are equivalent to diffeomorphisms. We now exploit thefact that our diffeormorphisms have a z -symmetry, to derivefurther structural properties. Explictly writing the diffeomor-phism φ t in terms of primed coordinates gives the followingfunctional dependencies for φ − t x ( x (cid:48) , y (cid:48) ) , y ( x (cid:48) , y (cid:48) ) , z ( x (cid:48) , y (cid:48) , z (cid:48) ) . (7)Both the x and y directions transform without dependence on z . In components, (6) tells us that α t transforms as ( α t ) i = ∂x j ∂ ( x (cid:48) ) i ( α ) j . (8)But since neither α nor α t depend on z (or z (cid:48) ), the Jacobianmatrix ∂x j /∂ ( x (cid:48) ) i also cannot depend on z . It therefore must have the form ∂x j ∂ ( x (cid:48) ) i = ∂x∂x (cid:48) ∂y∂x (cid:48) ∂z∂x (cid:48) ∂x∂y (cid:48) ∂y∂y (cid:48) ∂z∂y (cid:48) σ , (9)where σ is time-dependent positive constant. Now considerthe scalar m z , the z component of m . This is given by ( α t ) i e iz , where e z is the vector parallel to the z direction. Tak-ing the inverse of (9), we find the contravariant e z transformsunder φ t by a simple scale factor of /σ . Hence we have thepullback φ ∗ t ( m z ( t )) = φ ∗ t (cid:0) ( α t ) i e iz (cid:1) = ( φ ∗ t α t ) i φ ∗ t ( e z ) i = λ t σ ( α ) i e iz = λ t σm z (0) . (10)But both σ and λ t are positive, hence the sign of m z is pre-served under the diffeomorphism. Converting back to our 2Dvector v , we have the following characterization of the struc-tural stability of Skyrmions, or similar twisted solitons withnon-zero twisting energy density. The region S t is deformedover time. Within each region there are a set of (generically)closed curves Γ where v z = 0 . Over time these closed curvesmay move within the region S t , but they do not merge, norare any additional curves created. We include in the set Γ theboundary of the Skyrmion, ∂S t , so that one may give a pictureof the Skyrmion as a set of closed curves in the plane. This setof curves defines several regions, with v z a single sign withineach region, as shown in Fig. 3.This result directly separates a Skyrmionium (2 π twistedSkyrmion) from the vacuum state as any smooth pathwayfrom one to the other must pass through a configuration wheretwisting energy density (or contact condition) vanishes atsome point. Skyrmion Charge -
With the background field pointing up-wards, and positive twisting energy density (left-handed con-figurations), the charge Q of a highly twisted nπ Skyrmionis n mod 2 . This means that, in principle, the transformation n → n ± m does not change the Skyrmion charge, and hencecan be achieved via a non-singular transformation. However,requiring that twisting energy density remains non-zero withinthe Skyrmion, the topological stability ensures that this tran-sition cannot happen, and the number n ∈ N becomes a bona-fide topological invariant of the Skyrmion.The topological structure of highly-twisted Skyrmions alsogives structure to the Skyrmion charge density within theSkyrmion itself. Per our structural result, any Skyrmion canbe considered as a set of nested bands with a central disk andouter band, defined by the set Γ . In principle more exotic ar-rangements may exist, which we do not consider here. Ourresult implies that the Skyrmion charge of each band is well-defined – our structural stability result means that Skyrmioncharge cannot ‘leak’ between regions. Consider the spin field v as a map M → S , then the Skyrmion charge in a region R ⊂ M is the pullback of the area form ω under v integratedover R , Q R = (cid:90) R v ∗ ω = 14 π (cid:90) R (cid:15) abc v a (cid:15) ij ∂ i v b ∂ j v c dxdy. (11) FIG. 3. Anatomy of a chiral π Skyrmion. The set of lines Γ , com-prises the places where v z = 0 (black) as well as the boundary of theSkyrmion (green). The topology of the regions is preserved underarbitrary deformations, provided twisting energy density is non-zerothroughout. Moreover, the Skyrmion charge within each region isalso preserved. Alternatively, Q R is the oriented area of the image of v ( R ) on S . If the map v is deformed, then the change in Q R is thechange in this oriented area. By the Gauss-Bonnet theorem,if the boundary of v ( R ) ⊂ S is constant in time, then Q R must also be constant. Now take R to be one of the regionsof the Skyrmion S . These regions move over time, but thearea on S swept out must remain identical, since v on thebounding curves Γ does not change – on the boundary of theSkyrmion it points vertically, and on the curves defined by v z = 0 it sweeps out an equatorial great circle. It follows thatthe Skyrmion charge within each region is constant. This isillustrated in Fig. 3.More generally, for an nπ twisted Skyrmion there are threetypes of regions defined by Γ – a central disc, an outer band,and n − inner bands. Let σ ∈ {− , } be the sign of the z component of the background field, and τ ∈ {− , } the signof twisting energy density, then a short calculation shows thatthe total Skyrmion charge in the outer band is στ / , in the in-ner bands it is zero, and in the central disc it is στ ( − n − / .For example in Fig. 3 we have σ = +1 , τ = +1 , and n = 2 ,giving a total charge of zero. Discussions -
Our theorem shows that a whole class of previ-ously thought non-topological spin textures, such as Skyrmio-nium, are protected by non-vanishing DM interactions andcharacterized by a new N topological invariant. Stability fromthis topological protection mechanism makes highly-twistedSkyrmions great candidates for building racetrack memo-ries [50] as they will not decay when transported. Moreover,carrying zero net Skyrmion charge, motion of a highly-twistedSkrymion overcomes the obstacle of transverse deflection dueto the Skyrmion Hall effect[51].Besides ferromagnetic materials, our results equally apply to condensed matter systems exhibiting quantum Hall ferro-magnetism. For traditional quantum Hall systems (2D elec-tron gas), there is no spin-orbit coupling term, but impurities,in principle, can introduce such a term acting as a DM inter-action [52]. Quantum anomalous Hall ferromagnetic states,such as those found in moir´e systems[53–60], might providea more relevant platform since spin-orbit coupling phenomenaform an integral part of the physics of crystals. Furthermore,in the lowest Landau-level, spin degrees of freedom are inter-twined with charge degrees of freedom. Skyrmion charge den-sities from spin configurations(11) are proportional to elec-tric charge densities[16, 18]. Thus, highly-twisted Skrymions,such as Skyrmioniums, in quantum Hall ferromagnetic statescan give rise to stable charge neutral excitations. Under asmooth deformation, they can further evolve into neutral ex-citations with non-zero higher electric moments depending ontheir shape. Moreover, there are theoretical possibilities ofSkyrmions carrying fractionalized degrees of freedom, suchas electric charges or Majorana zero modes[17, 61], for po-tential applications in quantum information processing. Ourmechanism could thus provide an extra layer of topologicalprotection for quantum information stored and transported byhighly-twisted Skyrmions. Acknowledgements - We thank Randall Kamien and JianKang for useful discussions. This work is in part supportedby grant EP/S020527/1 from EPSRC (YH) . [1] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G.Niklowitz, and P. B¨oni, Topological hall effect in the a phase ofmnsi, Phys. Rev. Lett. , 186602 (2009).[2] S. Muhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch,A. Neubauer, R. Georgii, and P. Boni, Skyrmion lattice in achiral magnet, Science , 915 (2009).[3] A. Bogdanov and A. Hubert, The stability of vortex-like struc-tures in uniaxial ferromagnets, Journal of Magnetism and Mag-netic Materials , 182 (1999).[4] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Mat-sui, N. Nagaosa, and Y. Tokura, Real-space observation of atwo-dimensional skyrmion crystal, Nature , 901 (2010).[5] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubet-zka, R. Wiesendanger, G. Bihlmayer, and S. Bl¨ugel, Sponta-neous atomic-scale magnetic skyrmion lattice in two dimen-sions, Nature Physics , 713 (2011).[6] N. Nagaosa and Y. Tokura, Topological properties and dy-namics of magnetic skyrmions, Nature Nanotechnology , 899(2013).[7] S. Zhang, F. Kronast, G. van der Laan, and T. Hesjedal, Real-space observation of skyrmionium in a ferromagnet-magnetictopological insulator heterostructure, Nano Letters , 1057(2018).[8] I. F. Herbut and M. Oshikawa, Stable skyrmions in spinor con-densates, Phys. Rev. Lett. , 080403 (2006).[9] T. Kawakami, T. Mizushima, M. Nitta, and K. Machida, Stableskyrmions in su (2) gauged bose-einstein condensates, Phys.Rev. Lett. , 015301 (2012).[10] D.-W. Zhang, Y.-Q. Zhu, Y. X. Zhao, H. Yan,and S.-L. Zhu, Topological quantum matter with cold atoms, Advances in Physics , 253 (2018),https://doi.org/10.1080/00018732.2019.1594094.[11] P. J. Ackerman and I. I. Smalyukh, Diversity of knot solitons inliquid crystals manifested by linking of preimages in torons andhopfions, Phys. Rev. X , 011006 (2017).[12] B. G.-g. Chen, P. J. Ackerman, G. P. Alexander, R. D. Kamien,and I. I. Smalyukh, Generating the hopf fibration experimen-tally in nematic liquid crystals, Phys. Rev. Lett. , 237801(2013).[13] J. Pollard, G. Posnjak, S. ˇCopar, I. Muˇseviˇc, and G. P. Alexan-der, Point defects, topological chirality, and singularity theoryin cholesteric liquid-crystal droplets, Phys. Rev. X , 021004(2019).[14] Y. Bouligand, B. Derrida, V. Poenaru, Y. Pomeau, andG. Toulouse, Distortions with double topological character : thecase of cholesterics, Journal de Physique , 863 (1978).[15] J. Pollard and G. P. Alexander, Singular contact geometry andbeltrami fields in cholesteric liquid crystals, arXiv preprintarXiv:1911.10159 (2019).[16] K. Moon, H. Mori, K. Yang, S. M. Girvin, A. H. MacDonald,L. Zheng, D. Yoshioka, and S.-C. Zhang, Spontaneous inter-layer coherence in double-layer quantum hall systems: Chargedvortices and kosterlitz-thouless phase transitions, Phys. Rev. B , 5138 (1995).[17] A. C. Balram, U. Wurstbauer, A. W´ojs, A. Pinczuk, and J. K.Jain, Fractionally charged skyrmions in fractional quantumhall effect, Nature Communications , 10.1038/ncomms9981(2015).[18] S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi,Skyrmions and the crossover from the integer to fractionalquantum hall effect at small zeeman energies, Phys. Rev. B ,16419 (1993).[19] F. Jonietz, S. Muhlbauer, C. Pfleiderer, A. Neubauer, W. Mun-zer, A. Bauer, T. Adams, R. Georgii, P. Boni, R. A. Duine,K. Everschor, M. Garst, and A. Rosch, Spin transfer torques inMnSi at ultralow current densities, Science , 1648 (2010).[20] X. Yu, N. Kanazawa, W. Zhang, T. Nagai, T. Hara, K. Kimoto,Y. Matsui, Y. Onose, and Y. Tokura, Skyrmion flow near roomtemperature in an ultralow current density, Nature Communica-tions , 10.1038/ncomms1990 (2012).[21] S. S. P. Parkin, M. Hayashi, and L. Thomas, Magnetic domain-wall racetrack memory, Science , 190 (2008).[22] J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nu-cleation, stability and current-induced motion of isolated mag-netic skyrmions in nanostructures, Nature Nanotechnology ,839 (2013).[23] R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Carpen-tieri, and G. Finocchio, A strategy for the design of skyrmionracetrack memories, Scientific Reports , 10.1038/srep06784(2014).[24] W. Kang, Y. Huang, C. Zheng, W. Lv, N. Lei, Y. Zhang,X. Zhang, Y. Zhou, and W. Zhao, Voltage controlled magneticskyrmion motion for racetrack memory, Scientific Reports ,10.1038/srep23164 (2016).[25] W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch,F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang,O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, Blowingmagnetic skyrmion bubbles, Science , 283 (2015).[26] G. Chen, A. Mascaraque, A. T. N'Diaye, and A. K. Schmid,Room temperature skyrmion ground state stabilized throughinterlayer exchange coupling, Applied Physics Letters ,242404 (2015).[27] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio,C. A. F. Vaz, N. V. Horne, K. Bouzehouane, K. Garcia, C. De- ranlot, P. Warnicke, P. Wohlh¨uter, J.-M. George, M. Weigand,J. Raabe, V. Cros, and A. Fert, Additive interfacial chiral in-teraction in multilayers for stabilization of small individualskyrmions at room temperature, Nature Nanotechnology ,444 (2016).[28] O. Boulle, J. Vogel, H. Yang, S. Pizzini, D. de Souza Chaves,A. Locatelli, T. O. Mentes¸, A. Sala, L. D. Buda-Prejbeanu,O. Klein, M. Belmeguenai, Y. Roussign´e, A. Stashkevich,S. M. Ch´erif, L. Aballe, M. Foerster, M. Chshiev, S. Auffret,I. M. Miron, and G. Gaudin, Room-temperature chiral mag-netic skyrmions in ultrathin magnetic nanostructures, NatureNanotechnology , 449 (2016).[29] W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. B.Jungfleisch, J. E. Pearson, X. Cheng, O. Heinonen, K. L. Wang,Y. Zhou, A. Hoffmann, and S. G. E. te Velthuis, Direct observa-tion of the skyrmion hall effect, Nature Physics , 162 (2016).[30] A. Soumyanarayanan, M. Raju, A. L. G. Oyarce, A. K. C.Tan, M.-Y. Im, A. P. Petrovi´c, P. Ho, K. H. Khoo, M. Tran,C. K. Gan, F. Ernult, and C. Panagopoulos, Tunable room-temperature magnetic skyrmions in ir/fe/co/pt multilayers, Na-ture Materials , 898 (2017).[31] W. Jiang, G. Chen, K. Liu, J. Zang, S. G. te Velthuis, andA. Hoffmann, Skyrmions in magnetic multilayers, Physics Re-ports , 1 (2017).[32] S. Woo, K. Litzius, B. Kr¨uger, M.-Y. Im, L. Caretta, K. Richter,M. Mann, A. Krone, R. M. Reeve, M. Weigand, P. Agrawal,I. Lemesh, M.-A. Mawass, P. Fischer, M. Kl¨aui, and G. S. D.Beach, Observation of room-temperature magnetic skyrmionsand their current-driven dynamics in ultrathin metallic ferro-magnets, Nature Materials , 501 (2016).[33] Y. S. Lin, P. J. Grundy, and E. A. Giess, Bubble domains inmagnetostatically coupled garnet films, Applied Physics Letters , 485 (1973).[34] T. Garel and S. Doniach, Phase transitions with spontaneousmodulation-the dipolar ising ferromagnet, Phys. Rev. B , 325(1982).[35] S. Takao, A study of magnetization distribution of submicronbubbles in sputtered ho-co thin films, Journal of Magnetism andMagnetic Materials , 1009 (1983).[36] T. Okubo, S. Chung, and H. Kawamura, Multiple- q states andthe skyrmion lattice of the triangular-lattice heisenberg antifer-romagnet under magnetic fields, Phys. Rev. Lett. , 017206(2012).[37] I. Dzyaloshinsky, A thermodynamic theory of “weak” fer-romagnetism of antiferromagnetics, Journal of Physics andChemistry of Solids , 241 (1958).[38] T. Moriya, Anisotropic superexchange interaction and weak fer-romagnetism, Phys. Rev. , 91 (1960).[39] X. Zhang, J. Xia, Y. Zhou, D. Wang, X. Liu, W. Zhao, andM. Ezawa, Control and manipulation of a magnetic skyrmion-ium in nanostructures, Phys. Rev. B , 094420 (2016).[40] M. Finazzi, M. Savoini, A. R. Khorsand, A. Tsukamoto,A. Itoh, L. Du`o, A. Kirilyuk, T. Rasing, and M. Ezawa, Laser-induced magnetic nanostructures with tunable topological prop-erties, Phys. Rev. Lett. , 177205 (2013).[41] F. Zheng, H. Li, S. Wang, D. Song, C. Jin, W. Wei, A. Kov´acs,J. Zang, M. Tian, Y. Zhang, H. Du, and R. E. Dunin-Borkowski,Direct imaging of a zero-field target skyrmion and its polar-ity switch in a chiral magnetic nanodisk, Phys. Rev. Lett. ,197205 (2017).[42] H. Geiges, An Introduction to Contact Topology (CambridgeUniversity Press, 2008).[43] T. Machon, Contact topology and the structure and dynamics ofcholesterics, New J. Phys. , 113030 (2017). [44] M. Z. Hasan and C. L. Kane, Colloquium: Topological insula-tors, Rev. Mod. Phys. , 3045 (2010).[45] X.-L. Qi and S.-C. Zhang, Topological insulators and supercon-ductors, Rev. Mod. Phys. , 1057 (2011).[46] L. Fu and C. L. Kane, Topological insulators with inversionsymmetry, Phys. Rev. B , 045302 (2007).[47] X. S. Wang, H. Y. Yuan, and X. R. Wang, A theory on skyrmionsize, Communications Physics , 10.1038/s42005-018-0029-0(2018).[48] There is a further technical assumption that S t never touchesthe boundary of M .[49] J. W. Gray, Some global properties of contact structures, TheAnnals of Mathematics , 421 (1959).[50] A. G. Kolesnikov, M. E. Stebliy, A. S. Samardak, and A. V.Ognev, Skyrmionium–high velocity without the skyrmion halleffect, Sci. Rep , 1 (2018).[51] B. G¨obel, A. F. Sch¨affer, J. Berakdar, I. Mertig, and S. S. P.Parkin, Electrical writing, deleting, reading, and moving ofmagnetic skyrmioniums in a racetrack device, Scientific Re-ports , 10.1038/s41598-019-48617-z (2019).[52] A. A. Burkov, A. S. N´u˜nez, and A. H. MacDonald, Theoryof spin-charge-coupled transport in a two-dimensional electrongas with rashba spin-orbit interactions, Phys. Rev. B , 155308(2004).[53] A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watan-abe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon,Emergent ferromagnetism near three-quarters filling in twistedbilayer graphene, Science , 605 (2019).[54] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe,T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning su- perconductivity in twisted bilayer graphene, Science , 1059(2019).[55] M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu,K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Intrin-sic quantized anomalous hall effect in a moir´e heterostructure,Science , 900 (2019).[56] X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das,C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, A. Bachtold,A. H. MacDonald, and D. K. Efetov, Superconductors, orbitalmagnets and correlated states in magic-angle bilayer graphene,Nature , 653 (2019).[57] X. Liu, Z. Hao, E. Khalaf, J. Y. Lee, Y. Ronen, H. Yoo, D. H.Najafabadi, K. Watanabe, T. Taniguchi, A. Vishwanath, andP. Kim, Tunable spin-polarized correlated states in twisted dou-ble bilayer graphene, Nature , 221 (2020).[58] Y. Cao, D. Rodan-Legrain, O. Rubies-Bigorda, J. M. Park,K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunablecorrelated states and spin-polarized phases in twisted bi-layer–bilayer graphene, Nature , 215 (2020).[59] G. Chen, A. L. Sharpe, E. J. Fox, Y.-H. Zhang, S. Wang,L. Jiang, B. Lyu, H. Li, K. Watanabe, T. Taniguchi, Z. Shi,T. Senthil, D. Goldhaber-Gordon, Y. Zhang, and F. Wang, Tun-able correlated chern insulator and ferromagnetism in a moir´esuperlattice, Nature , 56 (2020).[60] G. Chen, L. Jiang, S. Wu, B. Lyu, H. Li, B. L. Chittari,K. Watanabe, T. Taniguchi, Z. Shi, J. Jung, Y. Zhang, andF. Wang, Evidence of a gate-tunable mott insulator in a trilayergraphene moir´e superlattice, Nature Physics , 237 (2019).[61] G. Yang, P. Stano, J. Klinovaja, and D. Loss, Majorana boundstates in magnetic skyrmions, Phys. Rev. B93