--- title: 'Twin-Skyrmions: Composite Spin Textures' url: https://www.emergentmind.com/topics/twin-skyrmions type: topic --- # Twin-Skyrmions: Composite Spin Textures Searching arXiv for papers on twin-skyrmions and closely related usages. arxiv_search(query="twin-skyrmions skyrmions twin", max_results=10, sort_by="relevance") Searching arXiv for the specific identifiers and topic variants. arxiv_search(query="1703.03227 OR 1906.07338 OR 2510.25415 OR twin skyrmion", max_results=10, sort_by="relevance") Twin-skyrmions are composite or paired skyrmionic spin textures in which two skyrmion-like objects, or a doubled topological wrapping, define the relevant structure. In the literature, the term does not denote a single universally fixed object. It has been used for a zero-field skyrmion crystal with topological number \(N_{\rm sk}=+2\) per magnetic unit cell in an itinerant triangular-lattice Kondo system, for bound pairs of mutually orthogonal isolated skyrmion tubes in chiral media, and for a three-dimensional bilayer object formed by two interlayer-coupled Néel skyrmions whose shape or helicities are modified by interlayer Dzyaloshinskii–Moriya interaction (IL-DMI) [1703.03227] [1906.07338] [2510.25415]. Closely related constructions include skyrmionic division into two unit-charge “twin” skyrmions, biskyrmions formed by merging two skyrmions, and vortex states embedding two skyrmions [2207.07337] [1902.10491] [2503.16078]. ## 1. Topological characterization and terminological scope The common invariant across these systems is the skyrmion number, written in continuum form as \[ Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy, \] or, for normalized localized spins \(\mathbf{n}(\mathbf r)\), as \[ N_{\rm sk}=\frac{1}{4\pi}\int d^2r\;\mathbf{n}\cdot\bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr). \] On lattices, the same quantity may be evaluated from plaquette solid angles, as in the triangular-lattice Kondo model where \(N_{\rm sk}=(1/4\pi)\sum_p \Omega_p\) [1703.03227]. | Usage in the literature | Constituent structure | Topological characterization | |---|---|---| | Itinerant twin SkX | Triple-\(Q\) skyrmion crystal on a triangular lattice | \(N_{\rm sk}=+2\) per 12-site magnetic unit cell | | Orthogonal twin cluster | Vertical and horizontal isolated skyrmion tubes | Two nonequivalent bound configurations | | Bilayer twin-skyrmion | Two IEC-locked Néel skyrmions in two layers with IL-DMI | \(Q=+1\) per layer in the Thiele description | | Skyrmionic division product | Two separated skyrmions after slicing | Each \(Q=-1\) | | Biskyrmion | Partially overlapping pair of skyrmions | \(N_{\rm Sk}=+2\) | | 2-skyrmion vortex | Vortex with two embedded skyrmions | \(Q_{\rm total}=-2.5\) for a specific sign convention | This diversity matters. A twin-skyrmion may be a single bound object, a crystal carrying doubled topological charge per cell, or a pair of skyrmions produced dynamically. This suggests that the label is descriptive rather than unique, and that the specific stabilization mechanism must be read from context. ## 2. Zero-field \(N_{\rm sk}=2\) twin-skyrmion crystals in itinerant magnets In itinerant magnets, a twin-skyrmion crystal arises in the minimal Kondo-lattice Hamiltonian on the two-dimensional triangular lattice, \[ H = -\sum_{\langle i,j\rangle,\sigma} t_{ij}\,c_{i\sigma}^\dagger c_{j\sigma} -J\sum_i \mathbf{s}_i\cdot\mathbf{S}_i -B\sum_i S_i^z, \] with \(t_1=1\) between nearest neighbors, \(t_3=-0.85\) for third neighbors, \(J=1.0\) in most simulations, and \(|\mathbf S_i|=1\) for classical localized spins. The itinerant-electron spin operator is \(\mathbf{s}_i=(1/2)\sum_{\sigma\sigma'}c_{i\sigma}^\dagger\boldsymbol\sigma_{\sigma\sigma'}c_{i\sigma'}\), and the field \(B\) couples only to the localized spins [1703.03227]. The zero-field state is a skyrmion crystal with unusually high topological number. Each magnetic unit cell of 12 sites carries exactly \(N_{\rm sk}=+2\). At \(B=0\), this twin SkX is exceptionally well approximated by the triple-\(Q\) ansatz \[ \mathbf S_i^{(2)} \propto \bigl( \cos(\mathbf Q_1\!\cdot r_i),\; \cos(\mathbf Q_2\!\cdot r_i),\; \cos(\mathbf Q_3\!\cdot r_i) \bigr), \] where the three ordering vectors \(\mathbf Q_\nu\) lie on the \(\Gamma\)–\(K\) lines of the Brillouin zone at angles \(0\), \(2\pi/3\), and \(4\pi/3\). Physically, the state is a periodic array of vorticity-2 cores surrounded by six meron walls, yielding net skyrmion number two per magnetic cell [1703.03227]. The stabilization mechanism is itinerant-electron-driven. For small \(J\), the RKKY interaction \(\sim J^2\chi_q^0\) favors helices at the maxima of the bare susceptibility \(\chi_q^0\); on the triangular lattice there are three symmetry-related \(Q\) vectors, degenerate at second order. Higher-order \(\sim J^4\) multi-spin interactions then lift this degeneracy in favor of the triple-\(Q\) superposition, lowering the energy by \(\sim10^{-3}\) per site compared to a single-\(Q\) state. This is a distinct route to a high-topological-number crystal because the ordering vectors are selected by the Fermi-surface effect rather than by a conventional DMI-dominated continuum theory [1703.03227]. The numerical evidence combines unrestricted large-scale real-space optimization with variational analysis. The modified KPM-LD simulation uses a Chebyshev expansion up to order \(M=2000\), \(16\times16\) correlated random vectors, a Langevin pseudo-time step \(\Delta\tau=2\), and system sizes up to \(N_s=96^2\) localized spins. The chemical potential \(\mu=-3.5\) fixes the electron filling at \(n\approx0.4\). Variational grand-potential calculations, based on analytic triple-\(Q\) spin configurations and exact diagonalization of the tight-binding electrons in the static spin background, agree with KPM-LD within \(10^{-3}\) [1703.03227]. A distinctive feature is field-controlled topological switching. As \(B\) is raised, the phase sequence is \(BB_{c2}\): \(N_{\rm sk}=0\) triple-\(Q\) collinear/conical or forced ferromagnet. At each transition, the skyrmion number and magnetization jump discontinuously, and the variational \(\Omega(B)\) curves cross at the same \(B_{c1,c2}\) where the KPM-LD data jump. The size of the skyrmions is also tunable because \(|Q_\nu|\) is set by Fermi-surface nesting and hence by \(\mu\), \(t_2\), and \(t_3\); increasing \(\mu\) from \(-4.05\) to \(-3.5\) shifts \(|Q|\) from \(0.18\pi\) to \(0.33\pi\), reducing the lattice constant \(\lambda\) from \(\sim11a\) to \(\sim6a\) [1703.03227]. The full simulations additionally resolve local solid angle, bond current density, itinerant-electron spin magnitude, and charge-density modulation. For the \(N_{\rm sk}=2\) phase, all local solid angles are positive, the bond currents form circular loops around each skyrmion core, \(s_i\) is largest where \(\Omega_p\) is smallest, and a weak 12-site charge-density wave correlates with the spin texture. These observables inherit the threefold rotational symmetry of the underlying SkX [1703.03227]. ## 3. Orthogonal twin-skyrmion clusters in chiral magnets and liquid crystals A second major usage of the term refers to bound pairs of isolated skyrmion tubes with mutually orthogonal axes. In this geometry, one distinguishes vertical tubes, whose central axis is parallel to the applied magnetic field \(\mathbf H\), and horizontal tubes, whose axis lies in the plane perpendicular to \(\mathbf H\). When a vertical isolated skyrmion (V-IS) and a horizontal isolated skyrmion (H-IS) become co-localized, their axes are mutually orthogonal and the H-tube thread may pass through the coil of the V-tube; this structure is described as a “twin” or orthogonal cluster [1906.07338]. The phenomenology is governed by the standard noncentrosymmetric ferromagnet energy density \[ w(\mathbf m)=A(\nabla\mathbf m)^2+D\,\mathbf m\cdot(\nabla\times\mathbf m) -\mu_0 M\,\mathbf m\cdot\mathbf H +w_{\rm surf}\,\delta_{\rm surf}, \] with Rapini–Papoular anchoring \(w_{\rm surf}=W(1-(\mathbf m\cdot\hat{\mathbf n})^2)\). The total free energy is \[ \mathcal F=\int (w_{\rm ex}+w_{\rm DMI}+w_{\rm Z}+w_{\rm surf})\,dV. \] In the liquid-crystal analogue, \(A\to K/2\), \(D\to Kq_0\), and the Zeeman term is replaced by \(-\tfrac12\varepsilon_0\Delta\varepsilon(\mathbf n\cdot\mathbf E)^2\), again with Rapini–Papoular anchoring [1906.07338]. The pair interaction is extracted from \[ U(r)=\mathcal F(r)-2\,\mathcal F_{\rm IS}, \] where \(\mathcal F(r)\) is the minimized energy at fixed center-to-center distance \(r\). The resulting profile is Lennard-Jones-like: a shallow repulsive core at very small \(r\), a pronounced minimum at the bound-state separation, and a small barrier at intermediate distances. The sign of the interaction depends on the host phase. In a conical background, each isolated skyrmion has a shell of positive excess energy, and shell overlap lowers the total energy, producing attraction. In the homogeneous state for \(H>H_D\), the far field is uniform and the skyrmions repel [1906.07338]. The orthogonal pair has two nonequivalent minima. Labeling the H-IS polarity by \((\downarrow\uparrow)\), one obtains \((\downarrow\uparrow\,V)\), in which the H-IS slides its positive shell into the coil of the V-IS, and \((V\,\downarrow\uparrow)\), in which the H-IS meets the V-IS core first. Numerically, \(U_a(r)\) for \((\downarrow\uparrow\,V)\) is deeper, with binding energy \(\sim0.8\,E_0\) at \(r_a\approx1.5\,L_D\), whereas \(U_b(r)\) for \((V\,\downarrow\uparrow)\) is shallower, \(\sim0.1\,E_0\), at \(r_b\approx2.5\,L_D\). As the field decreases and the cone angle increases, both separations shrink, but \(U_a\) strengthens much more than \(U_b\) [1906.07338]. In the extreme mutual-attraction regime, the structure ceases to resemble a simple bound pair. When the cone angle approaches \(90^\circ\), the H-IS deforms into an undulated path winding through successive loops of the V-IS coil. The winding angle satisfies approximately \[ \phi_{\rm wind}(z)\approx q_0 z\pm\Delta\phi, \] with \(\Delta\phi\sim\mathcal O(1)\), and the transverse modulation amplitude is \(a_\perp\sim0.3\,L_D\). This is still topologically rooted in the interlocking of skyrmion shells, but geometrically it is already a compact superstructure rather than a weakly bound dimer [1906.07338]. The experimental realization was demonstrated in a chiral nematic liquid crystal, 5CB doped with 3 wt% CB15, in cells of thickness \(d=10\,\mu{\rm m}\) with \(d/p_0\approx1.2\) and strong homeotropic anchoring \(W/K\sim10^5\,{\rm m}^{-1}\). Vertical and horizontal skyrmions were generated and manipulated with holographic optical tweezers using a 1064 nm laser and a spatial light modulator. Weakly bound twins with \(U\sim10^{-2}E_0\) and \(r\sim2.5L_D\) appear at small cone angles, whereas strongly bound twins with \(U\sim10^{-1}E_0\) and \(r\sim1.5L_D\) occur at large cone angles. The same study argues that analogous clusters are expected in the A-phase of cubic helimagnets such as MnSi, FeGe, and Cu\(_2\)OSeO\(_3\) [1906.07338]. ## 4. Bilayer twin-skyrmions stabilized by interlayer Dzyaloshinskii–Moriya interaction In magnetic bilayers, a twin-skyrmion is a three-dimensional bound object formed by two isolated Néel-type skyrmions, one in each ferromagnetic layer, coupled by interlayer exchange and IL-DMI. For layer \(l=1,2\), \[ \mathbf m_l(\rho,\varphi)= \bigl(\sin\Theta(\rho)\cos\Phi_l(\varphi),\; \sin\Theta(\rho)\sin\Phi_l(\varphi),\; \cos\Theta(\rho)\bigr), \] with \(\Phi_l(\varphi)=m\varphi+\psi_l\), \(m=+1\), \(\Theta(0)=\pi\), and \(\Theta(\infty)=0\). Without IL-DMI, the two skyrmions lock on top of each other by IEC so that \(\psi_1=\psi_2\) and their cores coincide. IL-DMI forces a finite tilt between spins at corresponding in-plane positions in the two layers and produces a twin-skyrmion in which the cores remain collinear but the radius, ellipticity, or helicities adapt to maximize the IL-DMI energy gain [2510.25415]. The interlayer chiral term is \[ E_{\rm IL\hyphen DMI} =-\sum_i \mathbf D_i^{\rm IL}\cdot \bigl(\mathbf S_i^{(1)}\times \mathbf S_i^{(2)}\bigr), \] or, in continuum form, \[ E_{\rm IL\hyphen DMI} =-\int d^2r\; \mathbf D^{\rm IL}\cdot \bigl(\mathbf m_1(\mathbf r)\times \mathbf m_2(\mathbf r)\bigr). \] The full continuum free energy is \[ E_{\rm tot} =\sum_{l=1}^2\int d^2r\; \Bigl[A|\nabla\mathbf m_l|^2 +D_{\rm IF}\,\mathbf m_l\cdot(\nabla\times\mathbf m_l) -\mu_0M_s\,\mathbf B\cdot\mathbf m_l\Bigr] -\int d^2r\; \Bigl[J_{\rm IL}\,\mathbf m_1\cdot\mathbf m_2 +\mathbf D^{\rm IL}\cdot(\mathbf m_1\times\mathbf m_2)\Bigr]. \] Minimization yields the equilibrium radius, the layer-resolved helicities, and the ellipticity [2510.25415]. The direction of \(\mathbf D^{\rm IL}\) determines the deformation mode. For in-plane \(\mathbf D^{\rm IL}\parallel\hat x\) or \(\hat y\), the background magnetizations in the two layers tilt in opposite transverse directions and the skyrmion elongates along a direction \[ \hat R =\pm \frac{1}{\sqrt{D_x^2+D_y^2}} \begin{pmatrix} \cos\psi & -\sin\psi\\ \sin\psi & \cos\psi \end{pmatrix} \begin{pmatrix} D_x\\ D_y \end{pmatrix}. \] For out-of-plane \(D_z^{\rm IL}\), the equilibrium helicities satisfy \(\psi_1=-\psi_2\neq0\), while the shape remains circular [2510.25415]. The dynamics under current-perpendicular-to-plane excitation are described by a Thiele equation, \[ \mathbf G\times\mathbf v+\alpha\,\mathfrak D\cdot\mathbf v =\mathbf F_{\rm STT}, \] with \(\mathbf G=-4\pi Q(\mu_s/\gamma)\hat e_z\), \(\mathfrak D_{\mu\nu}=\int (\partial_\mu\mathbf m\cdot\partial_\nu\mathbf m)\,d^2r\), and \(\mathbf F_{\rm STT}=\mathcal B\,\mathbf P\). For a circular skyrmion with helicity \(\psi\) and current polarization \(\mathbf P=(P_x,P_y)\), the velocity obeys the explicit relation given in the paper, and for \(\mathbf P\parallel\hat x\) the Hall angle satisfies \[ \theta_H(\psi)=\theta_H(0)-\psi,\qquad \theta_H(0)=\arctan\Bigl(\frac{G}{\alpha\mathcal D}\Bigr). \] The numerical findings are specific. Without IL-DMI, \(\theta_H\approx83^\circ\) and \(|v|\approx13\,a/{\rm ns}\). In-plane IL-DMI of magnitude \(1\) meV changes \(\theta_H\) by up to \(\sim23^\circ\), down to \(60^\circ\) or up to \(107^\circ\) along \(D_x=\pm D_y\), while \(|v|\) varies by up to \(70\%\) and is maximal along the elongation direction. Out-of-plane IL-DMI up to \(0.6\) meV increases \(|v|\) by about \(18\%\) through the larger skyrmion radius, whereas \(\theta_H\) changes by less than \(0.3\%\) [2510.25415]. This bilayer realization shows that a twin-skyrmion need not be a mere superposition of two static copies. The IL-DMI introduces internal shape and helicity degrees of freedom absent in a simple IEC-locked bilayer pair. At large in-plane IL-DMI, the twin-skyrmion undergoes an elliptic instability into stripes, which the source identifies as a possible mechanism for current-driven writing or erasure [2510.25415]. ## 5. Formation pathways: division into twins and merging into biskyrmions One route to twin-skyrmions is dynamical splitting. In a thin-film wire of lateral size approximately \(3072\times1024\) nm\(^2\), thickness \(1\) nm, and cell size \(2\times2\times1\) nm\(^3\), the micromagnetic dynamics are governed by \[ E[\mathbf m]=\int d^2x\; \Bigl\{ A(\nabla\mathbf m)^2 +D\,\mathbf m\cdot(\nabla\times\mathbf m) -\mu_0M_s\,\mathbf H_{\rm ext}\cdot\mathbf m +K_u(1-m_z^2) +E_{\rm demag}[\mathbf m] \Bigr\}, \] with \(A=20\) pJ m\(^{-1}\), \(D=3\) mJ m\(^{-2}\), \(K_u=0.53\) MJ m\(^{-3}\), \(M_s=580\) kA m\(^{-1}\), and \(\alpha=0.3\). The time evolution follows the Landau–Lifshitz–Gilbert equation with Slonczewski torque in CPP geometry [2207.07337]. The division protocol consists of inflation, cutting, detachment, and relaxation. A field \(\mathbf H_{\rm ext}=130\) mT in \(-\mathbf e_z\) is applied for a few nanoseconds to enlarge the skyrmion and reduce boundary curvature. A CPP current density \(j=40\) nA nm\(^{-2}\) along \(+z\) for about \(2\) ns drives the skyrmion onto a non-magnetic trench etched perpendicular to the wire axis. As the boundary wraps the trench, regions of opposite topological density \(q(x)\) form at the trench edges, the texture deforms from circular to figure-eight, and a pocket of negative charge grows. When the broken domain-wall segment reaches the trench end, a short jolt of \(j=80\) nA nm\(^{-2}\) for \(2\) ns pinches off the two halves, which relax into two separate stable skyrmions, each with \(Q=-1\). The source formulates the energetic criterion as \[ E_{S(1)}+E_{S(1)}+E_{\rm neg} < E_{\rm deformed\ single\ skyrmion}, \] so that cutting becomes favorable once the opposite-sign pocket is sufficiently developed [2207.07337]. A second route is controlled merging. In a centrosymmetric nanodisk without DMI, the free energy includes exchange, anisotropy, Zeeman, and full dipole–dipole terms, and the relevant bound pair is the biskyrmion. The analytical superposition ansatz for two Bloch skyrmions with opposite in-plane magnetizations shows that the dipolar term alone selects the bound state: the short-range approximation of the dipolar energy exhibits a pronounced minimum at \(\Delta r\approx2r_0\), corresponding to two partially overlapping skyrmions. Applying the standard topological integral to the superposed texture yields \(N_{\rm Sk}=+2\), and in the discrete simulation the result remains within \(1\)–\(2\%\) of exactly \(2\) [1902.10491]. The micromagnetic protocol uses a nanodisk of radius \(R=150\) nm and thickness \(d_z=3\) nm, discretized with cell size \(1\times1\times1\) nm\(^3\), and material parameters \(A=15\) pJ/m, \(K_u=1.2\) MJ/m\(^3\), \(M_s=1.4\) MA/m, \(\alpha=0.3\), and external field \(\mathbf B_{\rm ext}=-40\) mT \(\mathbf e_z\). Two auxiliary writing disks of radius \(25\) nm, centered at \(\pm120\) nm along \(x\), host in-plane vortices of helicity \(\pm\pi/2\). A spin-polarized current \(j_{\rm write}P=7\times10^8\) A/cm\(^2\) along \(+z\) for \(200\) ps imprints the textures; after about \(1\) ns of relaxation, two \(N_{\rm Sk}=+1\) Bloch skyrmions spiral toward the disk center and merge into a stable biskyrmion once their separation reaches the dipolar minimum. Helicity control determines whether the interaction is attractive or repulsive, and helicity-flipping pulses can reverse the bound state back into two skyrmions [1902.10491]. Taken together, these formation pathways show that “twin-skyrmion” behavior can emerge either by fission of a single skyrmion into two unit-charge descendants or by attraction and partial overlap of two initially separate skyrmions. The first mechanism exploits local nucleation of opposite-sign topological density; the second exploits a bound-state minimum selected by dipole–dipole interaction. ## 6. Related composite textures, distinctions, and functional implications Two neighboring categories sharpen the conceptual boundaries of twin-skyrmions. The first is the skyrmion-embedded vortex, or \(n\)-skyrmion vortex, where the total topological charge obeys \[ Q_{\rm total}=Q_{\rm vortex}+n\,Q_{\rm skyrmion}. \] In a circular disk of thickness \(t=2\) nm and radius \(R=200\) nm, with \(A=1.3\times10^{-11}\) J/m, \(M_s=1.1\times10^6\) A/m, \(D=2\times10^{-3}\) J/m\(^2\), \(K_u\approx(3\text{–}5)\times10^5\) J/m\(^3\), and \(H_z\approx0\)–\(50\) mT, a 2-skyrmion vortex becomes a local energy minimum for \(K_u\gtrsim2.8\times10^5\) J/m\(^3\). Under the conventional sign convention \(Q_{\rm vortex}=-1/2\) and \(Q_{\rm skyrmion}=-1\), the total charge is \(Q_{\rm total}=-2.5\). The state is metastable, higher in energy than the single-skyrmion vortex or pure vortex, but separated from collapse by an energy barrier [2503.16078]. The second is skyrmionium-based matter. A skyrmionium is a \(2\pi\) target texture with an inner core carrying \(Q_{\rm core}=+1\) and an outer ring carrying \(Q_{\rm ring}=-1\), so the total charge is \(Q=0\). Its double-wall structure is encoded by a profile \(\theta(r)\) with \(\theta(0)=\pi\), \(\theta(R_{\rm DW})=\pi/2\), and \(\theta(\infty)=0\), plus an additional oscillation at \(r\approx R_{\rm out}\). Pure skyrmionium lattices are unstable against elongation distortions and relax into the spiral phase, but mixed skyrmionium–skyrmion crystals can be stabilized by a dilute skyrmion sublattice acting as topological “pins.” These states are classified by topological stoichiometry such as SkmSk, Skm\(_2\)Sk, Skm\(_2\)Sk\(_3\), and SkmSk\(_2\), and they host hybrid excitations including deformation-assisted rotations and orbital modes [2511.15999]. These neighboring systems clarify a recurrent misconception: a doubled internal structure does not imply a unique topological charge or a unique stabilization mechanism. A twin-skyrmion crystal in an itinerant magnet has \(N_{\rm sk}=2\) per magnetic cell and can exist at zero magnetic field [1703.03227]. A bilayer twin-skyrmion consists of two skyrmions with cores kept collinear by IEC while IL-DMI reshapes or re-helicizes them [2510.25415]. An orthogonal twin cluster is defined geometrically by mutually perpendicular skyrmion tubes and by an interaction controlled by the conical host state [1906.07338]. By contrast, skyrmionium has total \(Q=0\), and a 2-skyrmion vortex carries a half-integer offset because of the vortex background [2503.16078] [2511.15999]. The functional implications also differ accordingly. In the itinerant \(N_{\rm sk}=2\) crystal, the topological number can be switched as \(2\leftrightarrow1\leftrightarrow0\) by tiny field changes \(\Delta B\sim10^{-3}\), the zero-field state has no net magnetization, and higher emergent magnetic fields proportional to \(N_{\rm sk}\) are expected to enhance topological Hall signals [1703.03227]. In IL-DMI bilayers, the tunable relation between elongation, helicity, velocity, and Hall angle provides additional control knobs for current-driven transport [2510.25415]. In orthogonal clusters, the attraction–repulsion crossover is controlled by host-state conicity or by liquid-crystal thickness and anchoring, making the geometry of the bound state itself an experimental control variable [1906.07338]. A plausible implication is that the scientific value of twin-skyrmions lies less in a single canonical texture than in a family of two-skyrmion constructions that expose different combinations of topology, dimensionality, chirality, and collective dynamics. Source: https://www.emergentmind.com/topics/twin-skyrmions