Bimerons on Möbius Surfaces

• The paper demonstrates that Möbius geometry alters the conventional topological charge, requiring a double cover formulation to account for local sign inversion.
• Micromagnetic simulations using FeGe parameters confirm metastable bimeron states with a two-lobe emergent field pattern and reversible Hall-like dynamics.
• The study explores spin-transfer torque effects and magnonic Aharonov–Bohm interference, highlighting new opportunities for curvature-engineered spintronic devices.

Bimerons are planar analogues of magnetic skyrmions characterized by in-plane magnetization winding from one direction at the core to the opposite in the far field. The confinement of bimerons to Möbius surfaces—two-dimensional nonorientable manifolds constructed by imparting a half-twist to a strip and joining the ends—profoundly alters their topological classification, conservation laws, stability, and quantum transport properties. The topology-induced inversion of surface normal across the Möbius strip reshapes both the effective topological charge and the soliton dynamics, enabling phenomenology inaccessible in orientable geometries (Saji et al., 14 Dec 2025).

1. Magnetization Field Structure on the Möbius Strip

The magnetization field $\mathbf{n}(\mathbf{x}) = \mathbf{M}/M_s$ on a Möbius strip must be described using curvilinear coordinates, with a standard parametrization $$\mathbf{r}(\rho, \varphi) = \left( (R + \rho \cos(\varphi/2)) \cos \varphi,\; (R + \rho \cos(\varphi/2)) \sin \varphi,\; \rho \sin(\varphi/2) \right)^T$$ for mid-radius $R$, width $w$, $\varphi \in [0, 2\pi)$, and $\rho \in [-w/2, w/2]$. The nonorientable nature is manifested in the local unit normal $\mathbf{\hat{n}}(\rho, \varphi)$, whose direction reverses under $\varphi \rightarrow \varphi + 2\pi$, precluding any global distinction between “up” and “down.” This property directly affects the possibility of defining topological invariants and continuity equations for the magnetization field.

2. Topological Charge on Nonorientable Manifolds

In planar or orientable systems, the topological charge $Q$ is given by $$Q = (1/4\pi) \int_\Omega \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})\, dx\,dy$$. For the Möbius geometry, due to orientation reversal, this integral is ill-defined: the two-form changes sign after encircling the strip. The proper formulation lifts the problem to the orientable double cover $$\mathbf{r}(\rho, \varphi) = \left( (R + \rho \cos(\varphi/2)) \cos \varphi,\; (R + \rho \cos(\varphi/2)) \sin \varphi,\; \rho \sin(\varphi/2) \right)^T$$0, where

$$\mathbf{r}(\rho, \varphi) = \left( (R + \rho \cos(\varphi/2)) \cos \varphi,\; (R + \rho \cos(\varphi/2)) \sin \varphi,\; \rho \sin(\varphi/2) \right)^T$$1

with $$\mathbf{r}(\rho, \varphi) = \left( (R + \rho \cos(\varphi/2)) \cos \varphi,\; (R + \rho \cos(\varphi/2)) \sin \varphi,\; \rho \sin(\varphi/2) \right)^T$$2. The factor $$\mathbf{r}(\rho, \varphi) = \left( (R + \rho \cos(\varphi/2)) \cos \varphi,\; (R + \rho \cos(\varphi/2)) \sin \varphi,\; \rho \sin(\varphi/2) \right)^T$$3 accounts for the twofold covering. Projecting back, the emergent field $$\mathbf{r}(\rho, \varphi) = \left( (R + \rho \cos(\varphi/2)) \cos \varphi,\; (R + \rho \cos(\varphi/2)) \sin \varphi,\; \rho \sin(\varphi/2) \right)^T$$4 exhibits local sign inversion after a full circuit. However, the globally defined $$\mathbf{r}(\rho, \varphi) = \left( (R + \rho \cos(\varphi/2)) \cos \varphi,\; (R + \rho \cos(\varphi/2)) \sin \varphi,\; \rho \sin(\varphi/2) \right)^T$$5 remains quantized (for a bimeron, $$\mathbf{r}(\rho, \varphi) = \left( (R + \rho \cos(\varphi/2)) \cos \varphi,\; (R + \rho \cos(\varphi/2)) \sin \varphi,\; \rho \sin(\varphi/2) \right)^T$$6) when referenced to $$\mathbf{r}(\rho, \varphi) = \left( (R + \rho \cos(\varphi/2)) \cos \varphi,\; (R + \rho \cos(\varphi/2)) \sin \varphi,\; \rho \sin(\varphi/2) \right)^T$$7.

3. Conservation Laws and Continuity Equations

Conservation of topological charge in orientable geometries is encoded in the continuity equation $$\mathbf{r}(\rho, \varphi) = \left( (R + \rho \cos(\varphi/2)) \cos \varphi,\; (R + \rho \cos(\varphi/2)) \sin \varphi,\; \rho \sin(\varphi/2) \right)^T$$8, where $$\mathbf{r}(\rho, \varphi) = \left( (R + \rho \cos(\varphi/2)) \cos \varphi,\; (R + \rho \cos(\varphi/2)) \sin \varphi,\; \rho \sin(\varphi/2) \right)^T$$9. For the Möbius strip, $R$0 reverses sign at the twist; the continuity equation must instead be written on the double cover as

$R$1

with $R$2. When projected onto the Möbius strip, apparent sources or sinks emerge at $R$3, though globally there is no net topological charge creation, reflecting the underlying nonorientability.

4. Micromagnetic Stabilization and Simulation Results

Micromagnetic simulations using Mumax³ and the Landau–Lifshitz–Gilbert (LLG) equation with FeGe parameters ($R$4 pJ/m, $R$5 mJ/m², $R$6 A/m, $R$7 H/m) confirm the stabilization of bimerons on Möbius geometries (Saji et al., 14 Dec 2025). The system geometry—length $R$8 nm, width $R$9 nm, thickness $w$0 nm, and mesh 1×1×1 nm³—incorporates free-surface boundary conditions and an external tangent field $w$1 mT. Nucleation is achieved by initializing a tangential magnetization and seeding a Bloch-type bimeron (20 nm diameter) at $w$2. After relaxation, the solution is metastable, with the emergent $w$3 field exhibiting a two-lobe (±) pattern that exhibits sign flipping after a full traversal, in agreement with $w$4 computed on the double cover.

Simulation Parameter	Value/Methodology	Role
Software	Mumax³	Micromagnetic solver
Geometry	$w$5 nm, $w$6 nm, $w$7 nm	Möbius ring parameters
Material	FeGe, DMI, exchange, dipolar, Zeeman	Realistic chiral magnet
Field	$w$8 mT (tangent)	Chiral configuration bias

Micromagnetic estimates show the requirements (field $w$9600 mT, current $\varphi \in [0, 2\pi)$0 A/m², speed tens of m/s) are within the experimental reach for FeGe. The Möbius geometry stabilizes chiral solitonic configurations via geometric constraints and topology-specific emergent fields.

5. Dynamics under Spin-Polarized Currents

When spin-transfer torque (Zhang–Li form) is added to the LLG equation, currents injected tangentially induce bimeron motion. In the collective-coordinate or Thiele approach, the steady-state equation is

$\varphi \in [0, 2\pi)$1

with $\varphi \in [0, 2\pi)$2, flipping direction upon $\varphi \in [0, 2\pi)$3.

Key dynamical properties:

• For $\varphi \in [0, 2\pi)$4 (nonadiabatic regime), local skyrmion Hall effects arise, but the sign of the Hall deflection follows the orientation of $\varphi \in [0, 2\pi)$5, reversing after one full circuit, so the net transverse deflection does not cross the strip globally.
• For $\varphi \in [0, 2\pi)$6, the bimeron propagates stably along a boundary-guided channel.
• For $\varphi \in [0, 2\pi)$7, the bimeron is driven to the edge and annihilated.

The emergent $\varphi \in [0, 2\pi)$8 also governs magnon scattering and spin-motive phenomena.

6. Quantum Interference: Magnonic Aharonov–Bohm Effect

Magnonic modes localized around the bimeron acquire a Berry phase when transported along the Möbius. The Bogoliubov–de Gennes (BdG) magnon Hamiltonian supports a gauge field (spin connection) $\varphi \in [0, 2\pi)$9, with the phase shift upon full traversal given by

$\rho \in [-w/2, w/2]$0

where $\rho \in [-w/2, w/2]$1, $\rho \in [-w/2, w/2]$2 is the magnetochirality. Therefore, the translational Goldstone mode associated with the bimeron acquires a quantized $\rho \in [-w/2, w/2]$3-phase after encircling the strip, resulting in path-dependent interference analogous to the conventional Aharonov–Bohm effect for charged particles on a ring.

7. Spintronic Implications and Curvature-Engineered Functionalities

The Möbius topology, by enforcing an inversion of the bimeron’s internal structure, yields distinctive features: (i) $\rho \in [-w/2, w/2]$4 is meaningful only on the orientable double cover, and (ii) Hall-like dynamics are reversed locally but globally unidirectional. The stabilization, dynamics, and quantum interference signals are robust for experimentally relevant fields and materials. The magnonic $\rho \in [-w/2, w/2]$5-phase is indicative of the possibility for Möbius-based spin wave interferometers. In summary, nonorientable geometries such as the Möbius strip enable soliton channels guided by geometry, topology-tunable Hall responses, and Aharonov–Bohm magnon interference—these regimes are unattainable in planar or orientable curved films, thereby offering pathways for new spintronic device concepts (Saji et al., 14 Dec 2025).