---
title: Bimerons on Möbius Surfaces
url: https://www.emergentmind.com/topics/bimerons-on-mobius-surfaces
type: topic
---

# Bimerons on Möbius Surfaces

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 [2512.12853].

## 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 $\widetilde{\mathcal{M}}$, where
$$
Q_{\text{top}} = \frac{1}{8\pi} \int_{\widetilde{\mathcal{M}}} \widetilde{\mathcal{F}}
$$
with $\widetilde{\mathcal{F}} = \mathbf{n} \cdot (\partial_{x^1} \mathbf{n} \times \partial_{x^2} \mathbf{n}) \; dx^1 \wedge dx^2$. The factor $1/8\pi$ accounts for the twofold covering. Projecting back, the emergent field $B^e = \epsilon^{\mu\nu\rho} \mathbf{n} \cdot (\partial_\nu \mathbf{n} \times \partial_\rho \mathbf{n})$ exhibits local sign inversion after a full circuit. However, the globally defined $Q_{\text{top}}$ remains quantized (for a bimeron, $Q_{\text{top}} = 1$) when referenced to $\widetilde{\mathcal{M}}$.

## 3. Conservation Laws and Continuity Equations

Conservation of topological charge in orientable geometries is encoded in the continuity equation $\partial_t q + \nabla \cdot \mathbf{j}_Q = 0$, where $q = \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})/(4\pi)$. For the Möbius strip, $q$ reverses sign at the twist; the continuity equation must instead be written on the double cover as
$$
\partial_t \widetilde{q} + \widetilde{\nabla} \cdot \widetilde{\mathbf{j}}_Q = 0,
$$
with $\widetilde{q} = \mathbf{n} \cdot (\partial_1 \mathbf{n} \times \partial_2 \mathbf{n})/(8\pi)$. When projected onto the Möbius strip, apparent sources or sinks emerge at $\varphi = 2\pi$, 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 ($A=8.78$ pJ/m, $D=2.9$ mJ/m², $M_s=3.84 \times 10^5$ A/m, $\mu_0=4\pi \times 10^{-7}$ H/m) confirm the stabilization of bimerons on Möbius geometries [2512.12853]. The system geometry—length $L=160\pi$ nm, width $w=40$ nm, thickness $t=4$ nm, and mesh 1×1×1 nm³—incorporates free-surface boundary conditions and an external tangent field $B_{\text{ext}}=600$ mT. Nucleation is achieved by initializing a tangential magnetization and seeding a Bloch-type bimeron (20 nm diameter) at $\varphi=\pi$. After relaxation, the solution is metastable, with the emergent $B^e_z$ field exhibiting a two-lobe (±) pattern that exhibits sign flipping after a full traversal, in agreement with $Q_{\text{top}}=1$ computed on the double cover.

| Simulation Parameter    | Value/Methodology                         | Role                        |
|------------------------|-------------------------------------------|-----------------------------|
| Software               | Mumax³                                    | Micromagnetic solver        |
| Geometry               | $L=160\pi$ nm, $w=40$ nm, $t=4$ nm        | Möbius ring parameters      |
| Material               | FeGe, DMI, exchange, dipolar, Zeeman      | Realistic chiral magnet     |
| Field                  | $B_{\text{ext}}=600$ mT (tangent)         | Chiral configuration bias   |

Micromagnetic estimates show the requirements (field $\sim$600 mT, current $\sim 2 \times 10^{12}$ 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
$$
\mathbf{G}(R) \times (\mathbf{v} - \mathbf{v}_s) + D\alpha(\mathbf{v} - \mathbf{v}_s) + \nabla U(R) = 0,
$$
with $\mathbf{G} = \int B^e d^3 r \approx (4\pi h Q_{\text{top}}/\gamma_0) \hat{n}(R)$, flipping direction upon $\varphi \to \varphi + 2\pi$.

Key dynamical properties:
- For $\alpha \neq \xi$ (nonadiabatic regime), local skyrmion Hall effects arise, but the sign of the Hall deflection follows the orientation of $\mathbf{G}$, reversing after one full circuit, so the net transverse deflection does not cross the strip globally.
- For $\alpha=0.13 > \xi=0.1$, the bimeron propagates stably along a boundary-guided channel.
- For $\alpha=0.08 < \xi$, the bimeron is driven to the edge and annihilated.

The emergent $B^e$ 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) $\mathbf{A}(R)$, with the phase shift upon full traversal given by
$$
\gamma = \oint \mathbf{A}(R) \cdot dR = \oint \tau_{\text{eff}} ds = \pi C \pmod{2\pi}
$$
where $\tau_{\text{eff}} = (\pi C)/L$, $C = \pm 1$ is the magnetochirality. Therefore, the translational Goldstone mode associated with the bimeron acquires a quantized $\pi$-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) $Q_{\text{top}}$ 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 $\pi$-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 [2512.12853].

Source: https://www.emergentmind.com/topics/bimerons-on-mobius-surfaces