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# Rotating Spin Wave Modes in Nanoscale Möbius Strips

Ashfaque Thonikkadavan, Massimiliano d'Aquino, Riccardo Hertel

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Ashfaque Thonikkadavan, Massimiliano d'Aquino, Riccardo Hertel. Rotating Spin Wave Modes in Nanoscale Möbius Strips. 2025. hal-05228333

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Rotating Spin Wave Modes in Nanoscale Möbius Strips

Ashfaque Thonikkadavan,¹ Massimiliano d’Aquino,²,\* and Riccardo Hertel¹,†

¹Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, F-67000 Strasbourg, France

²Department of Electrical Engineering and ICT, University of Naples Federico II, Naples, Italy

Curved and topologically nontrivial magnetic structures offer new pathways to control spin-wave behavior beyond planar geometries. Here, we study spin-wave dynamics in Möbius-shaped soft-magnetic nanostrips using micromagnetic simulations. By comparing single-, double-, and triple-twisted Möbius strips to a topologically trivial bent ring, we isolate the roles of helical twist and non-orientable topology. Möbius geometries exhibit non-degenerate mode doublets associated with counterpropagating spin waves, in contrast to the standing-wave doublets in the trivial case. This splitting arises from a twist-induced geometric (Berry) phase that breaks propagation symmetry, producing non-reciprocal dispersion relations. The Möbius topology further imposes antisymmetric boundary conditions, resulting in half-integer wavelength quantization. Local RF excitation allows for the selective generation of spin waves with defined frequency and direction. An analytical model reproduces the dispersion behavior with excellent agreement. These results highlight how geometric and topological design can be leveraged to engineer spin-wave transport in three-dimensional magnonic systems.

I. INTRODUCTION

The emergence of three-dimensional (3D) nanomagnetism has opened up new frontiers in the study of magnetization dynamics, extending beyond the well-charted territory of planar thin-film systems [1–3]. While magnetic thin films have been intensely investigated for decades, the ability to fabricate and probe 3D magnetic architectures [4–11] enables the exploration of physical phenomena that have no analogue in planar systems, where three-dimensional geometry, topology, and curvature can profoundly modify magnetic textures and spin-wave dynamics [12–17].

Within this broader context, 3D magnonics [2, 18, 19] has rapidly become a vibrant research field, seeking to understand and harness spin-wave phenomena in complex 3D geometries [3]. Parallel to this, curvilinear magnetism [20], where curvature and torsion act as effective interactions, has gained growing attention for the novel magnetic textures and dynamic responses it enables.

Due to their unique geometry and topology, Möbius ring structures have attracted interest across multiple areas of physics involving transport and oscillatory phenomena, including photonics [21], quantum systems [22], phonon dynamics [23], and microwave resonators [24]. In this work, we investigate spin-wave dynamics in Möbius-shaped soft-magnetic nanostructures, which combine three-dimensionality, curvature, and non-trivial topology. Möbius strips possess a non-orientable surface and intrinsic helical twist [25], distinguishing them from topologically trivial ring structures. We use full-scale finite-element (FEM) micromagnetic simulations to model spin-wave dynamics with high accuracy in both time [26] and frequency [27] domains, to capture

the effects of smooth curvature and topology on mode formation and propagation.

A central result is the helicity-dependent splitting, i.e., a lift of degeneracy, between clockwise and counterclockwise propagating modes—a phenomenon absent in topologically trivial strips, where such modes are locked into standing waves by symmetry. Remarkably, the Möbius geometry enables the selective excitation of circularly propagating spin-wave modes using a linearly oscillating magnetic field at the appropriate frequency. We numerically extract the magnon dispersion relations for Möbius strips with one, two, and three twists, which, due to the quantization by the circular boundary conditions, result in a distinct set of frequencies and wave vectors. We utilize an analytical model [28] that reproduces the essential spin-wave dynamics in the Möbius strips. The theoretical predictions show excellent agreement with micromagnetic simulations. These findings highlight how topology and 3D curvature can fundamentally alter magnonic behavior, offering new strategies for engineering spin-wave transport in next-generation nanoscale devices.

II. RESULTS

Previous studies on magnetic Möbius strips have focused primarily on static magnetic textures [29]. In particular, systems with perpendicular magnetic anisotropy have been shown to support topologically protected 180° Bloch-type domain walls stabilized by the strip's non-orientable geometry. Beyond the static regime, analytical work has predicted how curvature and torsion can affect spin-wave propagation in curved magnetic structures, including Möbius geometries [30, 31]. An experimental demonstration of nanoscale Möbius-shaped magnetic rings has also been reported [8], highlighting the feasibility of such structures for future magnonic applications.

In contrast to these earlier works, we investigate dy-

<footnote>* mdaquino@unina.it</footnote>

<footnote>† riccardo.hertel@ipcms.unistra.fr</footnote>
namic spin-wave behavior in soft-magnetic Möbius strips using full-scale micromagnetic simulations. The ground state is a topologically trivial annular vortex, and it is primarily the dynamic response, rather than the static configuration, that is affected by the strip's geometric and topological features. In particular, the perpendicular components of the dynamic magnetization are influenced by the geometry and topology.

<figure><img src="image_2.png" /><figcaption>FIG. 1. Soft-magnetic annular nanostrips with identical material, geometric parameters (thickness, radius, and width), but differing topology. (a) The Möbius strip features a non-orientable surface with constant torsion along the circumference. (b) The twist-compensated “bent” ring is topologically trivial and has a similar local torsion, but with alternating handedness. In both cases, the equilibrium magnetic state is a macroscopic vortex configuration, with the magnetization closely aligned along the tangential direction. Spin waves propagating along the strips experience antisymmetric boundary conditions in the Möbius strip (a) and ordinary, symmetric boundary conditions in the bent ring (b).</figcaption></figure>

As a model system, we consider a Permalloy Möbius strip with a diameter of 100 nm, a width of 20 nm, and a thickness of 2 nm, as shown in Fig. 1a). Due to its nonzero thickness, the structure is technically orientable and topologically equivalent to a torus. However, in a micromagnetic sense, its strong shape anisotropy and minimal thickness constrain the magnetization to lie in-plane, with negligible variation across the thickness. As a result, the system behaves as an effectively two-dimensional magnetic structure, with its topological characteristics directly influencing the dynamic magnetization field.

We expect that key features of the Möbius geometry will strongly influence spin-wave dynamics, particularly for modes propagating along the strip's circumference. First, unlike a flat strip of identical width and thickness, the Möbius strip exhibits a continuous helical twist with an intrinsic handedness defined by its geometry. Second, its reconnection creates a non-orientable surface, leading to nontrivial periodic boundary conditions. In addition, the finite size of the structure results in a discrete, quantized set of resonance modes and frequencies, in contrast to extended planar strips. As we will show later, these geometric and topological features produce distinct effects, including the emergence of geometric phases in the spin-wave modes. By contrast, the circular curvature introduced by closing the strip into a ring plays only a minor role.

To isolate the effects of twist and topology, we compare the Möbius strip to a control structure with the same material properties and geometric parameters (thickness, width, and radius), but differing in topology, as shown

in Fig. 1b. Specifically, we consider a topologically trivial “bent” ring—similar in spirit to the photonic system studied by Wang et al. [21]—in which the strip is twisted so that its local surface is perpendicular to the ring plane on one side and parallel on the opposite side. This twist-compensated ring closely matches the Möbius strip in local curvature over most of its length, but consists of two semicircular segments with opposite helicity and no topological twist.

This comparison allows us to disentangle the individual roles of curvature, local twist, and global topology in shaping spin-wave behavior. For brevity, we limit the discussion to the case of a left-handed Möbius strip; the right-handed version yields analogous, mirror-symmetric results. In both geometries, the equilibrium magnetic configuration (in the absence of external fields) is an annular vortex, with the magnetization lying in-plane and closely following the local tangential direction of the twisted strip.

### A. Resonant modes

We begin by analyzing the spin wave dynamics in the case of the bent, twist-compensated ring. To characterize its intrinsic high-frequency properties, we compute the first 30 eigenmodes, covering a frequency range up to approximately 45 GHz. Since our primary interest lies in understanding how twist and topology influence spin-wave behavior, considering only the first few modes is sufficient to analyze the system's magnonic properties. These modes exhibit well-defined azimuthal spin-wave patterns in the regime below about 20 GHz. In contrast, at higher frequencies, the mode patterns become increasingly complex as radial standing-wave patterns occur, with wavelengths comparable to the strip width.

<figure><img src="image_3.png" /><figcaption>FIG. 2. The first eight standing-wave modes in the bent, twist-compensated ring. The color code indicates the oscillation amplitude, rescaled for each panel, with blue indicating regions of lowest amplitudes (nodes) and red regions of maximum activity (antinodes). The lowest-frequency mode, an FMR-type oscillation without a wave character at 8.60 GHz, is omitted.</figcaption></figure>

We observe distinct standing-wave patterns characterized by a growing number of nodes and antinodes with increasing mode frequency. These results reflect the quan-
tization of spin-wave modes due to the finite ring circum-
ference and its periodic boundary conditions. While such
spin-wave quantization corresponds to what is expected
in flat, planar magnetic rings [32], we notice the appear-
ance of frequency doublets, each corresponding to a pair
of modes with the same number of nodes and antinodes,
but different wave profiles, see Fig. 3a). The modified
geometry of the bent ring leads to the appearance of
mode doublets—pairs of nearly degenerate frequencies.
This effect arises from the azimuthal asymmetry intro-
duced by the bend, causing the nodes and antinodes to
localize at well-defined regions of the ring. Specifically,
the modes develop either maxima or minima in the area
where the strip stands perpendicular or parallel to the
ring plane (the inflection points of the torsion), leading
to a two-fold appearance of modes of the same order.

<figure><img src="image_4.png" /><figcaption>FIG. 3. Frequency doublets of eigenmodes in the bent ring. (a) Eigenmode frequencies as a function of mode number, starting from the lowest-frequency mode. Except for the zero-frequency mode (the ferromagnetic resonance, FMR), all modes appear as nearly degenerate pairs—frequency doublets. A similar pattern is observed in the Möbius strip. (b) Spatial profiles of an example doublet: mode #5 and mode #6. Both modes have the same wavelength and identical numbers of nodes and antinodes, but differ in the spatial positioning of their nodes and antinodes, which are shifted relative to each other along the ring. Color indicates oscillation amplitude, with antinodes (maximum amplitude) shown in red and nodes (minimal amplitude) in blue.</figcaption></figure>

The lowest-frequency mode, at 8.60 GHz, is the FMR
mode in which the entire sample oscillates in phase.
While the phase of the oscillation is homogeneous, the
oscillation amplitude varies significantly across the ring,
with a clear maximum localized in the perpendicularly
bent region. The following two modes, at 9.09 GHz and
9.32 GHz, are long-wavelength standing modes whose
profile is not well-defined, as their wavelength corre-
sponds to the ring's perimeter. Their wave profiles are
difficult to interpret unambiguously as they are signifi-
cantly affected by the strip's torsion, with the wave un-
dergoing substantial changes in surface orientation over
its wavelength. Nevertheless, upon close inspection, one
sees that these modes represent a doublet, with one hav-

ing a maximum oscillation in the perpendicular and hor-
izontal regions and the other exhibiting maximum os-
cillation in the twisted branches. All subsequent modes
are clearly recognizable as doublets of standing-wave pat-
terns with an even number of (anti-)nodes. In Fig. 2, the
mode doublets are grouped columnwise, with the row on
the top displaying the modes where the horizontal seg-
ment on the right is a node. The bottom row shows the
modes complementary to those of the top row, in which
that region is an antinode.

A similar appearance of frequency doublets is also ob-
served in the eigenmodes of the Möbius strip, as shown
in Fig. 3a). In this case, too, the lowest-frequency mode
is an FMR-type oscillation of uniform phase at 8.60 GHz
with a frequency nearly identical to that of the bent ring.
Similar to the bent ring, the two next-higher frequency
modes don't display a clear wave pattern, as the mag-
netic modulation occurs on a length scale correspond-
ing to the ring's circumference. At higher frequencies,
however, the mode character becomes clearly discernible
and changes qualitatively compared to the topologically
trivial ring: the standing-wave patterns are replaced by
propagating modes. While these modes also form dou-
plets, the lifted degeneracy now reflects the direction of
propagation—clockwise or counterclockwise around the
Möbius strip, with one circulation direction consistently
showing a slightly higher frequency than the other, see
Fig. 4. The modes' sense of rotation alternates strictly
in the sequence of modes, revealing an apparent helicity-
dependent symmetry breaking absent in the topologically
trivial ring.

<figure><img src="image_5.png" /><figcaption>FIG. 4. Spin-wave eigenmodes in the Möbius strip. Modes #3 through #10 are shown, with each column representing a pair of modes with similar frequency (doublets). The top row shows modes propagating clockwise; the bottom row shows the corresponding counterclockwise-propagating modes. Each image displays the mode profile at a single phase in time. Color indicates the projection of the dynamic magnetization onto the local surface normal.</figcaption></figure>

Unlike the nearly degenerate mode doublets observed
in the bent ring, the spin-wave modes in the Möbius
strip appear in closely spaced pairs but with clearly dis-
tinct frequencies, cf. Fig. 5a). To understand this fre-
quency splitting, it is helpful to consider the standing-
wave modes in the bent ring as superpositions of two
counterpropagating spin waves with identical frequency
and wavelength. The resulting mode profiles differ only in their phase, i.e., in the positioning of nodes and antinodes around the ring—leading to mode doublets with nominally identical eigenfrequencies.

<figure><img src="image_6.png" /><figcaption>FIG. 5. Frequency splitting of spin-wave mode doublets in the Möbius strip. (a) Mode frequencies for the Möbius strip (orange circles) and the bent ring (blue diamonds). While the bent ring exhibits nearly degenerate doublets, the Möbius strip shows a clear frequency splitting. For example, modes #7 and #8 in the bent ring both resonate at 15.68 GHz, whereas the corresponding Möbius modes occur at 15.41 GHz and 15.92 GHz, yielding a frequency difference of $\Delta f \approx 510$ MHz. (b) Line scans of the dynamic magnetization projected along the central path of the Möbius strip, shown for several modes. The numbers above each scan indicate the number of wavelengths (in units of $2\pi$ phase) along the circular path. The scans exhibit odd half-integer mode numbers, consistent with the antisymmetric boundary conditions imposed by the Möbius topology. Clockwise and counterclockwise modes display visibly different wavelengths.</figcaption></figure>

In the Möbius strip, this degeneracy is lifted, resulting in pairs of eigenmodes with clearly separated frequencies. The formation of frequency doublets has been predicted in the literature in the case of planar magnetic nanorings with vortex configurations under perpendicular magnetic fields, where the splitting arises from field-induced symmetry breaking [33]. In contrast, the origin of the splitting in our case lies in the geometry: the continuous twist of the strip introduces a geometric phase [34] (a Berry phase [35] or Hannay angle [36]) that accumulates as the spin wave propagates along the structure. This geometric phase depends on the helicity of the strip and changes sign with the propagation direction. As a consequence, clockwise and counterclockwise spin waves experience different phase shifts over one complete circuit, which in turn modifies their effective wavelengths. Since the frequency of a spin-wave mode depends on its wavelength, this asymmetry lifts the degeneracy between the two propagation directions, resulting in the observed frequency splitting within each mode pair. As a result of this frequency splitting, a standing-wave mode, formed by the superposition of two counterpropagating waves of identical wavelength and frequency, is no longer possible in the Möbius strip.

In addition to the twist-induced Berry phase, spin-wave modes in the Möbius strip are constrained by quantization due to its non-orientable surface, which imposes antisymmetric boundary conditions. These conditions require the dynamic magnetization to change sign after one full circuit, leading to quantized modes with half-integer numbers of wavelengths along the strip's circumference. Figure 5b) shows line scans of the dynamic magnetization component projected onto the local surface normal, taken along the strip's central path. The scans reveal clear differences in wavelength between clockwise and counterclockwise modes within each doublet and confirm the appearance of odd-numbered half-integer wavelength modes, consistent with the Möbius geometry.

To disentangle the effects of topology-induced boundary conditions from those of the geometric phase introduced by the strip's helical twist, we extended the analysis to double- and triple-twisted Möbius geometries. Circulating spin-wave modes also emerge in these structures, with only minor qualitative differences compared to the single-twist Möbius strip. As before, clockwise and counterclockwise propagating modes appear as doublets; however, the frequency splitting between them increases slightly with the number of twists. This trend suggests that the emergence of circulating modes is not dictated by the boundary conditions associated with non-orientability—which are restored to symmetry in the double-twist case—but rather by the presence and degree of helical twist in the strip geometry.

## B. Dispersion relations

To deepen our understanding of spin-wave propagation in Möbius geometries, we analyze the dispersion relations of circulating spin-wave modes in single-, double-, and triple-twisted Möbius strips. Numerically extracting the wave vector for each eigenmode allows us to relate spin wave length and frequency and thereby to quantify how the geometry and twist affect spin-wave propagation.

The resulting dispersion relations, shown as cross symbols in Fig. 6, exhibit clear non-reciprocal behavior, with $\omega(k) \neq \omega(-k)$. This observation is consistent with previous analytical work on curved magnetic nanostrips [30, 31], which predicted non-reciprocal spin wave propagation due to geometric effects. The dispersion curves display a distinct asymmetry, including off-centered minima that do not occur at $k = 0$. Here, positive and negative wavevectors correspond to clockwise and counterclockwise propagation, respectively. As the number of twists increases—from the single to the double- and triple-twisted Möbius strips—the asymmetry in the dispersion becomes more pronounced. This trend is consistent with the twist-induced geometric phase discussed earlier, which breaks the degeneracy between propagation directions and modifies the effective wavelength.
<figure><img src="image_7.png" /><figcaption>FIG. 6. Comparison between theory and 3D FEM numerical results for single, double, and triple left-handed Möbius strips. Solid lines refer to analytical dispersion relation (II B), open symbols correspond to quantization rule (II B), and “cross” symbols refer to FEM calculations. The values of parameters are: $n = -1, -2, -3$, $R = 50$ nm, $w = 20$ nm, $d = 2$ nm, $N_1 = 0.1211$, $N_2 = 1 - N_1$. The other material parameters are the same as in the previous figures.</figcaption></figure>

In order to provide further interpretation of the finite element calculations, we compare them with analytical formulas for spin wave dispersion relations in twisted nanostrips. The nanostrips have rectangular cross sections with thickness $d$ comparable to the exchange length $\ell_{ex}$ of the material, their length $L$ is much larger than the width $w$, and the axis curve (middle line) has constant curvature $\kappa$ and torsion $\tau$.

The latter assumptions imply that the magnetization profile exhibits no change along the thickness and weak variation along the width, which means that the spatial dependence occurs mainly along the axis having a curvilinear abscissa $u \in [0, L]$. When the equilibrium magnetization is quasi-tangential to the strip axis, the spin wave oscillations at each abscissa $u$ will occur in the plane $(e_1(u), e_2(u))$ defined by the axis (in-plane) normal and (out-of-plane) binormal, respectively. Moreover, the small thickness $d \sim \ell_{ex}$ allows for treating magnetostatics in local form [37], which allows for introducing demagnetizing factors $N_1, N_2$ for the rectangular cross section [38].

A comprehensive analysis of spin wave propagation in twisted nanostrips (also including mutation due to magnetic inertia) has been performed in ref. [28]. The analyt-

ical formulas for spin wave dispersion relations relevant to the present work are readily obtained by neglecting inertial effects:

$$ 
\begin{aligned}
\omega_{\pm}(k) &= -2k\tau\ell_{ex}^2 \pm \sqrt{\omega_1(k)\omega_2(k)} & (1) \\
\omega_1(k) &= h_0 + N_1 + \ell_{ex}^2(k^2 + \tau^2 + \kappa^2), \\
\omega_2(k) &= h_0 + N_2 + \ell_{ex}^2(k^2 + \tau^2),
\end{aligned}
$$

where $k$ is the spin wave wavenumber and $h_0$ is the internal effective field at equilibrium along the strip axis.

It is apparent that the presence of a nonzero torsion $\tau$ produces a symmetry breaking that reflects into the non-reciprocal spin wave propagation and the asymmetric dispersion curve [30], while the curvature $\kappa$ only produces a positive spectral shift in the frequency response.

Moreover, it is possible to show [28] that the spin wave propagation along the nanostrip produces at the abscissa $u$ a geometric (Berry) phase accumulation $\theta(u)$ that arises from a continuous rotation of the spin-wave polarization plane $(e_1(u), e_2(u))$ directly connected with the torsion:

$$ \theta(u) = \int_{0}^{u} \tau(u')du' + C, $$

with $C$ being an arbitrary constant phase-shift. The chirality of the strip expressed by the sign of the torsion $\tau$ is responsible for the aforementioned lifted degeneracy of the spin wave doublets propagating in opposite directions.

The nontrivial topology of the Möbius strip with curvature $\kappa = 1/R$ ($R$ is the radius of the strip axis circle) and torsion $\tau = n/(2R)$ ($n \in \mathbb{Z}$, the sign positive/negative implies right/left-handed chirality) induces a special quantization of the wavenumber due to anti-periodic boundary conditions [28] that results in the following rule:

$$ k_h = \frac{n + 2h}{2R}, \quad h \in \mathbb{Z}. \qquad (2) $$

The dispersion curves for single, double, and triple (i.e. $n = -1, -2, -3$ left-handed Möbius strips) are reported in Fig.6. The excellent agreement with the theory is apparent both concerning the non-reciprocal character of the spin wave propagation and as well as the quantization rule (II B) for wavenumbers induced by the nontrivial topology of the Möbius strips (see the open and 'cross' symbols in Fig.6).

### C. Forced oscillations

While the eigenmode analysis reveals the system's intrinsic high-frequency characteristics, a more experimentally relevant scenario involves the response to an external oscillating field. In such cases, a high-frequency (RF) field induces forced, steady-state oscillations, leading to characteristic absorption peaks at resonant frequencies.
<figure><img src="image_8.png" /><figcaption>FIG. 7. Absorption spectra of the single-, double-, and triple-twisted Möbius strips under an externally applied oscillating magnetic field. The frequency range is chosen to highlight the dominant resonance peaks. The RF field is applied perpendicular to the ring plane (along the z axis). The top panels show the dynamic magnetization profiles corresponding to the labeled peaks [1], [2], and [3] in the spectra below. Color indicates the z component of the dynamic magnetization $\delta m$, and the snapshots are taken at the phase where the average z-component $\langle \delta m_z \rangle$ reaches its maximum.</figcaption></figure>

Fig. 7 shows the RF absorption spectra for the single-, double-, and triple-twisted Möbius strips. All three geometries exhibit a prominent resonance near 9 GHz, with only minor shifts in frequency. This main absorption peak corresponds to the lowest-frequency eigenmode, located at 8.60 GHz, 8.79 GHz, and 8.82 GHz for the single-, double-, and triple-twisted cases, respectively. This mode behaves as a ferromagnetic resonance (FMR), with the magnetization oscillating in phase across the entire structure, though with strongly inhomogeneous amplitude. The largest oscillation amplitudes occur in regions where the strip width is aligned along the z axis. Because this mode lacks propagating character and involves uniform phase oscillation, the geometric and topological differences between the structures have little effect on its dynamics, resulting in nearly identical FMR responses across all three Möbius variants.

In addition to the principal FMR peak, the double- and triple-twisted Möbius strips exhibit secondary resonance peaks at 9.9 GHz and 11.67 GHz, respectively. In the double-twisted case, this secondary peak corresponds to a propagating spin-wave mode (eigenmode #3, labeled as [2] in Fig. 7) whose profile shows in-phase oscillations in the two high-susceptibility regions—where the strip width is aligned along the z axis. A similar situation

arises in the triple-twisted Möbius strip, which features three such areas. There, mode #5 (labeled as [3] in Fig. 7) represents a propagating wave whose wavelength allows constructive, in-phase oscillations across all three high-susceptibility zones. These resonances illustrate how the geometry selectively enhances modes that match the spatial distribution of magnetic susceptibility.

<figure><img src="image_9.png" /><figcaption>FIG. 8. Absorption spectra of the Möbius strip and the bent ring under localized RF excitation. The excitation field is applied to a small, high-susceptibility region in each structure, as indicated by the colored rectangles in the schematics on the left (blue for the Möbius strip, orange for the bent ring). In both cases, the RF field is oriented along the z axis, perpendicular to the ring plane. The bent ring exhibits single-peak resonances at well-defined frequencies. In contrast, the Möbius strip displays double-peak structures with closely spaced frequencies, reflecting the non-degenerate nature of the counterpropagating spin-wave modes.</figcaption></figure>

However, aside from these geometry-matched cases, a globally applied RF field with uniform phase is generally ineffective at exciting the well-defined propagating spin-wave modes discussed in Section II A. To selectively excite such modes, we apply a localized RF field restricted to a specific region of the strip. This approach allows individual eigenmodes to be excited by tuning the driving frequency to match their resonance. An example of this mode-selective excitation is shown in Fig. 8, which presents the absorption spectra of the Möbius strip and the bent ring under time-harmonic excitation localized in the high-susceptibility region—where the strip is oriented such that its width lies along the z axis.

In the topologically trivial bent ring, we observe sharp resonance peaks at well-defined frequencies, each corresponding to one of the nearly degenerate standing-wave mode doublets. Because the excitation is spatially localized, only a subset of modes is activated, specifically those for which the field is applied at or near an oscillation antinode. In our example, these correspond to the bottom row of modes shown in Fig. 2. In contrast, the Möbius strips exhibit resonance doublets with closely spaced frequencies, consistent with the eigenmode structure discussed in Section II A, where each doublet corresponds to a pair of counterpropagating spin-wave modes. The use of a localized, linearly polarized RF field thus
enables the excitation of spin waves with well-defined
frequency and propagation direction, effectively selecting
between clockwise and counterclockwise modes.

### III. DISCUSSION

Our results demonstrate how geometric twist and non-
trivial topology fundamentally alter spin-wave dynam-
ics in three-dimensional magnetic nanostructures. Using
Möbius-shaped soft-magnetic strips, we show that the
interplay between helical curvature and non-orientable
topology leads to symmetry breaking in spin-wave prop-
agation. In contrast to the topologically trivial bent ring,
which supports degenerate standing-wave modes, Möbius
strips exhibit frequency-split doublets corresponding to
counterpropagating spin-wave modes. While the Möbius
strip is non-orientable, we find that it is not the topol-
ogy that drives this effect, but rather the presence of
a net geometric twist along the propagation path. The
bent ring, though curved, features twist-compensation
between its two halves, resulting in an effective zero net
twist and, correspondingly, no Berry-phase-induced split-
ting. In the Möbius geometries, the continuous helical
twist introduces a geometric (Berry) phase that modifies
the effective wavelength depending on direction, break-
ing propagation symmetry and lifting the degeneracy be-
tween clockwise and counterclockwise modes. As a re-
sult, the Möbius strips support directionally selective,
non-reciprocal spin-wave modes that cannot form stand-
ing waves. These effects grow more pronounced with in-
creasing twist, as confirmed by our analysis of double-
and triple-twisted Möbius geometries.

The non-orientable topology of the Möbius strip im-
poses antisymmetric boundary conditions, which result
in half-integer wavelength quantization. Spatially lo-
calized RF excitation enables controlled excitation of
specific eigenmodes and selection of propagation di-
rection—a mechanism that could be extended to pro-
grammable magnonic routing or phase-encoded informa-
tion transfer.

Our analytical model, which closely matches the nu-
merical dispersion data, provides a predictive framework
for designing twisted magnonic systems and interpreting
their spectral features.

These findings highlight the potential of using ge-
ometric and topological design principles to engineer
magnonic responses in 3D systems. Future work may
explore how these effects evolve in interacting Möbius
arrays, or systems with Dzyaloshinskii–Moriya interac-
tion [39]. Our approach also opens up opportunities
for integrating non-orientable magnetic structures into
on-chip spintronic architectures where control over spin-
wave phase, directionality, or non-reciprocity is desirable.

## IV. METHODS

### A. Mesh generation and static magnetization structure

Three-dimensional models of the Möbius-type geome-
tries and the bent ring were generated using the open-
source software FreeCAD [40]. Tetrahedral finite-element
meshes were produced with NETGEN [41], using a target
cell size of 1 nm. This discretization resulted in problem
sizes of slightly more than 50 000 elements for each geom-
etry. The zero-field static magnetization configurations
were computed with our GPU-accelerated open-source
code tetmag [26].

### B. Eigenmodes and forced oscillations

Eigenmodes were computed using a dedicated algo-
rithm described in Ref. [27], which implements the for-
malism originally developed by d’Aquino et al. [42].
The method is based on a linearized form of the Lan-
dau–Lifshitz–Gilbert equation. Initially designed for the
calculation of eigenmodes, the approach was later gen-
eralized [27] to solve micromagnetic problems involving
stationary magnetic oscillations in an applied RF field.
For the forced-oscillation calculations discussed in Sec-
tion IIC, we assumed a Gilbert damping constant of
$\alpha = 0.01$ and an RF field amplitude of 0.5 mT.

### C. Numerical extraction of dispersion relations

Unlike extended magnetic waveguides, Möbius-type
geometries form closed loops, which prevents the direct
application of standard techniques for extracting dis-
persion relations. To address this, we map the three-
dimensional eigenmode profiles onto an effective one-
dimensional coordinate that traces the central path of the
strip. This procedure conceptually “unrolls” the Möbius
geometry into a helical waveguide, rendering it amenable
to spatial Fourier analysis. By performing a Fourier
transform of the dynamic magnetization along this path,
we identify the dominant wavevectors associated with
each eigenmode and thereby construct the spin-wave dis-
persion relation. The *k*-values obtained from this analy-
sis correspond to the spin-wave wavelengths observed in
the line scans shown in Fig. 5b.

## ACKNOWLEDGMENTS

This work was funded by the France 2030 government
investment plan managed by the French National Re-
search Agency ANR under grant reference PEPR SPIN –
[SPINTHEORY] ANR-22-EXSP-0009. The authors ac-
knowledge the High Performance Computing Center of
the University of Strasbourg for supporting this work
by providing access to computing resources. M.d'A. ac-

knowledges support from the Italian Ministry of Univer-
sity and Research, PRIN2020 funding program, grant
number 2020PY8KTC.

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