Emergence of Nontrivial Topological Magnon States in Skyrmionium Lattices with Zero Topological Charge

Abstract: We predict the emergence of nontrivial topological magnon states in the skyrmionium lattice with zero topological charge. We propose the concept of weighted magnetic flux, which provides a clear physical picture for this anomalous phenomenon. We also map the skyrmionium lattice onto the Haldane model, offering an alternative framework for interpreting this. Our findings challenge the conventional wisdom that such states are linked to nonzero topological charge in skyrmion lattices, offering a new perspective in topological magnonics. To facilitate experimental validation, we propose two methods for preparing the skyrmionium lattice and calculate the induced magnon thermal Hall conductivity, which is a key indicator in transport measurements.

• The paper introduces a new mechanism where magnon bands exhibit nonzero Chern numbers in Q=0 skyrmionium lattices via weighted magnetic flux.
• Using a quantum Holstein-Primakoff framework, the authors map the skyrmionium lattice to an effective Haldane model, revealing chiral magnon edge states.
• Numerical simulations of thermal Hall conductivity validate the topological magnon transport properties and offer clear experimental pathways.

Nontrivial Topological Magnon States in Skyrmionium Lattices with Zero Topological Charge

Overview and Motivation

The paper explores the emergence of topologically nontrivial magnon states within skyrmionium lattices (SkMLs) possessing zero net topological charge. Traditionally, topological magnon bands in magnetic skyrmion lattices (SkLs) have been understood to arise from emergent magnetic fields proportional to the skyrmion topological charge, $Q$. The prevailing consensus held that $Q \neq 0$ was a necessary condition for magnonic topological phenomena, including chiral edge modes and the magnon thermal Hall effect. This work breaks from conventional assumptions by rigorously demonstrating nonzero Chern numbers and robust topological magnon edge states in SkMLs where $Q=0$, and by elucidating a new physical mechanism via the concept of weighted magnetic flux. These findings reframe theoretical understanding in topological magnonics, with direct implications for realizing novel magnonic transport properties in otherwise topologically trivial magnetic phases.

Model, Band Structure, and Topological Characterization

The authors model the SkML as a triangular spin lattice with Bloch-type DMI, uniaxial anisotropy, and Zeeman coupling. Skyrmionium lattices are constructed as periodic arrays where each unit cell comprises two concentric skyrmions of opposite charge, yielding a composite $Q=0$ configuration. Magnon excitations are analyzed within the quantum Holstein-Primakoff boson framework, leading to diagonalization of a quadratic bosonic Hamiltonian using para-unitary transformations.

A central result is the direct calculation of magnon band Chern numbers from Berry curvature integrals:

$$C_j = \frac{1}{2\pi} \int_{BZ} \Omega_j(\mathbf{k})\, d^2\mathbf{k},$$

where $\Omega_j(\mathbf{k})$ is the Berry curvature of the $j$-th magnon band. Contrary to expectations from vanishing topological charge, several bands in the SkML exhibit nonzero $C_j$—compelling evidence of intrinsic magnonic topology. Bulk-edge correspondence confirms the emergence of chiral magnon edge states localized at lattice boundaries whenever the cumulative Chern number below a gap is nonzero. Numerical simulations in strip geometries reveal spatial localization characteristics and the dependence of edge mode distribution on the sign of summed Chern number.

Physical Mechanism: Weighted Magnetic Flux and Real-Space Localization

The paper introduces weighted magnetic flux, $f$, as a physically transparent concept, defined by:

$$f = \int_{A_{uc}} \rho_{mag}(\mathbf{r})\, B_{em}(\mathbf{r})\, d^2\mathbf{r},$$

where $Q \neq 0$0 is the spatial magnon probability density and $Q \neq 0$1 is the emergent magnetic field (from local spin texture). This parameter provides insight into why magnons in a $Q \neq 0$2 SkML can experience a net effective flux depending on their spatial localization within the inner or outer skyrmion structures. The sign and magnitude of $Q \neq 0$3 correlate with the sign of the corresponding band Chern number, albeit not universally; local potentials and intersite couplings play non-negligible roles in full band topology.

Mapping to the Haldane Model

A rigorous mapping is constructed between the SkML and the Haldane model—a paradigmatic topological insulator on a hexagonal lattice with zero net magnetic flux but locally nontrivial flux arrangement. The SkML lattice is renormalized into an effective hexagonal tight-binding model with nearest and next-nearest neighbor couplings; the emergent structure displays the same Chern number assignment and band structure as the Haldane model when the third-nearest neighbor coupling is neglected. This mapping contextualizes SkML topology as a momentum-space phenomenon associated with the unit cell symmetry and coupling configuration, not strictly tied to real-space skyrmion charge.

Thermal Hall Conductivity and Experimental Implications

The magnon thermal Hall effect is calculated for SkMLs via Berry curvature-weighted integrations over the magnon band spectrum, incorporating Bose occupation factors. The primary numerical results are:

• The thermal Hall conductivity, $Q \neq 0$4, decreases monotonically with increasing magnetic field in both SkML and conventional SkL.
• SkMLs exhibit overall lower $Q \neq 0$5 compared to SkLs at comparable temperatures and fields.
• In SkMLs, the decrease of $Q \neq 0$6 with decreasing temperature is moderate, whereas it is precipitous in SkLs due to near-zero Chern numbers and Berry curvature for the lowest-energy magnon bands.

These transport signatures provide viable experimental observables for detecting topological magnon states in SkMLs, circumventing the need for direct edge-state spectroscopy.

Methods for SkML preparation are suggested, including nucleation strategies based on manipulating skyrmion lattices through optical, electrical, or X-ray beams, or tuning DMI and temperature protocols. Phase diagrams indicate SkML stability in low-temperature, low-field regimes.

Broader Implications and Future Directions

This work demonstrates that nontrivial magnon band topology can emerge even when the net topological charge vanishes, expanding the landscape of magnonic topological materials. The weighted flux picture enables generalization to other lattices and DMI types (square, Néel-type). These findings imply that magnonic logic and transport systems could exploit the robustness of topological edge modes without requiring skyrmion phases with $Q \neq 0$7, relaxing material and design constraints.

The Haldane model mapping suggests further exploration of synthetic magnonic lattices engineered for custom band topology, possibly incorporating external field or coupling modulation. Experimentally, the magnon thermal Hall effect in SkMLs can serve as a benchmark for discovery and validation.

Conclusion

The paper rigorously establishes the existence of nontrivial topological magnon bands and chiral edge states in skyrmionium lattices with zero net topological charge (2604.13451). By employing a weighted magnetic flux framework and mapping to the Haldane model, the authors clarify the origin and stability of these phenomena. The calculated magnon thermal Hall conductivity provides a realistic pathway for experimental validation. These advances shift theoretical paradigms in topological magnonics and indicate new directions for practical magnonic device engineering.