Topological Magnon Phase

• Topological magnon phase is a bosonic state in magnetic insulators where magnons form bands with nontrivial topological invariants, yielding robust boundary modes.
• Mechanisms like Dzyaloshinskii–Moriya interaction, Aharonov–Casher effect, strain, and photoirradiation generate Berry curvature in magnon bands, enabling chiral or helical propagation and measurable thermal and spin responses.
• Experimental signatures such as quantized thermal Hall conductivity and distinct magnon edge modes are validated via techniques like inelastic neutron scattering and light scattering, paving the way for advanced magnonic devices.

A topological magnon phase is a macroscopic bosonic state in magnetically ordered insulators where the quantized spin-wave excitations—magnons—form bands characterized by nontrivial topological invariants, resulting in robust boundary modes and anomalous transport phenomena. The topological classification of magnon bands, originally motivated by developments in electronic topological insulators and semimetals, is now established across a wide variety of quantum magnets, including both collinear and noncollinear orders, ferromagnets and antiferromagnets, in two and three dimensions. These phases are realized when mechanisms such as Dzyaloshinskii–Moriya interaction (DMI), electric field gradients (via the Aharonov–Casher effect), strain, or periodic driving (Floquet engineering) produce Berry curvature in magnon bands, leading to Chern numbers, $\mathbb{Z}_2$ invariants, and higher-order topology. The consequence is the emergence of chiral or helical magnon edge/surface states immune to backscattering and associated with measurable thermal and spin transport responses, including the magnon thermal Hall and spin Nernst effects.

1. Symmetry Protection and Topological Classification in Magnon Systems

Topological magnon phases are fundamentally protected by symmetries such as conserved spin components, effective time-reversal, or nonsymmorphic symmetries. In insulating antiferromagnets, the Néel order (opposite spin alignment on sublattices) leads to two decoupled magnon sectors, each behaving like a copy of a ferromagnetic magnon band but with opposite dipole moments. This setting enables a bosonic quantum spin Hall effect (QSHE): rather than time-reversal symmetry, it is the conservation of the spin-z component that protects the topological phase (Nakata et al., 2017). The classification extends to three dimensions using a bosonic analog of the Fu–Kane–Mele model, with $\mathbb{Z}_2$ indices distinguishing strong, weak, and trivial 3D magnon topological insulators (Kondo et al., 2019).

In systems with significant DMI, such as kagome and honeycomb magnets, Berry curvature is generated in the magnon bands, which are characterized by well-defined Chern numbers. The prototypical example is the honeycomb Heisenberg ferromagnet with next-nearest–neighbor DMI, which realizes a magnon version of the Haldane model, supporting magnon Chern insulators and Weyl or Dirac magnons (McClarty, 2021, Zhuo et al., 2023). The topological order may also be characterized by higher-order invariants (second-order topology), seen in van der Waals honeycomb ferromagnets with antiferromagnetic interlayer coupling where corner or hinge modes coexist with bulk and edge states (Li et al., 2022).

2. Mechanisms for Topological Magnon Band Engineering

Dzyaloshinskii–Moriya Interaction

DMI arises at bonds lacking inversion symmetry and acts as an effective spin-orbit coupling for magnons. In kagome, honeycomb, and Shastry–Sutherland lattices, the presence of a finite out-of-plane DMI opens a gap at otherwise degenerate Dirac crossings or quadratic band crossing points, leading to nontrivial Berry curvature and Chern numbers in magnon bands (Malki et al., 2018, Dias et al., 2022, Zhuo et al., 2023). In centrosymmetric materials such as Mn$_5$Ge$_3$, although the net DMI cancels by symmetry, locally noncentrosymmetric bonds generate a finite DMI that is crucial for opening a topological gap at the Dirac points. The direction of magnetization relative to the DMI axis can tune the band gap and control the topological phase transition (Dias et al., 2022).

Aharonov–Casher Effect and Electric Field Engineering

In antiferromagnetic insulators, electric field gradients couple to the magnon dipole moment by the Aharonov–Casher effect. The resulting synthetic vector potential mimics a magnetic field for charge-neutral magnons, causing the formation of Landau levels and the appearance of helical edge states (with up and down magnons propagating oppositely) as a bosonic analog of QSHE (Nakata et al., 2017).

Strain and Floquet (Photoirradiation) Engineering

Topological phase transitions in magnon bands can be induced via lattice distortions. In a strained kagome lattice, the critical strain

$$\delta_c = \frac{1}{2}\left[1 - \left( \frac{D}{J} \right)^2 \right]$$

(where $D$ is DMI and $J$ is the Heisenberg exchange) marks a switch of Chern numbers and topological order; the thermal Hall conductivity exhibits a discontinuous jump at the transition (Owerre, 2018). Time-dependent (high-frequency) strain fields also generate synthetic DMI via a Floquet mechanism, opening a topological gap and enabling spatial confinement or guiding of magnon states (Vidal-Silva et al., 2022).

Photoirradiation (Floquet-Bloch engineering) enables reversible tuning between Chern insulating and Dirac–Weyl magnon semimetal phases, controlled by light intensity and polarization (Owerre, 2018).

3. Topological Edge, Surface, and Higher-Order Modes

The bulk-boundary correspondence ensures that topological bulk magnon bands are accompanied by robust boundary states:

• Chiral edge states: In 2D systems with nonzero Chern number (Chern magnon insulators), unidirectional edge modes cross the gap, carrying heat and spin currents immune to elastic backscattering and disorder (Malki et al., 2018, Subramanian et al., 27 Nov 2024). For instance, ferromagnetic zigzag lattices and Shastry–Sutherland models with anisotropic couplings and DMI host these protected boundary channels.
• Helical edge states: Magnonic QSHE phases in antiferromagnets feature up and down magnons circulating in opposite directions along the sample edge, protected by a $\mathbb{Z}_2$ invariant (Nakata et al., 2017).
• Higher-order boundary modes: When a magnetic system exhibits higher-order topology, the traditional edge/surface states are replaced by modes localized at corners (in 2D) or hinges (in 3D), generally protected by crystalline symmetries or paraunitary Wilson loops. Such phases have been found in honeycomb stackings and monoclinic chromium trihalides, triggered by field-induced noncollinear order and the intrinsic non-Hermitian structure of the magnon BdG Hamiltonian (Park et al., 2021, Li et al., 2022).

4. Topological Invariants and Experimental Signatures

The nontrivial topology of magnon bands is quantified by invariant quantities:

• Chern number $$C_n = \frac{1}{2\pi} \int_{BZ} \Omega^z_n(\mathbf{k}) \, d^2k$$, computed from the Berry curvature $\Omega^z_n(\mathbf{k})$ of magnon band $n$ (Zhuo et al., 2023).
• $\mathbb{Z}_2$ invariant: Defined for systems with effective time-reversal or spin-z conservation, used to classify bosonic QSHE and higher-order topological magnon phases (Nakata et al., 2017, Li et al., 2022, Kondo et al., 2019).
• Spin Chern number: For antiferromagnetic systems with conserved $S_z$, difference between Chern numbers for $s_z = +1$ and $s_z = -1$ bands yields a quantized spin Hall conductance (Lee et al., 2017).

Experimental signatures directly tied to these invariants include:

• Thermal Hall conductivity $\kappa_{xy}$: Nonzero in chiral magnon phases and sharply features at topological phase transitions, with quantized jumps in $\kappa_{xy}$ arising from changes in the sum of Chern numbers or edge state structure (Owerre, 2018, Hofer et al., 2020).
• Spin Nernst and Ettinghausen effects: In AFs, bulk spin and heat currents attributed to Berry curvature, with sign changes as a function of DMI or stratification (Lee et al., 2017).
• Inelastic neutron scattering (INS): Direct measurement of magnon dispersions and gap openings/closings at Dirac points, crucial for verifying magnon band topology as in Mn$_5$Ge$_3$ (Dias et al., 2022).
• Infrared and Brillouin light scattering: Used to probe edge state propagation and topological transitions.

5. Phase Transitions, Topological Tuning, and Hybridization Effects

Topological magnon phase transitions typically occur upon gap closing and reopening events driven by external tuning parameters:

• Strain ($\delta$), electric/magnetic field (via Zeeman or Aharonov–Casher mechanisms), or temperature (renormalizing magnon-magnon interactions) can switch the system across topological boundaries, changing Chern numbers and associated edge state spectra (Owerre, 2018, Lu et al., 2021, Kim et al., 2022).
• Strong DMI regime: At large DMI, a transition from collinear ferromagnet to a $120^\circ$ noncollinear (triangular) phase occurs. The resulting spin structure supports a proliferation of magnon bands with complex Chern number configurations, and a Zeeman field can drive further noncoplanar (chiral) textures, yielding anomalous thermal Hall signals and multiple chiral edge channels (Dong et al., 17 Oct 2025).
• Magnon–phonon hybridization: In the strong DMI/noncollinear regime, quadratic magnon-phonon coupling arises, leading to hybrid bands which inherit topological characteristics from the underlying magnon structure. The degree of magnonic versus phononic character in these bands ($w$) and the associated Chern numbers are directly calculated post-diagonalization of a coupled bosonic Hamiltonian (Dong et al., 17 Oct 2025).

6. Interdisciplinary Links: Topological Transfer and Device Perspectives

The interaction of magnons with electrons enables the "topological transfer" of symmetry breaking and associated Berry curvature from the magnetic subsystem to a trivial electronic system. Electron–magnon interactions, especially in sandwiched heterostructures with proximity to ferromagnets displaying intrinsic magnon topology, induce topologically nontrivial spectral features in the electrons—including mass gaps and nonzero Chern numbers—without the need for intrinsic electron spin–orbit coupling or large magnetic fields (Fujiwara et al., 18 Jul 2025). This mechanism generalizes to the quantum spin Hall regime in time-reversal symmetric trilayer systems.

Table: Manifestations and Control of Topological Magnon Phases

Mechanism or Order	Topological Invariant	Experimental Signature
Dzyaloshinskii–Moriya interaction	Chern number	Chiral edge states, thermal Hall effect
Aharonov–Casher effect (AFs)	$\mathbb{Z}_2$	Helical edge modes, spin Nernst effect
Strain/Floquet/photoirradiation	Chern number	Phase-tunable thermal Hall response
Noncollinear/higher-order topology	$\mathbb{Z}_2$, Z	Hinge/corner modes
Magnon–phonon hybridization	Inherited Chern number	Topological hybrid bands
Magnon–electron coupling (proximity)	Transferred Chern/$\mathbb{Z}_2$	Topological Hall/spin Hall in electrons

7. Outlook and Prospects

Topological magnon phases are crucial for next-generation magnonic and spintronic devices due to their intrinsic immunity to disorder and backscattering, enabling dissipationless information transport. Rapid advances in material engineering—including van der Waals magnets, strain-tunable lattices, and photonic/magnonic crystals—have expanded the experimental platforms for detecting, exploiting, and manipulating magnon topological states. Open research directions include:

• Direct imaging and dynamic control of edge/hinge modes and the impact of interactions on magnon band topology.
• Exploration of non-Hermitian and Floquet magnonic systems.
• Extending "topological transfer" to more complex heterostructures and antiferromagnetic backgrounds for multifunctional device applications.

The field continues to bridge concepts from topological condensed matter physics to quantum magnetism and magnonics, with strong prospects for device-level exploitation of topological magnons (Nakata et al., 2017, Kondo et al., 2019, McClarty, 2021, Dias et al., 2022, Dong et al., 17 Oct 2025, Fujiwara et al., 18 Jul 2025, Zhuo et al., 2023).