Dirac Magnons: Topological Bosonic Excitations

• Dirac magnons are bosonic spin-wave excitations exhibiting linear band crossings similar to graphene’s Dirac cones, arising from symmetry-protected nodal degeneracies.
• They appear in various systems—from 2D honeycomb ferromagnets to 3D PT-symmetric antiferromagnets—and can shift from massless to massive states with interactions like DMI.
• Theoretical models such as linear spin-wave theory and Green’s function methods effectively correlate neutron scattering data with the topological characteristics of magnon bands.

Dirac magnons are bosonic spin-wave excitations whose band structures exhibit linear crossings analogous to Dirac fermions in electronic systems. In the simplest realization, a ferromagnet on a honeycomb lattice yields two magnon branches that touch linearly at the Brillouin-zone corners $K$ and $K'$; in three-dimensional antiferromagnets with combined inversion and time-reversal symmetry, $PT$, a Dirac magnon node can appear as a fourfold crossing of two doubly degenerate magnon branches (Fransson et al., 2015, Bao et al., 2017). Across the literature, the term therefore denotes a symmetry-protected bosonic nodal excitation rather than a single lattice type or dimensionality, and it extends naturally to gapped topological Dirac magnons, nodal loops, nodal lines, nodal planes, and related multicomponent crossings (Li et al., 2017, Owerre, 2018).

1. Definition and elementary band-theoretic structure

In honeycomb ferromagnets, the defining structural ingredient is the two-site basis. Linear spin-wave theory then produces a $2\times 2$ magnon Hamiltonian in sublattice pseudospin space, directly paralleling graphene. For the nearest-neighbor Heisenberg model, one obtains two magnon branches

$$\varepsilon_{\pm}(\mathbf k)=JS\left(3\pm |\gamma_{\mathbf k}|\right),$$

with $\gamma_{\mathbf k}$ the honeycomb structure factor. Because $\gamma_{\mathbf K}=\gamma_{\mathbf K'}=0$, the two bands touch linearly at $K$ and $K'$, yielding Dirac magnons at finite positive energy rather than at a Fermi level (Pershoguba et al., 2017).

The same logic can be written in a form convenient for topological perturbations. In the honeycomb ferromagnet with next-nearest-neighbor Dzyaloshinskii–Moriya interaction (DMI), the single-magnon Hamiltonian becomes

$$M_{\mathbf k}= \begin{pmatrix} 3JS-2DS\,\beta_{\mathbf k} & -JS\,\gamma_{\mathbf k}\ -JS\,\gamma_{\mathbf k}^* & 3JS+2DS\,\beta_{\mathbf k} \end{pmatrix},$$

with magnon energies

$K'$0

For $K'$1, the bands touch at $K'$2 and $K'$3; for $K'$4, a gap

$K'$5

opens at the Dirac points (Sun et al., 2022). This distinction underlies the common classification into massless and massive Dirac magnons.

The concept is not restricted to honeycomb ferromagnets. The earliest general honeycomb analysis showed that localized spins on a two-dimensional honeycomb lattice naturally generate magnon bands with Dirac-like crossings, with the Dirac points at $K'$6 and $K'$7 in the ferromagnetic case and at $K'$8 in the antiferromagnetic case (Fransson et al., 2015). A common misconception is therefore that Dirac magnons are intrinsically tied to ferromagnetic honeycomb layers; the literature instead treats them as a broader class of bosonic linear band touchings whose detailed form is fixed by lattice geometry, magnetic order, and symmetry.

2. Symmetry protection and nodal taxonomy

The simplest honeycomb Dirac magnons are protected by the two-sublattice structure and the vanishing of the inter-sublattice structure factor at $K'$9 and $PT$0. In three-dimensional antiferromagnets, however, the central protecting mechanism can instead be magnetic $PT$1 symmetry. For a collinear $PT$2-symmetric antiferromagnet with spin $PT$3 symmetry, the quadratic magnon Hamiltonian decomposes as

$PT$4

so a Weyl point in one spin sector is accompanied by an opposite-charge partner in the other sector at the same momentum. The full crossing is then a fourfold Dirac magnon node (Li et al., 2017). In this setting, Dirac magnons are not merely accidental point degeneracies but composites of symmetry-related Weyl magnons.

Once $PT$5 is broken while $PT$6 remains intact, the same three-dimensional crossings need not gap. Instead, the generic descendants are nodal lines with nontrivial $PT$7 charge, and the Dirac point becomes the zero-radius limit of a nodal line (Li et al., 2017). This mechanism is conceptually distinct from the honeycomb DMI gap problem: the same label, “Dirac magnon,” covers both a two-band graphene-like cone and a fourfold $PT$8-protected three-dimensional crossing.

The taxonomy becomes broader in lower dimensions as well. On the two-dimensional CaVO lattice, a collinear antiferromagnet with $PT$9 symmetry yields eight doubly degenerate magnon branches whose linear crossings form one-dimensional closed loops in the two-dimensional Brillouin zone. These are Dirac nodal loop magnons rather than isolated Dirac points, and their topology is diagnosed by a parity-based $2\times 2$0 invariant with $2\times 2$1 (Owerre, 2018). On the anisotropic square-hexagon-octagon lattice, type-II Dirac magnon points arise without requiring DMI, and their overtilted cones coexist with Dirac nodal loops whose protection is characterized by a $2\times 2$2 invariant and Wilson-loop analysis (Zhang et al., 2023).

A further extension concerns the internal spin texture of the magnon eigenmodes. In a kagome $2\times 2$3 antiferromagnet with $2\times 2$4 order, finite-energy Dirac points at $2\times 2$5 and $2\times 2$6 are accompanied by a momentum-space magnon spin vortex with winding number

$2\times 2$7

whereas a related kagome model exhibits a $2\times 2$8 vortex around $2\times 2$9 (Okuma, 2017). This establishes that Dirac magnons may be characterized not only by linear dispersion but also by a Dirac-like winding of the one-magnon spin expectation value.

3. Microscopic Hamiltonians and calculational frameworks

The microscopic starting point is usually an exchange Hamiltonian of Heisenberg type,

$$\varepsilon_{\pm}(\mathbf k)=JS\left(3\pm |\gamma_{\mathbf k}|\right),$$0

sometimes augmented by anisotropy, DMI, or Zeeman terms. For Cu$$\varepsilon_{\pm}(\mathbf k)=JS\left(3\pm |\gamma_{\mathbf k}|\right),$$1TeO$$\varepsilon_{\pm}(\mathbf k)=JS\left(3\pm |\gamma_{\mathbf k}|\right),$$2, the experimentally fitted model is an isotropic Heisenberg Hamiltonian extending to sixth neighbors,

$$\varepsilon_{\pm}(\mathbf k)=JS\left(3\pm |\gamma_{\mathbf k}|\right),$$3

with best-fit parameters

$$\varepsilon_{\pm}(\mathbf k)=JS\left(3\pm |\gamma_{\mathbf k}|\right),$$4

and $$\varepsilon_{\pm}(\mathbf k)=JS\left(3\pm |\gamma_{\mathbf k}|\right),$$5 the dominant scale (Bao et al., 2017). For elemental Gd, the basic topological magnon structure is already present in a two-band hcp ferromagnetic model with exchange constants $$\varepsilon_{\pm}(\mathbf k)=JS\left(3\pm |\gamma_{\mathbf k}|\right),$$6, $$\varepsilon_{\pm}(\mathbf k)=JS\left(3\pm |\gamma_{\mathbf k}|\right),$$7, and $$\varepsilon_{\pm}(\mathbf k)=JS\left(3\pm |\gamma_{\mathbf k}|\right),$$8, where the band splitting is controlled by an inter-sublattice structure factor $$\varepsilon_{\pm}(\mathbf k)=JS\left(3\pm |\gamma_{\mathbf k}|\right),$$9 and nodal manifolds occur where $\gamma_{\mathbf k}$0 (Scheie et al., 2021).

The standard calculational route is linear spin-wave theory. After a Holstein–Primakoff expansion about the ordered state, one diagonalizes either an ordinary quadratic bosonic Hamiltonian or, in antiferromagnets, a bosonic Bogoliubov problem. The CaVO antiferromagnet is a representative example: its quadratic spin-wave Hamiltonian contains the usual pairing terms $\gamma_{\mathbf k}$1, so diagonalization requires a Bogoliubov transformation rather than a simple unitary one (Owerre, 2018). For noncollinear orders, one first rotates to local spin frames and then performs the same expansion (Li et al., 2017).

When comparison to inelastic neutron scattering is required, the theory must also produce the neutron cross section. In Cu$\gamma_{\mathbf k}$2TeO$\gamma_{\mathbf k}$3, the measured intensity is compared with

$\gamma_{\mathbf k}$4

where

$\gamma_{\mathbf k}$5

These expressions are not topological invariants, but they are the quantitative link between symmetry-based magnon band theory and spectroscopic evidence (Bao et al., 2017).

Beyond linear theory, several many-body frameworks have been developed. For honeycomb Dirac magnons with DMI, a spinor Green’s-function formalism yields a Dyson equation up to second order,

$\gamma_{\mathbf k}$6

capturing static self-energy renormalization, momentum-dependent band reshaping, and damping (Sun et al., 2022). A separate pumping-based treatment of a honeycomb ferromagnet constructs a bosonic analog of the Cooper ladder for opposite-momentum Dirac magnons, with a pump-induced pairing scale

$\gamma_{\mathbf k}$7

at the Dirac resonance $\gamma_{\mathbf k}$8 (Zyuzin, 2020).

4. Material realizations and spectroscopic evidence

Experimental realizations now span two-dimensional, quasi-two-dimensional, and three-dimensional magnets. The table summarizes representative systems discussed in the literature.

System	Magnetic setting	Salient Dirac-magnon phenomenology
Cu$\gamma_{\mathbf k}$9TeO$\gamma_{\mathbf K}=\gamma_{\mathbf K'}=0$0 (Bao et al., 2017)	3D collinear antiferromagnet with $\gamma_{\mathbf K}=\gamma_{\mathbf K'}=0$1 symmetry	Dirac nodes at $\gamma_{\mathbf K}=\gamma_{\mathbf K'}=0$2 and coexistence with triply degenerate magnons
CoTiO$\gamma_{\mathbf K}=\gamma_{\mathbf K'}=0$3 (Yuan et al., 2019)	Stacked honeycomb quantum $\gamma_{\mathbf K}=\gamma_{\mathbf K'}=0$4 magnet	Gapless Dirac cone at $\gamma_{\mathbf K}=\gamma_{\mathbf K'}=0$5 near $\gamma_{\mathbf K}=\gamma_{\mathbf K'}=0$6 meV
CrCl$\gamma_{\mathbf K}=\gamma_{\mathbf K'}=0$7 (Chen et al., 2021)	Quasi-2D honeycomb ferromagnet	Massless Dirac magnons; phase interpreted as not topological
CrBr$\gamma_{\mathbf K}=\gamma_{\mathbf K'}=0$8 (Nikitin et al., 2022)	Honeycomb ferromagnet	Ideal Dirac crossing at $\gamma_{\mathbf K}=\gamma_{\mathbf K'}=0$9, isospin winding, $K$0 thermal renormalization
CrI$K$1 (Yao et al., 7 May 2026)	Quasi-2D honeycomb ferromagnet	Gapped Dirac magnons near $K$2 with winding feature around the gap
Gd (Scheie et al., 2021)	hcp ferromagnet	Dirac magnons at $K$3, nodal lines along $K$4-$K$5-$K$6, nodal planes at half-integer $K$7

Cu$K$8TeO$K$9 established the three-dimensional antiferromagnetic case experimentally. Inelastic neutron scattering at $K'$0 K identified a Dirac node at $K'$1 near $K'$2 meV and triply degenerate nodes at $K'$3 near $K'$4 meV and $K'$5 meV, with the spectrum well described by linear spin-wave theory based on the fitted $K'$6-$K'$7 model (Bao et al., 2017). The paper is explicit that what is directly measured is the magnon dispersion and the nodal crossings in $K'$8 space; Berry curvature, Chern numbers, and surface arcs are inferred theoretically rather than observed directly.

CoTiO$K'$9 provided a distinct route: a stacked honeycomb lattice magnet with strong easy-plane exchange anisotropy and effective pseudospin-$$M_{\mathbf k}= \begin{pmatrix} 3JS-2DS\,\beta_{\mathbf k} & -JS\,\gamma_{\mathbf k}\ -JS\,\gamma_{\mathbf k}^* & 3JS+2DS\,\beta_{\mathbf k} \end{pmatrix},$$0 physics. Its low-energy magnons exhibit a gapless Dirac cone at the honeycomb $$M_{\mathbf k}= \begin{pmatrix} 3JS-2DS\,\beta_{\mathbf k} & -JS\,\gamma_{\mathbf k}\ -JS\,\gamma_{\mathbf k}^* & 3JS+2DS\,\beta_{\mathbf k} \end{pmatrix},$$1 points, within an overall magnon bandwidth of approximately $$M_{\mathbf k}= \begin{pmatrix} 3JS-2DS\,\beta_{\mathbf k} & -JS\,\gamma_{\mathbf k}\ -JS\,\gamma_{\mathbf k}^* & 3JS+2DS\,\beta_{\mathbf k} \end{pmatrix},$$2 meV, and the fitted nearest-neighbor intralayer exchange is almost purely $$M_{\mathbf k}= \begin{pmatrix} 3JS-2DS\,\beta_{\mathbf k} & -JS\,\gamma_{\mathbf k}\ -JS\,\gamma_{\mathbf k}^* & 3JS+2DS\,\beta_{\mathbf k} \end{pmatrix},$$3,

$$M_{\mathbf k}= \begin{pmatrix} 3JS-2DS\,\beta_{\mathbf k} & -JS\,\gamma_{\mathbf k}\ -JS\,\gamma_{\mathbf k}^* & 3JS+2DS\,\beta_{\mathbf k} \end{pmatrix},$$4

with smaller interlayer couplings of about $$M_{\mathbf k}= \begin{pmatrix} 3JS-2DS\,\beta_{\mathbf k} & -JS\,\gamma_{\mathbf k}\ -JS\,\gamma_{\mathbf k}^* & 3JS+2DS\,\beta_{\mathbf k} \end{pmatrix},$$5 meV (Yuan et al., 2019).

The chromium trihalides separate the massless and gapped regimes. CrCl$$M_{\mathbf k}= \begin{pmatrix} 3JS-2DS\,\beta_{\mathbf k} & -JS\,\gamma_{\mathbf k}\ -JS\,\gamma_{\mathbf k}^* & 3JS+2DS\,\beta_{\mathbf k} \end{pmatrix},$$6 is described by a nearly two-dimensional Heisenberg model with fitted in-plane exchanges

$$M_{\mathbf k}= \begin{pmatrix} 3JS-2DS\,\beta_{\mathbf k} & -JS\,\gamma_{\mathbf k}\ -JS\,\gamma_{\mathbf k}^* & 3JS+2DS\,\beta_{\mathbf k} \end{pmatrix},$$7

while the estimated interlayer coupling and anisotropy,

$$M_{\mathbf k}= \begin{pmatrix} 3JS-2DS\,\beta_{\mathbf k} & -JS\,\gamma_{\mathbf k}\ -JS\,\gamma_{\mathbf k}^* & 3JS+2DS\,\beta_{\mathbf k} \end{pmatrix},$$8

are too small to matter at the Dirac scale. The Dirac crossing is inferred at $$M_{\mathbf k}= \begin{pmatrix} 3JS-2DS\,\beta_{\mathbf k} & -JS\,\gamma_{\mathbf k}\ -JS\,\gamma_{\mathbf k}^* & 3JS+2DS\,\beta_{\mathbf k} \end{pmatrix},$$9 and $K'$00 meV, with no observable gap and no anomalous broadening; the phase is therefore interpreted as massless and not topological (Chen et al., 2021). CrBr$K'$01, by contrast, shows an ideal Dirac crossing at $K'$02 at

$K'$03

together with the neutron-scattering signature of isospin winding,

$K'$04

around the Dirac point (Nikitin et al., 2022).

CrI$K'$05 realizes the gapped case. A recent single-crystal study resolved a Dirac magnon gap near $K'$06 centered around $K'$07 meV with size

$K'$08

and, crucially, observed a momentum-space winding of neutron intensity around $K'$09: spectral weight lies outside the first Brillouin zone above the gap and inside it below the gap, with angular cuts fit by phase-shifted cosine functions (Yao et al., 7 May 2026). In structurally disordered CrI$K'$10, neutron scattering on a mixed rhombohedral/monoclinic sample showed that antiferromagnetic and ferromagnetic stackings can coexist, with the monoclinic antiferromagnetic component shifting magnon energies downward by just under $K'$11 meV while leaving a gap at the Dirac-point energy; the paper therefore argued cautiously that the nontrivial magnon topology of rhombohedral CrI$K'$12 may persist in the monoclinic phase as well (Schneeloch et al., 2023).

A recurring experimental issue is momentum resolution. In CrBr$K'$13, broad in-plane integration windows in time-of-flight analysis can generate an apparent splitting at $K'$14, whereas narrow integration windows recover the continuous Dirac crossing (Nikitin et al., 2022). In CrCl$K'$15, the evidence is powder averaged and therefore identifies the Dirac crossing through density-of-states fingerprints rather than a directly imaged cone (Chen et al., 2021). These cases illustrate that the phrase “Dirac magnon” can refer either to a fully momentum-resolved cone or to a more indirect but model-consistent spectroscopic signature.

5. Interactions, pumping, and finite-temperature evolution

A central issue is the fate of the Dirac cone under magnon–magnon interactions. In ferromagnetic honeycomb layers, extending the theory beyond standard Dyson theory produces strongly momentum-dependent renormalization of the bare magnon bands and strongly momentum-dependent lifetimes, with qualitative agreement to unexplained anomalies in old neutron data for CrBr$K'$16 (Pershoguba et al., 2017). A later experimental reinvestigation of CrBr$K'$17 found that both the thermal band renormalization and the magnon linewidth follow the expected universal $K'$18 evolution, with fitted exponents

$K'$19

although the measured momentum dependence lacks the sharp van Hove features predicted by the available low-order interacting spin-wave theory (Nikitin et al., 2022). In CrI$K'$20, the temperature evolution of several magnon modes likewise fits

$K'$21

with exponents $K'$22, $K'$23, and $K'$24, consistent with $K'$25-type renormalization from magnon–magnon interactions (Yao et al., 7 May 2026).

More elaborate many-body effects arise when DMI and nonequilibrium populations are present. For topological honeycomb Dirac magnons, the first-order self-energy can renormalize the Dirac velocity-like term, shift the Dirac points in momentum space through an interaction-generated field $K'$26, and modify the effective Haldane mass through an interaction-induced term $K'$27. The renormalized spectrum takes the form

$K'$28

so interactions reshape not only energies but also Berry curvature and, under strong pumping, can even generate topological flat bands and Chern-number inversion (Sun et al., 2022).

Pumping opens a separate nonequilibrium route. A theory of a uniformly pumped honeycomb ferromagnet showed that a direct one-photon process cannot create a single Dirac magnon because the pump carries zero crystal momentum, but a second-order Suhl process can create pairs of opposite-momentum Dirac magnons. The resonant frequency is the Dirac-point energy,

$K'$29

and the resulting instability was interpreted as formation of a Dirac magnon paired state with zero or reduced magnetization, with CrBr$K'$30 and CrCl$K'$31 proposed as candidate materials near the Curie temperature (Zyuzin, 2020). This does not describe an equilibrium condensate; it is a pump-induced instability of a driven bosonic Dirac system.

A particularly important current controversy concerns thermally driven topological phase transitions in gapped honeycomb Dirac magnons. A Hartree-based treatment predicted that, in a 2D honeycomb ferromagnet with DMI and Zeeman field, increasing temperature or field can close and reopen the Dirac gap at a critical temperature $K'$32, reverse the Chern numbers, and induce a sign change of the thermal Hall conductivity $K'$33 (Lu et al., 2021). A later microscopic theory based on nonlinear spin-wave theory with T-matrix resummation reached the opposite conclusion for CrI$K'$34, CrSiTe$K'$35, and CrGeTe$K'$36: it found a thermally induced reduction of the topological magnon gap but no evidence of thermally driven topological transitions, and it identified two-magnon bound-state resonances as an important source of low-temperature damping (Eto et al., 17 Sep 2025). This suggests that the thermal fate of topological Dirac magnons is approximation-sensitive and that linewidth broadening should not be conflated with gap closing.

6. Topological responses, emergent gauge fields, and open directions

Once a gap opens at a Dirac point, the magnon bands acquire Berry curvature and may support Chern-band responses. In Cu$K'$37TeO$K'$38, a magnetic field is expected to split a Dirac point into two Weyl points with monopole charges $K'$39 and $K'$40, leading to thermal Hall conductivity; similarly, gapping the triply degenerate nodes is expected to produce thermal Hall contributions from bands carrying Chern numbers $K'$41 and $K'$42 (Bao et al., 2017). In honeycomb ferromagnets with DMI, the thermal Hall conductivity is commonly written as

$K'$43

and under interaction-driven band reshaping its sign can be tuned by the pumped magnon population (Sun et al., 2022).

Several works extend the response theory beyond thermal Hall transport. In the honeycomb Dirac-magnon continuum coupled to emergent gauge fields, the effective Euclidean Lagrangian

$K'$44

yields an induced current

$K'$45

and in the DC long-wavelength limit the transverse spin conductivity becomes

$K'$46

a magnonic analog of the quantum Hall effect (Fernández et al., 15 Dec 2025). In the optical limit, the same theory predicts a sharp resonance when

$K'$47

interpreted as an interband transition of gapped Dirac magnons (Fernández et al., 15 Dec 2025).

Elastic degrees of freedom provide another route. In a honeycomb ferromagnet, strain modulates the exchange and generates elastic gauge fields

$K'$48

which couple to the Dirac magnons like valley-odd pseudogauge fields. For deformations producing constant pseudomagnetic fields, the Dirac-magnon spectrum splits into pseudo-Landau levels

$K'$49

When DMI opens a topological gap, integrating out the thermally occupied Dirac magnons generates a Chern–Simons term for the elastic gauge field and a phonon Hall viscosity

$K'$50

which vanishes at $K'$51 and grows with temperature (Ferreiros et al., 2017).

Surface and boundary manifestations remain an active theme. In Gd, linear spin-wave theory on finite slabs yields a surface magnon mode associated with the nodal lines, while the neutron intensity around the Dirac cone exhibits a sinusoidal angular modulation that reverses above and below the crossing, reflecting the momentum-dependent internal phase of the magnon eigenvectors (Scheie et al., 2021). In CrBr$K'$52 and CrI$K'$53, the experimentally resolved intensity winding around $K'$54 similarly probes the pseudospin texture of the Dirac magnons rather than only the eigenvalues (Nikitin et al., 2022, Yao et al., 7 May 2026). A plausible implication is that future progress will depend increasingly on probes sensitive to eigenvectors, not only dispersions.

Open directions stated across the literature are relatively consistent. They include direct detection of surface arc states in three-dimensional topological magnon materials, quantitative tests of field-induced Weyl-magnon scenarios, controlled separation of thermal broadening from genuine topological gap evolution, and exploitation of stacking, strain, pumping, or gauge fields as external knobs for band engineering (Bao et al., 2017, Schneeloch et al., 2023, Eto et al., 17 Sep 2025, Fernández et al., 15 Dec 2025). The broader record already makes one point clear: Dirac magnons are not a single phenomenon but a family of symmetry-protected bosonic nodal excitations whose manifestations range from graphene-like honeycomb cones to $K'$55-protected fourfold nodes, nodal loops, and gauge-responsive topological magnon bands.