Nonreciprocal Magnons

• Nonreciprocal magnons are spin-wave excitations with broken inversion and time-reversal symmetry, resulting in asymmetric energy dispersion.
• Microscopic mechanisms such as Dzyaloshinskii–Moriya interaction, anisotropic exchange, and dipolar fields drive their chiral propagation and nonreciprocal damping.
• These phenomena enable practical applications like magnonic diodes and isolators, and support hybrid quantum systems by tailoring directional lifetimes and coupling strengths.

Nonreciprocal magnons are spin-wave excitations for which reciprocity is broken, so that the magnon energy at a given wavevector differs from that at the opposite wavevector, $\omega(\mathbf q)\neq \omega(-\mathbf q)$. In the literature, the same term also encompasses direction-dependent group velocity, linewidth, lifetime, transmission, and hybridization, including cases with $S_{21}(\omega)\neq S_{12}(\omega)$ or asymmetric magnon blockade and entanglement. The phenomenon therefore denotes a family of chiral magnon responses rather than a single microscopic mechanism, spanning bulk noncentrosymmetric magnets, antiferromagnets, metallic two-dimensional crystals, domain-wall waveguides, ferromagnet/superconductor hybrids, cavity magnonics, and magnomechanical platforms (Costa et al., 2020, Gitgeatpong et al., 2017, Zhang et al., 2020, Dobrovolskiy et al., 2021).

1. Definition, reciprocity, and symmetry constraints

The minimal reciprocity condition for a magnon branch is $\omega(\mathbf q)=\omega(-\mathbf q)$. A standard measure of asymmetry is

$$\Delta\omega(\mathbf q)\equiv \omega(\mathbf q)-\omega(-\mathbf q),$$

which is odd in $\mathbf q$. The same asymmetry can be expressed through the group velocity $$\mathbf v(\mathbf q)=\nabla_{\mathbf q}\omega(\mathbf q)$$, for which nonreciprocity implies $\mathbf v(\mathbf q)\neq -\mathbf v(-\mathbf q)$. In transport-oriented settings, the relevant observable may instead be a directional scattering or transmission asymmetry, such as $S_{21}\neq S_{12}$, or a direction-dependent second-order coherence $g^{(2)}(0)$ (Costa et al., 2020, Zhang et al., 2020, Wang et al., 2021).

At the symmetry level, nonreciprocity generally requires the absence of inversion symmetry and magnetic time-reversal breaking, or an equivalent loss of the dynamical symmetries that would map $+\mathbf q$ to $S_{21}(\omega)\neq S_{12}(\omega)$0. In the noncentrosymmetric antiferromagnet $S_{21}(\omega)\neq S_{12}(\omega)$1-Cu$S_{21}(\omega)\neq S_{12}(\omega)$2V$S_{21}(\omega)\neq S_{12}(\omega)$3O$S_{21}(\omega)\neq S_{12}(\omega)$4, the crystal structure breaks $S_{21}(\omega)\neq S_{12}(\omega)$5, antiferromagnetic order breaks $S_{21}(\omega)\neq S_{12}(\omega)$6, and the studied state also breaks combined $S_{21}(\omega)\neq S_{12}(\omega)$7, enabling finite-$S_{21}(\omega)\neq S_{12}(\omega)$8 magnon minima and field-tunable asymmetry (Gitgeatpong et al., 2017). By contrast, in MnPS$S_{21}(\omega)\neq S_{12}(\omega)$9 the centrosymmetric $\omega(\mathbf q)=\omega(-\mathbf q)$0 lattice and collinear Néel order preserve effective constraints that enforce reciprocal spectra to experimental accuracy, so that no significant nonreciprocity was resolved at opposite Brillouin-zone corners (Wildes et al., 2020).

A more microscopic symmetry language has been formulated in terms of magnetic toroidal multipoles. One bond-level criterion is the activation of a bond magnetic toroidal dipole through a finite parallel component between the Dzyaloshinskii–Moriya vector and the averaged spin moments at the bond ends. In that formulation, a bond magnetic toroidal dipole is active when $\omega(\mathbf q)=\omega(-\mathbf q)$1, and higher-rank cluster toroidal multipoles then organize the angular dependence of asymmetric dispersions, including band-bottom shifts and valley splitting (Matsumoto et al., 2021).

These symmetry statements also clarify a frequent ambiguity: “nonreciprocal magnons” may refer to intrinsic band asymmetry, but it may also denote asymmetric coupling, damping, detection, or hybridization. In cavity magnonics the defining object can be the scattering matrix rather than $\omega(\mathbf q)=\omega(-\mathbf q)$2, while in thermally generated transport the asymmetry can arise from direction-dependent coupling to a third magnetic terminal even when fundamental four-terminal reciprocity remains intact under full reversal of the time-reversal-breaking fields (Zhang et al., 2020, Cosset-Chéneau et al., 2024).

2. Microscopic mechanisms and theoretical descriptions

The most common microscopic starting point is a spin Hamiltonian containing isotropic exchange, antisymmetric Dzyaloshinskii–Moriya interaction, and anisotropy,

$\omega(\mathbf q)=\omega(-\mathbf q)$3

or closely related multisublattice generalizations. In ultrathin interfacial systems, linear spin-wave theory often yields

$\omega(\mathbf q)=\omega(-\mathbf q)$4

so that the odd-in-$\omega(\mathbf q)=\omega(-\mathbf q)$5 term directly encodes chiral propagation. In the two-dimensional metallic ferromagnet Fe$\omega(\mathbf q)=\omega(-\mathbf q)$6GeTe$\omega(\mathbf q)=\omega(-\mathbf q)$7, however, the odd component is strongly path dependent: the acoustic mode near $\omega(\mathbf q)=\omega(-\mathbf q)$8 fits almost perfectly to $\omega(\mathbf q)=\omega(-\mathbf q)$9 with $$\Delta\omega(\mathbf q)\equiv \omega(\mathbf q)-\omega(-\mathbf q),$$0, while clear nonreciprocity appears only at finite $$\Delta\omega(\mathbf q)\equiv \omega(\mathbf q)-\omega(-\mathbf q),$$1 along $$\Delta\omega(\mathbf q)\equiv \omega(\mathbf q)-\omega(-\mathbf q),$$2, not along $$\Delta\omega(\mathbf q)\equiv \omega(\mathbf q)-\omega(-\mathbf q),$$3 (Costa et al., 2020).

Dzyaloshinskii–Moriya interaction is not the only route. Matsumoto and Hayami showed that a bond-dependent symmetric anisotropic exchange on the honeycomb lattice produces valley-type nonreciprocal magnons under staggered antiferromagnetic order, even in the absence of nearest-neighbor DM interaction. In their formulation, the nonreciprocal direction is manipulable by an in-plane rotating magnetic field, and the effect is accounted for by magnetic toroidal multipoles (Matsumoto et al., 2020). A broader microscopic synthesis later established that odd orders of an effective antisymmetric DM interaction and even orders of an effective symmetric anisotropic interaction in the spin-rotated frame can both generate antisymmetric dispersions, and that products of the Bogoliubov Hamiltonian suffice to identify the momentum dependence and essential couplings without explicitly solving the eigenvalue problem (Hayami et al., 2021).

In itinerant magnets the relevant theory is not a pure spin model but the transverse spin susceptibility,

$$\Delta\omega(\mathbf q)\equiv \omega(\mathbf q)-\omega(-\mathbf q),$$4

with spin-flip spectral density $$\Delta\omega(\mathbf q)\equiv \omega(\mathbf q)-\omega(-\mathbf q),$$5. In Fe$$\Delta\omega(\mathbf q)\equiv \omega(\mathbf q)-\omega(-\mathbf q),$$6GeTe$$\Delta\omega(\mathbf q)\equiv \omega(\mathbf q)-\omega(-\mathbf q),$$7, $$\Delta\omega(\mathbf q)\equiv \omega(\mathbf q)-\omega(-\mathbf q),$$8 is computed within RPA from a PAO tight-binding Hamiltonian derived from DFT+SOC,

$$\Delta\omega(\mathbf q)\equiv \omega(\mathbf q)-\omega(-\mathbf q),$$9

so that collective magnon poles and the Stoner continuum are treated self-consistently. This framework predicts not only nonreciprocal energies but also nonreciprocal damping and lifetimes because $\mathbf q$0 is itself chiral (Costa et al., 2020).

Dipolar mechanisms form another major class. In Bloch-like domain walls in perpendicular-anisotropy Fe/Gd multilayers, the nonreciprocity is not attributed to interfacial DMI but to dynamic dipolar fields of the wall-confined Winter mode. In surface magnetoelastic systems, Rayleigh surface acoustic waves carry rotation–momentum locking, and precessing nanomagnets exert rotating edge forces that couple more strongly to one sign of the surface-phonon wavevector than to the other. The resulting coupling constant contains an explicit $\mathbf q$1 term, so that $\mathbf q$2 and complete chirality is obtained at a material-dependent critical angle $\mathbf q$3 (Che et al., 2023, Yu, 2020).

Driven hybrid systems realize nonreciprocity through spatiotemporal modulation or reservoir engineering rather than static band asymmetry. In ferromagnet/superconductor hybrids, a moving Abrikosov vortex lattice acts as a time-dependent magnetic grating, and the magnon bandgaps shift by

$\mathbf q$4

with the sign set by the relative direction of vortex drift and spin-wave propagation. In a magnon-based hybrid quantum system, a coherent Jaynes–Cummings coupling and a phase-sensitive dissipative coupling combine to yield an effective non-Hermitian Hamiltonian

$\mathbf q$5

so that the linewidths of the dressed ladders become direction dependent and the resulting magnon blockade is nonreciprocal (Dobrovolskiy et al., 2021, Wang et al., 2021).

3. Representative material platforms and spectral phenomenology

System	Dominant mechanism	Characteristic signature
Fe$\mathbf q$6GeTe$\mathbf q$7 monolayer (Costa et al., 2020)	SOC-enabled intrinsic DMI in a stand-alone metallic 2D crystal	Nonreciprocity along $\mathbf q$8, $\mathbf q$9 meV, acoustic lifetime $$\mathbf v(\mathbf q)=\nabla_{\mathbf q}\omega(\mathbf q)$$0 ps
$$\mathbf v(\mathbf q)=\nabla_{\mathbf q}\omega(\mathbf q)$$1-Cu$$\mathbf v(\mathbf q)=\nabla_{\mathbf q}\omega(\mathbf q)$$2V$$\mathbf v(\mathbf q)=\nabla_{\mathbf q}\omega(\mathbf q)$$3O$$\mathbf v(\mathbf q)=\nabla_{\mathbf q}\omega(\mathbf q)$$4 (Gitgeatpong et al., 2017)	Competition of anisotropic exchange and DM in a noncentrosymmetric antiferromagnet	Minima at $$\mathbf v(\mathbf q)=\nabla_{\mathbf q}\omega(\mathbf q)$$5, gap $$\mathbf v(\mathbf q)=\nabla_{\mathbf q}\omega(\mathbf q)$$6 meV
MnSi (Weber et al., 2018)	Bulk chiral DM interaction in $$\mathbf v(\mathbf q)=\nabla_{\mathbf q}\omega(\mathbf q)$$7	Full asymmetry of $$\mathbf v(\mathbf q)=\nabla_{\mathbf q}\omega(\mathbf q)$$8 across helical, conical, skyrmion, and field-polarized phases
LiFe$$\mathbf v(\mathbf q)=\nabla_{\mathbf q}\omega(\mathbf q)$$9O$\mathbf v(\mathbf q)\neq -\mathbf v(-\mathbf q)$0 (Iguchi et al., 2015)	Bulk DMI in a noncentrosymmetric ferrimagnet	Nonreciprocal microwave response near $\mathbf v(\mathbf q)\neq -\mathbf v(-\mathbf q)$1 GHz, reversed by magnetization reversal
Fe/Gd domain walls (Che et al., 2023)	Dynamic dipolar nonreciprocity of Bloch-like wall modes	$\mathbf v(\mathbf q)\neq -\mathbf v(-\mathbf q)$2 nm near $\mathbf v(\mathbf q)\neq -\mathbf v(-\mathbf q)$3 GHz, $\mathbf v(\mathbf q)\neq -\mathbf v(-\mathbf q)$4 nm
VPX$\mathbf v(\mathbf q)\neq -\mathbf v(-\mathbf q)$5 ($\mathbf v(\mathbf q)\neq -\mathbf v(-\mathbf q)$6) (Du et al., 8 Sep 2025)	2NN DMI, interlayer coupling, and magnon–magnon interactions	$\mathbf v(\mathbf q)\neq -\mathbf v(-\mathbf q)$7, $\mathbf v(\mathbf q)\neq -\mathbf v(-\mathbf q)$8, and $\mathbf v(\mathbf q)\neq -\mathbf v(-\mathbf q)$9 meV for VPS$S_{21}\neq S_{12}$0, VPSe$S_{21}\neq S_{12}$1, and VPTe$S_{21}\neq S_{12}$2

Fe$S_{21}\neq S_{12}$3GeTe$S_{21}\neq S_{12}$4 is notable because the nonreciprocity is intrinsic to a stand-alone metallic two-dimensional crystal with out-of-plane magnetization rather than interfacial in-plane magnetization. The itinerant calculation yields an acoustic magnon with anisotropy gap $S_{21}\neq S_{12}$5 meV and bandwidth $S_{21}\neq S_{12}$6 meV, a broad non-bonding magnon around $S_{21}\neq S_{12}$7 meV, and a high-energy incoherent feature at $S_{21}\neq S_{12}$8 meV. The acoustic lifetime reaches “hundreds of picoseconds” in the small-$S_{21}\neq S_{12}$9 regime, and the lifetime itself is nonreciprocal, with magnons around $g^{(2)}(0)$0 predicted to have both higher energies and longer lifetimes than those around $g^{(2)}(0)$1 (Costa et al., 2020).

In $g^{(2)}(0)$2-Cu$g^{(2)}(0)$3V$g^{(2)}(0)$4O$g^{(2)}(0)$5, the magnon minimum is shifted away from the static ordering vector because the anisotropic exchange $g^{(2)}(0)$6 stabilizes a collinear antiferromagnet while the uniform DM component $g^{(2)}(0)$7 favors a helical structure. The measured minima lie at $g^{(2)}(0)$8 and $g^{(2)}(0)$9, the gap is $+\mathbf q$0 meV, and the field dependence is linear with opposite slopes at the two $+\mathbf q$1-points, closing at $+\mathbf q$2 T (Gitgeatpong et al., 2017). MnSi presents a different phenomenology: the DM-shifted parabolas evolve through helical, conical, skyrmion-lattice, and field-polarized phases, with $+\mathbf q$3, $+\mathbf q$4, and $+\mathbf q$5 (Weber et al., 2018).

Recent antiferromagnetic van der Waals predictions place VPX$+\mathbf q$6 among the clearest layered candidates. In the monolayer limit, the odd-in-$+\mathbf q$7 term arises from 2NN DMI in an easy-axis honeycomb antiferromagnet, and the valley asymmetry obeys

$+\mathbf q$8

so that the nonreciprocity depends asymmetrically and periodically on the Néel-vector orientation. The effect is large in VPTe$+\mathbf q$9 and VPSe$S_{21}(\omega)\neq S_{12}(\omega)$00, but it is strongly layer dependent: an AFM-coupled bilayer restores inversion and becomes reciprocal, whereas odd-layer stacks remain nonreciprocal (Du et al., 8 Sep 2025).

At the nanoscale, Fe/Gd multilayers demonstrate that confined nonreciprocal magnons need not rely on DM interaction. Scanning transmission x-ray microscopy detected coherent domain-wall-guided magnons near $S_{21}(\omega)\neq S_{12}(\omega)$01 GHz with wavelengths down to $S_{21}(\omega)\neq S_{12}(\omega)$02 nm inside walls as narrow as $S_{21}(\omega)\neq S_{12}(\omega)$03 nm. Bloch points were identified as topological defects that disrupt phase evolution and generate different adjacent wavelengths without removing the underlying wall-mode nonreciprocity (Che et al., 2023).

Not all proposed materials exhibit a measurable effect. A careful neutron three-axis study of MnPS$S_{21}(\omega)\neq S_{12}(\omega)$04 found no significant difference between opposite Brillouin-zone corners within the experimental sensitivity. The apparent energy differences were $S_{21}(\omega)\neq S_{12}(\omega)$05 meV and within the $S_{21}(\omega)\neq S_{12}(\omega)$06 meV instrumental resolution, leading to the conclusion that any DMI responsible for $S_{21}(\omega)\neq S_{12}(\omega)$07 must be much smaller than $S_{21}(\omega)\neq S_{12}(\omega)$08 meV (Wildes et al., 2020).

4. Experimental probes and evidentiary standards

Inelastic neutron scattering has provided the clearest direct access to spectral nonreciprocity in bulk magnets. In $S_{21}(\omega)\neq S_{12}(\omega)$09-Cu$S_{21}(\omega)\neq S_{12}(\omega)$10V$S_{21}(\omega)\neq S_{12}(\omega)$11O$S_{21}(\omega)\neq S_{12}(\omega)$12, single-crystal neutron measurements established the finite-$S_{21}(\omega)\neq S_{12}(\omega)$13 minima, the field-linear gap asymmetry, and an explicit test of detailed balance in a state with broken $S_{21}(\omega)\neq S_{12}(\omega)$14 and $S_{21}(\omega)\neq S_{12}(\omega)$15 (Gitgeatpong et al., 2017). In MnSi, cold-neutron triple-axis spectroscopy fully mapped the field-dependent evolution of the nonreciprocal dynamical structure factor in all ordered phases, showing that flipping $S_{21}(\omega)\neq S_{12}(\omega)$16, $S_{21}(\omega)\neq S_{12}(\omega)$17, or $S_{21}(\omega)\neq S_{12}(\omega)$18 individually changes the spectrum, whereas simultaneous reversal of two variables restores it (Weber et al., 2018).

Because false positives are possible when opposite wavevectors are measured under different resolution conditions, negative results have played an important methodological role. In MnPS$S_{21}(\omega)\neq S_{12}(\omega)$19, the sample was rotated by $S_{21}(\omega)\neq S_{12}(\omega)$20 to access corresponding $S_{21}(\omega)\neq S_{12}(\omega)$21 and $S_{21}(\omega)\neq S_{12}(\omega)$22 positions with effectively identical instrumental settings, and the analysis explicitly convolved the dynamical structure factor with the experimental resolution function. The null result, together with the bound $S_{21}(\omega)\neq S_{12}(\omega)$23 meV, illustrates that theoretical proposals for nonreciprocity must be separated from experimentally resolvable band asymmetry (Wildes et al., 2020).

Optical and x-ray probes are especially powerful in confined geometries. Brillouin light scattering on permalloy thin films on oxide/Si substrates showed that hybridization between magnetostatic surface spin waves and the first perpendicular standing spin wave transfers surface localization and direction selectivity between branches. In the 44 nm film, only the dipole-dominated branch was detected on anti-Stokes and only the exchange-dominated branch on Stokes beyond the anticrossing, and reversing the in-plane field exchanged the two patterns (Song et al., 2020). Time-resolved STXM at the Gd $S_{21}(\omega)\neq S_{12}(\omega)$24 edge then extended direct imaging to domain-wall channels, with $S_{21}(\omega)\neq S_{12}(\omega)$25 nm spatial and $S_{21}(\omega)\neq S_{12}(\omega)$26 ps temporal resolution, making it possible to reconstruct phase maps, wavelengths, and propagation directions of individual wall-confined modes (Che et al., 2023).

Microwave transmission and nonlocal electrical detection broaden the experimental notion of nonreciprocity. In LiFe$S_{21}(\omega)\neq S_{12}(\omega)$27O$S_{21}(\omega)\neq S_{12}(\omega)$28, lithographic meander antennae with $S_{21}(\omega)\neq S_{12}(\omega)$29 excited exchange-regime magnons with $S_{21}(\omega)\neq S_{12}(\omega)$30, and the measured $S_{21}(\omega)\neq S_{12}(\omega)$31 and $S_{21}(\omega)\neq S_{12}(\omega)$32 differed near $S_{21}(\omega)\neq S_{12}(\omega)$33 GHz at $S_{21}(\omega)\neq S_{12}(\omega)$34, reversing upon magnetization reversal. Centrosymmetric YIG, measured in the same geometry, remained reciprocal at $S_{21}(\omega)\neq S_{12}(\omega)$35, isolating the bulk DMI contribution in LiFe$S_{21}(\omega)\neq S_{12}(\omega)$36O$S_{21}(\omega)\neq S_{12}(\omega)$37 (Iguchi et al., 2015). In ultrathin YIG with an intermediate Py wire, nonlocal first- and second-harmonic signals demonstrated nonreciprocal transport of electrically and thermally generated incoherent magnons. The normalized second-harmonic contrast reached approximately $S_{21}(\omega)\neq S_{12}(\omega)$38, and the symmetry matched remote dipolar coupling controlled by the chirality of YIG magnon stray fields (Cosset-Chéneau et al., 2024).

Cavity systems use the same language of reciprocity but probe it through scattering rather than $S_{21}(\omega)\neq S_{12}(\omega)$39. In an irregular quadrant-stadium resonant cavity loaded with a YIG wafer, forward and backward transmission maps displayed three anticrossings involving the FMR mode and higher-order FVMSW modes. The reported isolation ratios reached $S_{21}(\omega)\neq S_{12}(\omega)$40 dB for the FMR–cavity hybridization and $S_{21}(\omega)\neq S_{12}(\omega)$41 dB and $S_{21}(\omega)\neq S_{12}(\omega)$42 dB for higher-order couplings, while reversing the magnetic field interchanged the $S_{21}(\omega)\neq S_{12}(\omega)$43 and $S_{21}(\omega)\neq S_{12}(\omega)$44 asymmetry (Zhang et al., 2020).

5. Hybrid, driven, and quantum regimes

A distinct route to nonreciprocal magnons uses driven magnetic textures as moving gratings. In Py/Nb ferromagnet/superconductor hybrids, the Abrikosov vortex lattice forms a periodic magnetic grating with spacing

$S_{21}(\omega)\neq S_{12}(\omega)$45

and Bragg gaps open at $S_{21}(\omega)\neq S_{12}(\omega)$46. When the vortex lattice moves, the bandgaps shift by $S_{21}(\omega)\neq S_{12}(\omega)$47. At $S_{21}(\omega)\neq S_{12}(\omega)$48 mT, the experiments reported $S_{21}(\omega)\neq S_{12}(\omega)$49–$S_{21}(\omega)\neq S_{12}(\omega)$50 nm, $S_{21}(\omega)\neq S_{12}(\omega)$51, tunability of approximately $S_{21}(\omega)\neq S_{12}(\omega)$52, a polarity-induced asymmetry of approximately $S_{21}(\omega)\neq S_{12}(\omega)$53 GHz at $S_{21}(\omega)\neq S_{12}(\omega)$54, and $S_{21}(\omega)\neq S_{12}(\omega)$55 ns reconfiguration under a $S_{21}(\omega)\neq S_{12}(\omega)$56 MHz ac current with power $S_{21}(\omega)\neq S_{12}(\omega)$57 nW (Dobrovolskiy et al., 2021).

Cavity magnonics realizes nonreciprocity through polarization-selective magnon–photon coupling. In the irregular resonant cavity system, the gyrotropic permeability tensor of YIG,

$S_{21}(\omega)\neq S_{12}(\omega)$58

combines with spatially asymmetric cavity polarization content so that the ports sample different chiral photon admixtures. The multimode coupling matrix then contains unequal $S_{21}(\omega)\neq S_{12}(\omega)$59 and $S_{21}(\omega)\neq S_{12}(\omega)$60, and elimination of the cavity modes yields photon-mediated indirect magnon–magnon couplings of the form

$S_{21}(\omega)\neq S_{12}(\omega)$61

This indirect coupling broadens the field window over which isolation remains high (Zhang et al., 2020).

In the quantum few-excitation regime, nonreciprocal statistics rather than linear transmission become central. In the dissipation-induced magnon blockade proposal, a YIG Kittel mode is coherently coupled to a superconducting qubit through a detuned cavity and dissipatively coupled through a waveguide. The shared reservoir generates the phase-sensitive term $S_{21}(\omega)\neq S_{12}(\omega)$62, and the resulting direction-dependent broadening turns the single-magnon blockade on for one drive direction and off for the other. For the representative parameters $S_{21}(\omega)\neq S_{12}(\omega)$63 GHz, $S_{21}(\omega)\neq S_{12}(\omega)$64 MHz, and $S_{21}(\omega)\neq S_{12}(\omega)$65 MHz, the maximum contrast was $S_{21}(\omega)\neq S_{12}(\omega)$66, with $S_{21}(\omega)\neq S_{12}(\omega)$67 and $S_{21}(\omega)\neq S_{12}(\omega)$68 at $S_{21}(\omega)\neq S_{12}(\omega)$69 MHz (Wang et al., 2021).

Magnomechanical systems add Kerr and squeezing control. In cavity-magnon optomechanics, the magnon Kerr effect produces both a frequency shift $S_{21}(\omega)\neq S_{12}(\omega)$70 and a two-magnon term $S_{21}(\omega)\neq S_{12}(\omega)$71, and changing the magnetic-field orientation from $S_{21}(\omega)\neq S_{12}(\omega)$72 to $S_{21}(\omega)\neq S_{12}(\omega)$73 reverses the sign of $S_{21}(\omega)\neq S_{12}(\omega)$74, $S_{21}(\omega)\neq S_{12}(\omega)$75, and $S_{21}(\omega)\neq S_{12}(\omega)$76. This yields nonreciprocal bipartite and tripartite entanglement, quantified by bidirectional contrast ratios and tunable with bath temperature (Chen et al., 2023). A related two-cavity magnomechanical proposal used the sign change of the self-Kerr coefficient between $S_{21}(\omega)\neq S_{12}(\omega)$77 and $S_{21}(\omega)\neq S_{12}(\omega)$78 to define $S_{21}(\omega)\neq S_{12}(\omega)$79 and $S_{21}(\omega)\neq S_{12}(\omega)$80 configurations, with different optimal detunings and different effective couplings $S_{21}(\omega)\neq S_{12}(\omega)$81 versus $S_{21}(\omega)\neq S_{12}(\omega)$82 (Ahmed et al., 2024). An alternative squeezed-magnon scheme showed that the phase-tunable quantities $S_{21}(\omega)\neq S_{12}(\omega)$83 and $S_{21}(\omega)\neq S_{12}(\omega)$84 can yield ideal nonreciprocity of entanglement, $S_{21}(\omega)\neq S_{12}(\omega)$85 or $S_{21}(\omega)\neq S_{12}(\omega)$86, in experimentally feasible parameter regimes (Imara et al., 9 Aug 2025). In a spinning WGM cavity coupled to two YIG spheres, the Sagnac–Fizeau shift

$S_{21}(\omega)\neq S_{12}(\omega)$87

combines with magnon Kerr nonlinearity to produce nonreciprocal magnon–magnon entanglement that remains robust up to about $S_{21}(\omega)\neq S_{12}(\omega)$88 mK (Xu et al., 14 May 2026).

6. Applications, limitations, and recurrent misconceptions

The device implications are diverse but structurally similar. Direction-dependent $S_{21}(\omega)\neq S_{12}(\omega)$89, $S_{21}(\omega)\neq S_{12}(\omega)$90, or transmission can be used for magnonic diodes, isolators, filters, interferometers, and nonreciprocal logic elements. Fe$S_{21}(\omega)\neq S_{12}(\omega)$91GeTe$S_{21}(\omega)\neq S_{12}(\omega)$92 is attractive because it combines intrinsic chiral magnons with unusually long acoustic lifetimes in a conducting 2D ferromagnet (Costa et al., 2020). Fluxonic ferromagnet/superconductor hybrids enable electrically reconfigurable microwave filters with fast $S_{21}(\omega)\neq S_{12}(\omega)$93 ns modulation (Dobrovolskiy et al., 2021). Surface magnetoelastic dynamics predict a nearly complete phonon diode effect by several high-quality or tens of ordinary magnetic nanowires, suggesting directional transducers rather than purely magnonic components (Yu, 2020). In layered VPX$S_{21}(\omega)\neq S_{12}(\omega)$94, the asymmetric periodic dependence of the valley asymmetry on the Néel vector suggests a route to probing antiferromagnetic order parameters in the two-dimensional limit (Du et al., 8 Sep 2025).

Several open questions recur across the field. In Fe$S_{21}(\omega)\neq S_{12}(\omega)$95GeTe$S_{21}(\omega)\neq S_{12}(\omega)$96, the qualitative role of second-neighbor DMI on the B sublattice is established, but a quantitative microscopic extraction of the DMI magnitude and range remains to be refined (Costa et al., 2020). In VPX$S_{21}(\omega)\neq S_{12}(\omega)$97, the experimentally practical extraction of $S_{21}(\omega)\neq S_{12}(\omega)$98, $S_{21}(\omega)\neq S_{12}(\omega)$99, and $\omega(\mathbf q)=\omega(-\mathbf q)$00, and the effect of twist or stacking registry beyond idealized stackings, remain unresolved (Du et al., 8 Sep 2025). Hybrid platforms introduce their own constraints: the Nb-based fluxonic scheme is cryogenic and sensitive to vortex disorder near depinning (Dobrovolskiy et al., 2021), while cavity and magnomechanical schemes remain sensitive to polarization purity, mode crowding, thermal noise, and stability thresholds (Zhang et al., 2020, Chen et al., 2023).

Two misconceptions are especially persistent. First, nonreciprocity does not always mean that a bare magnon band satisfies $\omega(\mathbf q)=\omega(-\mathbf q)$01. In the NiFe thin-film hybridization experiment, the essential observation is unidirectional visibility caused by opposite surface localization and top-surface-sensitive BLS, and the authors explicitly state that frequency-level nonreciprocity is not required to explain the measurements (Song et al., 2020). In the ultrathin YIG/Py transport experiment, fixed-configuration nonreciprocity coexists with the four-terminal Landauer–Büttiker reciprocity relation once all time-reversal-breaking fields are reversed (Cosset-Chéneau et al., 2024). Second, the existence of theoretical proposals or indirect thermoelectric anomalies does not guarantee a resolvable spectral asymmetry. The MnPS$\omega(\mathbf q)=\omega(-\mathbf q)$02 null neutron result constrains any relevant DMI to be much smaller than $\omega(\mathbf q)=\omega(-\mathbf q)$03 meV and shows that proposed magnon Nernst mechanisms need not correspond to an experimentally accessible $\omega(\mathbf q)=\omega(-\mathbf q)$04 splitting in bulk material (Wildes et al., 2020).

Taken together, these results indicate that nonreciprocal magnons constitute a broad and internally differentiated class of phenomena. The same umbrella term covers intrinsic odd-in-$\omega(\mathbf q)=\omega(-\mathbf q)$05 magnon bands generated by Dzyaloshinskii–Moriya interaction or symmetric anisotropic exchange, nonreciprocal linewidths and lifetimes in itinerant magnets, dipolar asymmetries in domain walls and thermally generated transport, driven bandgaps in vortex lattices, and direction-dependent quantum correlations in cavity-based hybrids. This suggests that the most useful classification is not by material family alone, but by the physical object that becomes asymmetric: dispersion, damping, scattering matrix, pumped current, or quantum correlation function.