---
title: Nonreciprocal Magnons
url: https://www.emergentmind.com/topics/nonreciprocal-magnons
type: topic
---

# Nonreciprocal Magnons

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 [2004.14179][1702.04889][2012.14765][2107.04095].

## 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 \(g^{(2)}(0)\) [2004.14179][2012.14765][2112.02351].

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 \(-\mathbf q\). In the noncentrosymmetric antiferromagnet \(\alpha\)-Cu\(_2\)V\(_2\)O\(_7\), the crystal structure breaks \(\mathcal P\), antiferromagnetic order breaks \(\mathcal T\), and the studied state also breaks combined \(\mathcal{PT}\), enabling finite-\(k\) magnon minima and field-tunable asymmetry [1702.04889]. By contrast, in MnPS\(_3\) the centrosymmetric \(C\,2/m\) 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 [2010.02898].

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 \( \mathbf D^{(ij)}\cdot \mathbf M^{(ij)}\neq 0 \), and higher-rank cluster toroidal multipoles then organize the angular dependence of asymmetric dispersions, including band-bottom shifts and valley splitting [2107.12743].

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 \(\omega(\mathbf q)\), 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 [2012.14765][2408.15595].

## 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,
\[
H = -\sum_{\langle i,j\rangle} J_{ij}\,\mathbf S_i\cdot \mathbf S_j
+ \sum_{\langle i,j\rangle} \mathbf D_{ij}\cdot(\mathbf S_i\times \mathbf S_j)
+ \sum_i K (S_i^z)^2,
\]
or closely related multisublattice generalizations. In ultrathin interfacial systems, linear spin-wave theory often yields
\[
\omega(\mathbf q)\approx \Delta_\Gamma + A q^2 \pm D q_\parallel,
\]
so that the odd-in-\(q\) term directly encodes chiral propagation. In the two-dimensional metallic ferromagnet Fe\(_3\)GeTe\(_2\), however, the odd component is strongly path dependent: the acoustic mode near \(\Gamma\) fits almost perfectly to \( \omega_{\rm ac}(\mathbf q)\approx \Delta_\Gamma + A q^2 \) with \( \Delta_\Gamma=2.93\ {\rm meV} \), while clear nonreciprocity appears only at finite \(q\) along \(K\!-\!\Gamma\!-\!K'\), not along \(\Gamma\!-\!M\) [2004.14179].

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 [2005.13729]. 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 [2112.07071].

In itinerant magnets the relevant theory is not a pure spin model but the transverse spin susceptibility,
\[
\chi^{+-}(\mathbf q,\omega)= -i\int_0^\infty dt\, e^{i\omega t}
\langle [S^+_{\mathbf q}(t),S^-_{-\mathbf q}(0)]\rangle,
\]
with spin-flip spectral density \(A(\mathbf q,\omega)\propto {\rm Im}\,\chi^{+-}(\mathbf q,\omega)\). In Fe\(_3\)GeTe\(_2\), \( \chi^{+-} \) is computed within RPA from a PAO tight-binding Hamiltonian derived from DFT+SOC,
\[
\chi^{+-}(\mathbf q,\omega)=
[1-U\chi_0^{+-}(\mathbf q,\omega)]^{-1}\chi_0^{+-}(\mathbf q,\omega),
\]
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 \( {\rm Im}\,\chi_0^{+-}(\mathbf q,\omega) \) is itself chiral [2004.14179].

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 \({\rm sgn}(k)\) term, so that \(|g_{|k|}|\neq |g_{-|k|}|\) and complete chirality is obtained at a material-dependent critical angle \(\varphi_c\) [2311.06186][2007.12240].

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
\[
\Delta f_n \approx \frac{n G v}{2\pi},
\]
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
\[
H_{\rm eff} =
(\omega_q-i\gamma)\sigma_+\sigma_- + (\omega_b-i\kappa)b^\dagger b
+ (\lambda-i\Gamma e^{i\Theta})(\sigma^+ b + b^\dagger \sigma^-),
\]
so that the linewidths of the dressed ladders become direction dependent and the resulting magnon blockade is nonreciprocal [2107.04095][2112.02351].

## 3. Representative material platforms and spectral phenomenology

| System | Dominant mechanism | Characteristic signature |
|---|---|---|
| Fe\(_3\)GeTe\(_2\) monolayer [2004.14179] | SOC-enabled intrinsic DMI in a stand-alone metallic 2D crystal | Nonreciprocity along \(K\!-\!\Gamma\!-\!K'\), \(\Delta_\Gamma=2.93\) meV, acoustic lifetime \(\tau \approx 100\) ps |
| \(\alpha\)-Cu\(_2\)V\(_2\)O\(_7\) [1702.04889] | Competition of anisotropic exchange and DM in a noncentrosymmetric antiferromagnet | Minima at \((0,2\pm 0.25,0)\), gap \(\Delta=0.75(6)\) meV |
| MnSi [1805.08750] | Bulk chiral DM interaction in \(P2_13\) | Full asymmetry of \(S(q,E,\mu_0H_{\rm int})\) across helical, conical, skyrmion, and field-polarized phases |
| LiFe\(_5\)O\(_8\) [1505.01942] | Bulk DMI in a noncentrosymmetric ferrimagnet | Nonreciprocal microwave response near \(5.5\) GHz, reversed by magnetization reversal |
| Fe/Gd domain walls [2311.06186] | Dynamic dipolar nonreciprocity of Bloch-like wall modes | \(\lambda = 281\pm 44\) nm near \(1\) GHz, \(\delta_{\rm DW}=52\pm 10\) nm |
| VPX\(_3\) (\(X={\rm S,Se,Te}\)) [2509.06538] | 2NN DMI, interlayer coupling, and magnon–magnon interactions | \(\Delta\omega(K)\approx 0.4\), \(2.0\), and \(3.2\) meV for VPS\(_3\), VPSe\(_3\), and VPTe\(_3\) |

Fe\(_3\)GeTe\(_2\) 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 \(2.93\) meV and bandwidth \(\approx 120\) meV, a broad non-bonding magnon around \(\approx 300\) meV, and a high-energy incoherent feature at \(\gtrsim 400\) meV. The acoustic lifetime reaches “hundreds of picoseconds” in the small-\(q\) regime, and the lifetime itself is nonreciprocal, with magnons around \(K\) predicted to have both higher energies and longer lifetimes than those around \(K'\) [2004.14179].

In \(\alpha\)-Cu\(_2\)V\(_2\)O\(_7\), the magnon minimum is shifted away from the static ordering vector because the anisotropic exchange \(G_1\) stabilizes a collinear antiferromagnet while the uniform DM component \(D_{1a}\) favors a helical structure. The measured minima lie at \((0,1.75,0)\) and \((0,2.25,0)\), the gap is \(0.75(6)\) meV, and the field dependence is linear with opposite slopes at the two \(k\)-points, closing at \(\mu_0H_{c1}=\pm 6.61(2)\) T [1702.04889]. MnSi presents a different phenomenology: the DM-shifted parabolas evolve through helical, conical, skyrmion-lattice, and field-polarized phases, with \(k_h\approx 0.036\ {\rm \AA}^{-1}\), \( \mu_0H_{c1}\approx 0.1\ {\rm T}\), and \( \mu_0H_{c2}\approx 0.55\ {\rm T} \) [1805.08750].

Recent antiferromagnetic van der Waals predictions place VPX\(_3\) among the clearest layered candidates. In the monolayer limit, the odd-in-\(k\) term arises from 2NN DMI in an easy-axis honeycomb antiferromagnet, and the valley asymmetry obeys
\[
\varepsilon_r = 6\sqrt{3}\left(D_x\sin\theta\cos\phi + D_y\sin\theta\sin\phi + D_z\cos\theta\right),
\]
so that the nonreciprocity depends asymmetrically and periodically on the Néel-vector orientation. The effect is large in VPTe\(_3\) and VPSe\(_3\), but it is strongly layer dependent: an AFM-coupled bilayer restores inversion and becomes reciprocal, whereas odd-layer stacks remain nonreciprocal [2509.06538].

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 \(1\) GHz with wavelengths down to \(281\pm 44\) nm inside walls as narrow as \(52\pm 10\) nm. Bloch points were identified as topological defects that disrupt phase evolution and generate different adjacent wavelengths without removing the underlying wall-mode nonreciprocity [2311.06186].

Not all proposed materials exhibit a measurable effect. A careful neutron three-axis study of MnPS\(_3\) found no significant difference between opposite Brillouin-zone corners within the experimental sensitivity. The apparent energy differences were \(\le 0.2\) meV and within the \(\approx 0.9\) meV instrumental resolution, leading to the conclusion that any DMI responsible for \( \omega(J)\neq \omega(N) \) must be much smaller than \(0.01\) meV [2010.02898].

## 4. Experimental probes and evidentiary standards

Inelastic neutron scattering has provided the clearest direct access to spectral nonreciprocity in bulk magnets. In \(\alpha\)-Cu\(_2\)V\(_2\)O\(_7\), single-crystal neutron measurements established the finite-\(k\) minima, the field-linear gap asymmetry, and an explicit test of detailed balance in a state with broken \(\mathcal P\) and \(\mathcal T\) [1702.04889]. 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 \(q\), \(E\), or \(\mu_0H_{\rm int}\) individually changes the spectrum, whereas simultaneous reversal of two variables restores it [1805.08750].

Because false positives are possible when opposite wavevectors are measured under different resolution conditions, negative results have played an important methodological role. In MnPS\(_3\), the sample was rotated by \(180^\circ\) to access corresponding \(+Q\) and \(-Q\) 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 \(|\Delta\omega|<0.2\) meV, illustrates that theoretical proposals for nonreciprocity must be separated from experimentally resolvable band asymmetry [2010.02898].

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 [2003.00230]. Time-resolved STXM at the Gd \(M_5\) edge then extended direct imaging to domain-wall channels, with \(20\) nm spatial and \(50\) ps temporal resolution, making it possible to reconstruct phase maps, wavelengths, and propagation directions of individual wall-confined modes [2311.06186].

Microwave transmission and nonlocal electrical detection broaden the experimental notion of nonreciprocity. In LiFe\(_5\)O\(_8\), lithographic meander antennae with \(\lambda=20\ \mu{\rm m}\) excited exchange-regime magnons with \(k\approx 0.314\ \mu{\rm m}^{-1}\), and the measured \(S_{21}\) and \(S_{12}\) differed near \(5.5\) GHz at \(\theta=0^\circ\), reversing upon magnetization reversal. Centrosymmetric YIG, measured in the same geometry, remained reciprocal at \(\theta=0^\circ\), isolating the bulk DMI contribution in LiFe\(_5\)O\(_8\) [1505.01942]. 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 \(23\%\), and the symmetry matched remote dipolar coupling controlled by the chirality of YIG magnon stray fields [2408.15595].

Cavity systems use the same language of reciprocity but probe it through scattering rather than \(\omega(\mathbf q)\). 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 \(|{\rm ISO}|\approx 40\) dB for the FMR–cavity hybridization and \(25\) dB and \(15\) dB for higher-order couplings, while reversing the magnetic field interchanged the \(S_{21}\) and \(S_{12}\) asymmetry [2012.14765].

## 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
\[
a(B)=\left(\frac{2\Phi_0}{\sqrt{3}B}\right)^{1/2},\qquad G=\frac{2\pi}{a},
\]
and Bragg gaps open at \(k\approx nG/2\). When the vortex lattice moves, the bandgaps shift by \( \Delta f_n\approx nGv/(2\pi) \). At \(H_\perp=7.2\) mT, the experiments reported \(a\approx 576\)–\(585\) nm, \(G\approx 10.9\ {\rm rad}/\mu{\rm m}\), tunability of approximately \(10\ {\rm GHz/mA}\), a polarity-induced asymmetry of approximately \(2\) GHz at \(|I|=100\ \mu{\rm A}\), and \(10\) ns reconfiguration under a \(100\) MHz ac current with power \(\approx 2\) nW [2107.04095].

Cavity magnonics realizes nonreciprocity through polarization-selective magnon–photon coupling. In the irregular resonant cavity system, the gyrotropic permeability tensor of YIG,
\[
\mu_{ij}=
\begin{pmatrix}
\mu_\parallel & -i\kappa & 0\\
i\kappa & \mu_\parallel & 0\\
0 & 0 & \mu_\perp
\end{pmatrix},
\]
combines with spatially asymmetric cavity polarization content so that the ports sample different chiral photon admixtures. The multimode coupling matrix then contains unequal \(g_{+,n}\) and \(g_{-,n}\), and elimination of the cavity modes yields photon-mediated indirect magnon–magnon couplings of the form
\[
g_{\rm eff}^{(m_0\leftrightarrow m_n)}\approx
\sum_{p\in\{+,-\}}\frac{g_{p,0}\,g_{p,n}}{\Delta_p}.
\]
This indirect coupling broadens the field window over which isolation remains high [2012.14765].

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 \( -i\Gamma e^{i\Theta} \), 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 \( \omega_b/2\pi=\omega_q/2\pi=5\) GHz, \( \lambda/2\pi=10\) MHz, and \( \Gamma/2\pi=5\) MHz, the maximum contrast was \(C_{\max}=0.895\), with \(g_{\rightarrow}^{(2)}(0)=1.615\) and \(g_{\leftarrow}^{(2)}(0)=0.089\) at \(\Delta/2\pi=-10.2\) MHz [2112.02351].

Magnomechanical systems add Kerr and squeezing control. In cavity-magnon optomechanics, the magnon Kerr effect produces both a frequency shift \( \Delta_K \) and a two-magnon term \( (K/2)(m^2+m^{\dagger 2}) \), and changing the magnetic-field orientation from \([100]\) to \([110]\) reverses the sign of \(K_0\), \(K\), and \( \Delta_K \). This yields nonreciprocal bipartite and tripartite entanglement, quantified by bidirectional contrast ratios and tunable with bath temperature [2305.03325]. A related two-cavity magnomechanical proposal used the sign change of the self-Kerr coefficient between \([100]\) and \([110]\) to define \(\Delta_K>0\) and \(\Delta_K<0\) configurations, with different optimal detunings and different effective couplings \(G/\Gamma\approx 1.0\) versus \(1.4\) [2405.16221]. An alternative squeezed-magnon scheme showed that the phase-tunable quantities \( \Delta_\theta=\Upsilon\sin\theta \) and \( \kappa_\theta=\Upsilon\cos\theta \) can yield ideal nonreciprocity of entanglement, \(C_E=1\) or \(C_R=1\), in experimentally feasible parameter regimes [2508.06850]. In a spinning WGM cavity coupled to two YIG spheres, the Sagnac–Fizeau shift
\[
\Delta_F=\pm \frac{\Omega n r\omega_a}{c}
\left(1-\frac{1}{n^2}-\frac{\lambda}{n}\frac{dn}{d\lambda}\right)
\]
combines with magnon Kerr nonlinearity to produce nonreciprocal magnon–magnon entanglement that remains robust up to about \(100\) mK [2605.14394].

## 6. Applications, limitations, and recurrent misconceptions

The device implications are diverse but structurally similar. Direction-dependent \(\omega(\mathbf q)\), \(\tau(\mathbf q)\), or transmission can be used for magnonic diodes, isolators, filters, interferometers, and nonreciprocal logic elements. Fe\(_3\)GeTe\(_2\) is attractive because it combines intrinsic chiral magnons with unusually long acoustic lifetimes in a conducting 2D ferromagnet [2004.14179]. Fluxonic ferromagnet/superconductor hybrids enable electrically reconfigurable microwave filters with fast \(10\) ns modulation [2107.04095]. 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 [2007.12240]. In layered VPX\(_3\), 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 [2509.06538].

Several open questions recur across the field. In Fe\(_3\)GeTe\(_2\), 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 [2004.14179]. In VPX\(_3\), the experimentally practical extraction of \(D_x\), \(D_y\), and \(D_z\), and the effect of twist or stacking registry beyond idealized stackings, remain unresolved [2509.06538]. Hybrid platforms introduce their own constraints: the Nb-based fluxonic scheme is cryogenic and sensitive to vortex disorder near depinning [2107.04095], while cavity and magnomechanical schemes remain sensitive to polarization purity, mode crowding, thermal noise, and stability thresholds [2012.14765][2305.03325].

Two misconceptions are especially persistent. First, nonreciprocity does not always mean that a bare magnon band satisfies \( \omega(\mathbf q)\neq \omega(-\mathbf q) \). 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 [2003.00230]. 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 [2408.15595]. Second, the existence of theoretical proposals or indirect thermoelectric anomalies does not guarantee a resolvable spectral asymmetry. The MnPS\(_3\) null neutron result constrains any relevant DMI to be much smaller than \(0.01\) meV and shows that proposed magnon Nernst mechanisms need not correspond to an experimentally accessible \( \omega(\mathbf q)-\omega(-\mathbf q) \) splitting in bulk material [2010.02898].

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-\(k\) 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.

Source: https://www.emergentmind.com/topics/nonreciprocal-magnons