Longitudinal Magnons in Magnetic Systems

• Longitudinal magnons are distinct excitation modes characterized by oscillations in the magnitude of magnetic order, manifesting as gapped amplitude modes, density waves, or bound spin-reversal states.
• They are analyzed via effective field theories and many-body spin-wave treatments, with examples ranging from TlCuCl₃ amplitude modes to propagating reversal modes in anisotropic large-S systems.
• Their study informs experimental insights on decay processes, transport phenomena such as the spin Seebeck effect, and the role of anisotropy and disorder in magnetic heat and spin currents.

Longitudinal magnons are not a single universally defined quasiparticle class. In the literature, the term denotes several related but distinct objects: gapped amplitude modes of an ordered antiferromagnet near symmetry breaking; magnon-density waves generated by longitudinal spin-density fluctuations; propagating full-spin-reversal bound states in strongly anisotropic large-$S$ magnets; and, in transport work, magnon-mediated spin or heat responses measured in a longitudinal geometry, meaning parallel to an applied gradient or probe axis. The shared feature is the existence of a distinguished axis—set by the order parameter, an easy axis, a magnetic field, or the measurement geometry—relative to which the excitation or response is called longitudinal (Kulik et al., 2011, Xian, 2011, Mardele et al., 2024, Arakawa et al., 2017).

1. Terminology and conceptual scope

In broken-symmetry magnets, the most standard use of “longitudinal” contrasts fluctuations of the magnitude of an ordered moment with transverse fluctuations of its orientation. Near an $O(3)$-symmetric antiferromagnetic ordering transition, the longitudinal magnon is therefore the gapped excitation associated with oscillations along the direction selected by the staggered order parameter, whereas the transverse modes are the two Goldstone magnons (Kulik et al., 2011).

A second usage, developed in microscopic many-body treatments of ordered antiferromagnets, identifies longitudinal excitations with magnon-density waves. In this construction, the Fourier transform of $S^z$ is treated as a density operator, and the longitudinal mode is a collective oscillation of magnon density rather than an ordinary one-magnon spin flip (Xian, 2011, Xian et al., 2013, 1901.11007).

A third usage appears in strongly anisotropic easy-axis magnets with $S\ge 1$, where longitudinal magnons are propagating spin reversals. In this setting they carry $S^z=\pm 2S$, can be viewed as bound states of $2S$ conventional magnons, and are fundamentally different from small-angle transverse spin waves with $S^z=\pm 1$ (Mardele et al., 2024, Mendili et al., 5 Aug 2025).

Transport papers employ the same adjective differently. There, “longitudinal” often refers to a measurement channel such as $\kappa_{xx}$, a longitudinal spin Seebeck geometry, or a current-dependent longitudinal resistance, without implying a separate longitudinal excitation branch. This distinction is explicit in the Rotated Ferromagnetic Heisenberg model literature, where the low-energy modes are classified as C-C$_0$, C-C$_\pi$, and C-IC magnons, while “longitudinal” is reserved for susceptibilities or probe directions rather than a separate magnon species (Arakawa et al., 2017, Arakawa et al., 2017, Sun et al., 2016).

2. Amplitude modes near ordered phases and quantum criticality

In three-dimensional quantum antiferromagnets near an $O(3)$0 quantum critical point, the longitudinal magnon is the canonical amplitude mode. For TlCuCl$O(3)$1, described by an effective Landau–Ginzburg field theory for the staggered magnetization $O(3)$2, the disordered phase has a triply degenerate gapped spectrum, while the ordered phase develops a condensate $O(3)$3 and splits into two transverse gapless Goldstone modes and one longitudinal gapped mode with

$O(3)$4

This is the standard broken-symmetry structure: transverse modes rotate the order parameter, whereas the longitudinal mode modulates its magnitude (Kulik et al., 2011).

The same framework explains why the longitudinal mode in pressure-induced ordered TlCuCl$O(3)$5 is broad. After shifting $O(3)$6, the quartic interaction generates the cubic term

$O(3)$7

which permits the decay process “one longitudinal magnon $O(3)$8 two transverse magnons.” At zero momentum the width obeys

$O(3)$9

so $S^z$0. Using the parameters fixed from the Bose condensation of magnons in field, the estimate

$S^z$1

accounts quantitatively for the substantial experimental width. A small easy-plane anisotropy, introduced through

$S^z$2

reduces the available decay phase space and suppresses the width when the longitudinal mode cannot produce two transverse magnons (Kulik et al., 2011).

This amplitude-mode interpretation is highly specific. It applies to nearly $S^z$3-symmetric ordered antiferromagnets close to criticality, and it should not be conflated with the distinct multipolar spin-reversal modes found in strongly anisotropic large-$S^z$4 magnets.

3. Magnon-density waves in ordered quantum antiferromagnets

A separate microscopic tradition treats longitudinal excitations as magnon-density waves. The defining ansatz is Feynman-like: $S^z$5 with excitation energy

$S^z$6

where $S^z$7 is a double commutator and $S^z$8 is the longitudinal structure factor. In this approach, transverse magnons are one-magnon spin-wave modes, whereas longitudinal magnons are collective density oscillations of the ordered state (Xian, 2011, Xian et al., 2013, 1901.11007).

For bipartite Heisenberg antiferromagnets, this framework yields a clear dimensional distinction. In quasi-1D and quasi-2D systems with long-range order, the longitudinal mode has a nonzero gap. In the pure isotropic 1D limit, the longitudinal spectrum becomes gapless and merges with the transverse spectrum; for the spin-$S^z$9 chain the result reduces to the des Cloizeaux–Pearson form

$S\ge 1$0

This merging is central to the many-body picture: in one dimension the longitudinal mode is no longer a separate gapped amplitude excitation, whereas in ordered higher-dimensional systems it remains distinct (Xian, 2011).

Higher-order corrections are quantitatively important. For bipartite lattices, second-order large-$S\ge 1$1 terms reduce the longitudinal gaps by about $S\ge 1$2–$S\ge 1$3, improving agreement with experiments on quasi-1D compounds. In KCuF$S\ge 1$4, for example, the calculated minimum gap changes from $S\ge 1$5 meV to $S\ge 1$6 meV, close to the experimental value of about $S\ge 1$7 meV. Similar improvements occur for CsNiCl$S\ge 1$8 and RbNiCl$S\ge 1$9 (Xian et al., 2013).

Noncollinear triangular antiferromagnets add further structure. Because the ordered state has a $S^z=\pm 2S$0 three-sublattice pattern, the longitudinal spectrum folds into two branches, $S^z=\pm 2S$1 and $S^z=\pm 2S$2. The triangular geometry also permits a cubic interaction $S^z=\pm 2S$3, absent in collinear antiferromagnets, and the inclusion of cubic and quartic terms reduces the longitudinal spectrum by about $S^z=\pm 2S$4 at the zone boundaries in the two-dimensional model. In quasi-1D triangular or hexagonal antiferromagnets, the cubic correction is numerically very small because it originates mainly from weak planar couplings, while the quartic term provides the dominant renormalization (1901.11007).

4. Multipolar spin-reversal longitudinal magnons in easy-axis magnets

Strong easy-axis anisotropy produces a qualitatively different longitudinal sector. In FePSe$S^z=\pm 2S$5, a collinear honeycomb antiferromagnet with large spins $S^z=\pm 2S$6, two conventional transverse magnons coexist with two higher-energy modes interpreted as multimagnon hexadecapole excitations. These longitudinal magnons correspond to full reversals of Fe spins propagating coherently through the up-down antiferromagnetic structure. Their key spectroscopic fingerprint is an anomalous effective $S^z=\pm 2S$7 factor approximately equal to four times the $S^z=\pm 2S$8 factor of a single Fe$S^z=\pm 2S$9 ion, reflecting the fact that they carry

$2S$0

rather than the $2S$1 of an ordinary magnon. The reported zero-field energies are approximately $2S$2 meV and $2S$3 meV for the symmetric and antisymmetric longitudinal branches, alongside transverse gaps near $2S$4 meV and $2S$5 meV (Mardele et al., 2024).

The same physical idea can be formulated generically for large-$2S$6 easy-axis magnets with single-ion anisotropy: $2S$7 In the large-$2S$8 limit, the low-energy on-site doublet $2S$9 maps onto an effective pseudospin-$S^z=\pm 1$0 model, and the longitudinal mode becomes a pseudospin flip between the two extremal states. The resulting quasiparticle carries $S^z=\pm 1$1 and can be viewed as a bound state of $S^z=\pm 1$2 conventional magnons. Strong-coupling analysis gives the rough crossover scales

$S^z=\pm 1$3

marking, respectively, the point where the longitudinal gap meets the transverse gap and the point where overlap with the $S^z=\pm 1$4-magnon continuum begins to destabilize the mode (Mendili et al., 5 Aug 2025).

For $S^z=\pm 1$5, the problem can be analyzed particularly explicitly. The linked-cluster expansion refines the antiferromagnetic crossover estimates to

$S^z=\pm 1$6

while multiboson theory tracks the mode until roughly

$S^z=\pm 1$7

The multiboson representation is important because it yields not only the renormalized spectrum but also a decay rate,

$S^z=\pm 1$8

showing explicitly how a longitudinal magnon loses coherence once it enters the two-transverse-magnon continuum (Mendili et al., 5 Aug 2025).

This family of modes is neither a Goldstone excitation nor the near-critical amplitude mode of an approximately $S^z=\pm 1$9-symmetric magnet. It is a bound, high-angular-momentum object stabilized by strong easy-axis anisotropy.

5. Longitudinal transport geometries, spin Seebeck physics, and current-driven magnon populations

In transport studies, “longitudinal” commonly designates the geometry rather than the quasiparticle type. A representative case is the longitudinal spin Seebeck effect in Pt/YIG, where pulsed microwave heating of the Pt layer generates a vertical thermal gradient and the resulting spin current is detected electrically through the inverse spin Hall effect in the same Pt layer. Time-resolved measurements on YIG films of different thicknesses show that the signal is governed by the thermal gradient near the YIG/Pt interface and is consistent with ballistic propagation of quasi-acoustic magnons in room-temperature YIG. These magnons, with energies above $\kappa_{xx}$0 K, move with constant group velocity and decay exponentially with an effective propagation length that is practically independent of film thickness, which the authors interpret as strong support for a ballistic transport scenario (Noack et al., 2018).

Antiferromagnetic longitudinal spin Seebeck measurements reveal a more complex division between magnonic and critical regimes. In epitaxial FeF$\kappa_{xx}$1(110)/Pt heterostructures, the low-temperature longitudinal SSE peak at about $\kappa_{xx}$2 K is attributed to antiferromagnetic magnons: an applied field along the easy axis splits the degenerate $\kappa_{xx}$3- and $\kappa_{xx}$4-mode branches, creating an imbalance in their opposite angular momenta and thus a net spin current. A second SSE peak appears at $\kappa_{xx}$5 K and extends well into the paramagnetic phase. Near and above $\kappa_{xx}$6, data taken up to $\kappa_{xx}$7 T collapse onto the critical scaling form for the magnetic susceptibility with the three-dimensional Ising exponents $\kappa_{xx}$8 and $\kappa_{xx}$9, indicating that the high-temperature longitudinal SSE is governed by critical spin fluctuations rather than ordered-AFM magnons (Li et al., 2019).

Longitudinal transport can also probe current-driven changes in magnon population. In ultrathin YIG/Pt bilayers, spin Hall conversion in Pt creates a spin accumulation at the interface that either creates or annihilates magnons depending on whether the interfacial spin is parallel or antiparallel to the magnetization. The induced change in magnetization amplitude is written phenomenologically as

$_0$0

and it produces additional nonlinear longitudinal and transverse magnetoresistances in any channel that depends on magnetization. Harmonic measurements on a $_0$1 nm YIG / $_0$2 nm Pt bilayer show that the second-harmonic longitudinal and transverse signals are dominated by these magnon creation-annihilation terms; the inferred $_0$3 reaches about $_0$4 at $_0$5 mT and $_0$6 mA, and the critical current for damping compensation is reported as $_0$7 mA below $_0$8 mT (Noël et al., 2024).

6. Conductivity, localization, and waveguiding

Longitudinal thermal conductivity is a natural observable for magnon transport in insulating magnets. In a disordered two-dimensional Heisenberg antiferromagnet, the longitudinal thermal conductivity $_0$9 can be derived within linear-response theory and linear spin-wave approximation. The key result is a magnon analogue of weak localization: particle-particle-type multiple impurity scattering enhances backscattering, producing a logarithmic correction

$_\pi$0

The negative logarithmic term implies that coherent disorder scattering suppresses longitudinal magnon heat transport in two dimensions (Arakawa et al., 2017).

A closely related mechanism survives in noncollinear systems. For a disordered screw-type spiral magnet on a square lattice, the in-plane longitudinal thermal conductivities are anisotropic, $_\pi$1, because the Dzyaloshinsky–Moriya interaction affects propagation along the spiral axis and the antiferromagnetic direction differently. At the same time, the main disorder correction remains logarithmic and negative, again signaling weak localization through critical coherent backscattering. The anisotropy scales as

$_\pi$2

so longitudinal transport becomes a possible probe of the Dzyaloshinsky–Moriya scale (Arakawa et al., 2017).

Longitudinal transport can also be formulated in clean three-dimensional ferromagnets. For the easy-axis XXZ Heisenberg ferromagnet on the face-centered cubic lattice, linear spin-wave theory yields four magnon branches, of which two are flat and two are dispersive. Only the dispersive branches carry current, and the easy-axis gap is crucial for the convergence of both longitudinal spin and thermal conductivities. In the low-temperature regime the conductivities are activated,

$_\pi$3

and their ratio obeys a magnon analog of the Wiedemann–Franz law,

$_\pi$4

or simply $_\pi$5 in the paper’s dimensionless units (Parymuda, 5 Jul 2025).

A different usage appears in chains of YIG particles coupled by magnetic dipole–dipole interactions. There the longitudinal mode is the branch whose dynamic magnetic dipole moment is parallel to the chain direction. In the quasistatic approximation $_\pi$6, the relevant tensor element is $_\pi$7-independent, so the longitudinal branch is dispersionless. Its group velocity therefore vanishes in that approximation, making it a spectral mode rather than an efficient power-transmission channel, unlike the dispersive transverse and elliptically polarized branches (Pike et al., 2016).

7. Ultracold-gas realizations and conceptual boundaries

Longitudinal magnons are not restricted to crystalline magnets. In a ferromagnetic $_\pi$8 $_\pi$9Rb spinor Bose gas prepared in the fully polarized state $O(3)$00, magnons are injected by a pulsed rf field that transfers atoms primarily into $O(3)$01. Here “longitudinal” refers to the initial spin polarization along the magnetic-field axis, while the condensation of magnons is detected through the spontaneous emergence of transverse magnetization. The experiment shows that the magnons behave thermodynamically as free bosons in an effective flat potential

$O(3)$02

inside the ferromagnetic condensate, and that near-conservation of net longitudinal magnetization allows evaporative cooling into a magnon condensate. Above threshold, images of transverse magnetization directly reveal spontaneous symmetry breaking, quasi-condensation, and phase singularities identified as Mermin–Ho spin textures (Fang et al., 2015).

This ultracold-gas case is useful because it clarifies the role of language. The magnons originate in a longitudinally magnetized background, but the ordered phase that signals condensation is transverse. The same caution applies in spin-orbit-coupled lattice models such as the Rotated Ferromagnetic Heisenberg model, where the excitation content is organized by commensurate and incommensurate magnon minima rather than by a distinct longitudinal branch. In that setting, transverse fields convert the exact collinear Y-x state into canted phases, and the C-C$O(3)$03, C-C$O(3)$04, and C-IC magnons become relativistic and enter the quantum ground state, but the paper does not define a separate longitudinal magnon species (Sun et al., 2016).

The surveyed literature therefore uses “longitudinal magnon” in a strictly relational sense. Depending on context, it may mean an amplitude mode of a broken-symmetry order parameter, a magnon-density wave, a propagating spin reversal with $O(3)$05, or a transport response measured along the driving axis. Any precise use of the term requires specification of the underlying axis, symmetry, and observable.