Phonon Polariton Hall Effect in Terahertz Systems

• The phonon polariton Hall effect is a hybrid phenomenon where coupled phonon and photon modes create a transverse energy current under a magnetic field and coherent terahertz drive.
• It arises from magnetic-field-induced splitting of circularly polarized optical phonons, resulting in chiral polariton branches with unequal group velocities.
• PbTe, as a model system, shows predicted Hall angles up to 10⁻⁴, offering potential for tunable terahertz beam steering and polarization-controlled routing.

The phonon polariton Hall effect is a theoretically proposed Hall-like transport phenomenon of hybrid light–matter quasiparticles in which phonon polaritons—formed by coupling infrared-active optical phonons to terahertz photons—support a transverse energy flow under coherent drive in an applied magnetic field. In the formulation developed for PbTe, the magnetic field lifts the degeneracy of circularly polarized optical-phonon components, the resulting phonon-polariton branches acquire unequal group velocities, and the mixed phononic-photonic excitation carries a transverse current in addition to its longitudinal propagation. The effect is presented as an extension of the conventional phonon Hall effect from thermally excited lattice vibrations to coherently driven terahertz polaritons, and as a mechanism for magnetically controlled bending of light in the terahertz regime (Yaniv et al., 19 Sep 2025).

1. Definition and conceptual scope

In its conventional form, the phonon Hall effect describes a transverse heat current generated by a longitudinal thermal gradient in a magnetic field. The phonon polariton Hall effect retains the Hall structure but changes both the quasiparticle and the drive: the relevant excitations are propagating phonon polaritons rather than thermally populated acoustic phonons, and the excitation is a coherent terahertz pulse rather than a temperature gradient (Yaniv et al., 19 Sep 2025).

This distinction is central. The proposed effect does not rely on thermal phonon populations. Instead, a directed terahertz wavepacket excites a phonon-polariton mode whose phonon component couples to the magnetic field and whose photonic component provides finite group velocity and long-range propagation. In this sense, the phenomenon is not simply a thermal Hall effect of optical phonons, but a coherent transport response of a hybrid bosonic mode.

The scope of the concept is narrower than the broader phonon Hall literature. Several works on phonon thermal Hall transport explicitly do not treat phonon polaritons or phonon-photon hybridization, even when they discuss direct magnetic-field coupling of lattice motion, Berry-curvature formulations, or Hall-viscosity responses (Jin et al., 2024). The phonon polariton Hall effect therefore occupies a specific conceptual niche: it is a Hall response of coupled phonon-photon bands rather than of pure lattice modes.

2. Microscopic mechanism and polaritonic band structure

The microscopic starting point is the optical-phonon Zeeman response. For a propagation geometry with the terahertz pulse along $y$, magnetic field $\mathbf B = B\hat y$, and linear polarization at $\phi=45^\circ$ in the $xz$ plane, the incident pulse decomposes into equal left- and right-handed circular components. In zero field, the two transverse optical phonon branches are degenerate. The field couples to the phonon magnetic moment and shifts the circularly polarized phonon frequencies according to

$$\Omega_\sigma=\Omega_T-s_\sigma\frac{\mu_{\mathrm{ph}} B}{\hbar},$$

with $s_l=+1$ and $s_r=-1$. Left- and right-handed circularly polarized phonons are therefore split in opposite directions (Yaniv et al., 19 Sep 2025).

Hybridization with the terahertz photon is described by a $2\times2$ avoided-crossing Hamiltonian,

$$H = \hbar\sum_{\mathbf k} \begin{pmatrix} a_{\mathbf{k},\sigma}^\dagger & c_{\mathbf{k}}^\dagger \end{pmatrix} \begin{pmatrix} \Omega_{\sigma} & g(\mathbf{k}) \ g^*(\mathbf{k}) & \omega_\nu(\mathbf{k}) \end{pmatrix} \begin{pmatrix} a_{\mathbf{k},\sigma} \ c_{\mathbf{k}} \end{pmatrix},$$

where $a_{\mathbf{k},\sigma}$ and $\mathbf B = B\hat y$0 are the phonon and photon operators and $\mathbf B = B\hat y$1. Diagonalization yields the chiral polariton branches

$\mathbf B = B\hat y$2

Because the field-split phonons hybridize with the photon, the upper and lower polariton branches each separate into left- and right-handed sectors, giving four nondegenerate branches in total. The phonon content of each branch is

$\mathbf B = B\hat y$3

so the Hall response depends not only on the polariton dispersion but also on how strongly phononic a given branch remains at a given wavevector. Near strong mixing, the polariton simultaneously inherits the phonon’s magnetic response and the photon’s propagation efficiency.

3. Energy-current formulation and origin of the transverse response

A central result of the theory is the derivation of energy-current operators for propagating phonon polaritons. The current expectation value separates into a longitudinal contribution $\mathbf B = B\hat y$4 and a transverse contribution $\mathbf B = B\hat y$5. The longitudinal term is associated with the branch group velocity $\mathbf B = B\hat y$6, whereas the transverse term arises from interbranch overlap terms involving derivatives of the polarization vectors. In the language of the paper, $\mathbf B = B\hat y$7 originates from the mixed-mode structure and the magnetic-field-induced chiral splitting (Yaniv et al., 19 Sep 2025).

Two linked ingredients generate the transverse flow. First, the split chiral branches have different dispersions and therefore different group velocities. Second, the polariton state carries a $\mathbf B = B\hat y$8-dependent phonon fraction, so the magnetic-field-induced modification of the phononic part redistributes the energy current across the hybrid branches. The transverse current is therefore neither purely photonic nor purely phononic; it is a consequence of the hybrid mode structure.

The coherent nature of the drive introduces an additional control parameter absent in thermal Hall transport. The transverse current can be reversed by changing the sign of the pulse polarization angle $\mathbf B = B\hat y$9, equivalently by rotating the input polarization by $\phi=45^\circ$0, which flips the relative phase between left- and right-circular components. This makes the Hall response polarization-selective in a direct dynamical sense rather than only through static band geometry.

The mixed character of the quasiparticle is not incidental but constitutive. Pure phonons can carry magnetic moment and chiral structure but have limited propagation speed. Pure photons propagate efficiently but do not directly realize the phonon Zeeman physics invoked here. Phonon polaritons combine both features, which is why the effect appears as a transverse deflection of coherently driven terahertz energy flow.

4. PbTe as the model system and predicted signatures

PbTe is used as the model platform because it is a narrow-bandgap semiconductor with strong infrared-active optical phonons and strongly coupled phonon polaritons. The calculation employs density-functional-theory inputs obtained with VASP, the frozen-phonon method, and phonopy, using PBEsol exchange-correlation, PAW pseudopotentials, a plane-wave cutoff of $\phi=45^\circ$1 eV, a $\phi=45^\circ$2 $\phi=45^\circ$3-centered Monkhorst-Pack mesh, and a $\phi=45^\circ$4 supercell. The reported parameters are $\phi=45^\circ$5 THz, $\phi=45^\circ$6 THz, mode effective charge $\phi=45^\circ$7, $\phi=45^\circ$8, and $\phi=45^\circ$9 (Yaniv et al., 19 Sep 2025).

The strong-hybridization point occurs near $xz$0, where the phonon fraction is approximately $xz$1. This is the regime in which the polariton is roughly half phonon and half photon, and therefore most favorable for the Hall effect. The calculated transverse current $xz$2 peaks near $xz$3 for the upper branch and near $xz$4 for the lower branch. Longitudinal propagation is significant mainly for $xz$5.

The reported Hall angles are approximately

$xz$6

for the lower branch and

$xz$7

for the upper branch. The lower-branch Hall angle is about an order of magnitude larger, although the same analysis notes that this is partly because the longitudinal current becomes small as the branch evolves toward surface-like phonon states. The most robust Hall response is therefore expected between the maxima of $xz$8 and $xz$9, rather than at the point of largest nominal ratio alone.

In phenomenological terms, the predicted observables are polarization-resolved polariton dispersions, branch splitting into left- and right-handed circular sectors, and a measurable transverse energy flow in pump-probe terahertz experiments. The proposal explicitly frames the effect as a route to controllable transverse transport of terahertz radiation and to magnetically tunable beam steering and polarization-controlled routing.

5. Relation to phonon Hall transport, Hall viscosity, and other bosonic Hall effects

The phonon polariton Hall effect is best understood against the background of several distinct Hall-like responses of neutral bosons. Conventional phonon thermal Hall transport has been reported in non-magnetic insulators and semiconductors including SrTiO$$\Omega_\sigma=\Omega_T-s_\sigma\frac{\mu_{\mathrm{ph}} B}{\hbar},$$0, SiO$$\Omega_\sigma=\Omega_T-s_\sigma\frac{\mu_{\mathrm{ph}} B}{\hbar},$$1, MgO, MgAl$$\Omega_\sigma=\Omega_T-s_\sigma\frac{\mu_{\mathrm{ph}} B}{\hbar},$$2O$$\Omega_\sigma=\Omega_T-s_\sigma\frac{\mu_{\mathrm{ph}} B}{\hbar},$$3, Si, and Ge, with the claim that a universal intrinsic phonon thermal Hall effect obeys the scaling law $$\Omega_\sigma=\Omega_T-s_\sigma\frac{\mu_{\mathrm{ph}} B}{\hbar},$$4. That literature concerns transverse heat transport of phonons in magnetic field, including planar geometries, and does not explicitly discuss phonon polaritons (Jin et al., 2024).

A complementary theoretical tradition formulates the phonon Hall effect in terms of Berry curvature and band topology. In a dielectric with Raman spin-phonon coupling, the Hall conductivity can be written as a Berry-curvature sum over phonon bands, and the Hall response can exhibit nonmonotonic field dependence and topological transitions associated with Chern-number changes (Zhang et al., 2010). This provides a template for hybrid-mode generalization: the relevant Berry curvature would belong to phonon-photon polariton bands rather than to pure phonon bands. This suggests a natural formal bridge between topological phonon Hall theory and phonon polariton Hall transport.

Hall viscosity offers a second bridge. In $$\Omega_\sigma=\Omega_T-s_\sigma\frac{\mu_{\mathrm{ph}} B}{\hbar},$$5-RuCl$$\Omega_\sigma=\Omega_T-s_\sigma\frac{\mu_{\mathrm{ph}} B}{\hbar},$$6, ultrasonic measurements of the acoustic Faraday effect were interpreted as evidence that phonons possess Hall viscosity, which rotates transverse sound polarizations and yields an intrinsic thermal Hall conductivity $$\Omega_\sigma=\Omega_T-s_\sigma\frac{\mu_{\mathrm{ph}} B}{\hbar},$$7 (Shragai et al., 7 Oct 2025). In magnetic topological insulators, a surface phonon Hall viscosity localized at an interface produces circular birefringence, an interface-localized phonon mode below the bulk continuum, a phonon polarization-filter mechanism, and surface acoustic Faraday rotation (Chatterjee et al., 18 Mar 2026). These results concern phonons rather than phonon polaritons, but they establish that Hall-odd viscoelastic responses can generate chirality-dependent propagation and polarization selection in lattice systems.

By contrast, some phonon Hall mechanisms are explicitly extrinsic. In rare-earth garnets, the phonon Hall effect has been attributed to resonant skew scattering of phonons from the crystal-field states of superstoichiometric Tb$$\Omega_\sigma=\Omega_T-s_\sigma\frac{\mu_{\mathrm{ph}} B}{\hbar},$$8 ions, with impurity correlations required to avoid angular cancellation in $$\Omega_\sigma=\Omega_T-s_\sigma\frac{\mu_{\mathrm{ph}} B}{\hbar},$$9 (Mori et al., 2014). The phonon polariton Hall effect as formulated in PbTe is not of this type: it is a coherent branch-splitting and current-operator effect of hybrid modes, not a defect-scattering mechanism.

There is also a broader polaritonic precedent. In cavity-coupled transition-metal dichalcogenides, a polariton Hall effect was analyzed in terms of Berry curvature inherited from excitons and photons and enhanced by momentum-dependent light-matter coupling. There the transverse motion is an anomalous-velocity effect of exciton-polariton bands rather than a phonon Zeeman splitting of phonon polaritons (Gutiérrez-Rubio et al., 2018). The comparison is instructive because both cases assign the Hall response to the geometry of a hybrid bosonic quasiparticle, but the microscopic constituents and coupling structures are different.

6. Interpretive issues, limits, and prospective developments

The phonon polariton Hall effect remains, in the available literature, a theoretical prediction rather than an experimental observation. Its present formulation is materially specific, using PbTe as an example of a strongly coupled phonon-polariton system with a sizable phonon Zeeman effect, and it is tailored to coherent terahertz excitation rather than to equilibrium or near-equilibrium transport (Yaniv et al., 19 Sep 2025). Any generalization beyond that setting remains a matter for further theory and experiment.

One interpretive issue is terminological. The phenomenon is adjacent to, but not identical with, the conventional phonon Hall effect. It does not describe a thermal Hall conductivity extracted from antisymmetrized transverse temperature gradients, nor does it require the scaling relations discussed for non-magnetic crystals. It is instead a transverse energy-current response of coherently driven hybrid modes. Confusion between these two meanings of “phonon Hall effect” obscures the role of coherence, branch selectivity, and explicit phonon-photon mixing.

A second issue concerns microscopic unification. The broader phonon Hall literature still contains unresolved questions about direct magnetic coupling of lattice vibrations, the role of Berry curvature, Hall viscosity, local symmetry breaking, and extrinsic skew scattering. This suggests that a complete theory of the phonon polariton Hall effect may eventually need to integrate several strands: hybrid-band geometry, viscoelastic Hall responses, and material-specific phonon magnetic moments. The existing work on universal phonon thermal Hall transport, Berry-curvature formulations, and Hall-viscosity-induced polarization rotation indicates that such an overview is plausible, but it is not yet explicitly carried out for coupled phonon-photon bands (Jin et al., 2024).

The device-oriented implications are already explicit. Because the response is magnetic-field tunable, polarization-selective, and strongest in systems with strong infrared-active phonons and phonon magnetic moments, it has been proposed as a route to terahertz polaritonic devices based on beam steering, polarization-controlled routing, and nonreciprocal energy flow (Yaniv et al., 19 Sep 2025). More broadly, the surrounding literature on polarization filtering, acoustic Faraday rotation, and Hall transport of other lattice-derived bosons implies that Hall physics of hybrid vibrational excitations may extend beyond PbTe and beyond the specific geometry originally analyzed.