Circularly Polarized Phonons

• Circularly polarized phonons are vibrational modes characterized by circular or elliptical ionic displacements that generate intrinsic angular momentum and an associated orbital magnetic moment.
• They can be formed via superposition of orthogonal linearly polarized eigenmodes or arise intrinsically in chiral crystals, with detection methods including Raman scattering, RIXS, and THz polarimetry.
• Their angular momentum couples to electronic topology and magnetism, influencing nonequilibrium transport phenomena and offering design principles for optimizing phonon magnetons in materials.

Circularly polarized phonons are lattice vibrational modes in which the ionic displacement field traces circular or elliptical trajectories, so that the mode carries angular momentum; in ionic materials, the same motion of charged ions also generates an orbital magnetic moment (Juraschek et al., 2018). In current usage, the topic spans several related but nonidentical notions: real phonon angular momentum from actual rotating atomic motion, pseudo-angular momentum defined by crystal symmetry, helicity, and circular-dichroic optical selection rules (Zhang et al., 28 Mar 2025). The literature therefore treats circularly polarized phonons both as a kinematic property of phonon eigenvectors and as a symmetry-constrained collective degree of freedom that couples to magnetism, topology, valley physics, and nonequilibrium transport (Juraschek et al., 2018, Pols et al., 2024, Ueda et al., 2023).

1. Definition and scope

A general elliptically polarized phonon may be formed by superposing two orthogonal linearly polarized phonon eigenmodes. Writing the normal-mode coordinate as

$$\mathbf{Q}(t)= \left( \begin{array}{c} Q_1(t)\ Q_2(t)\ 0 \end{array} \right) = \left( \begin{array}{c} Q_1 \sin(\Omega_1 t + \varphi)\ Q_2 \sin(\Omega_2 t)\ 0 \end{array} \right),$$

the mode is circularly polarized when

$$\Omega_1=\Omega_2\equiv\Omega_0,\qquad Q_1=Q_2\equiv Q,\qquad \varphi=\pi/2,$$

so that the ionic motion is circular rather than linear (Juraschek et al., 2018).

At finite wavevector, a second construction is also common. In chiral crystals or at special points such as $K/K'$, the phonon eigenvector itself can be complex, so that a single branch is intrinsically circular. In that case, the same magnetic-moment formalism may be used after reinterpreting the two orthogonal components as the real and imaginary parts of one complex eigenvector rather than as two distinct degenerate branches (Juraschek et al., 2018). This distinction underlies the frequent contrast between coherently prepared zone-center circular phonons and intrinsically circular finite-$\mathbf{k}$ chiral phonons (Juraschek et al., 2018, Ueda et al., 2023).

The stricter literature on chirality adds a geometric condition. In quartz, chiral phonons are defined as vibrational modes in which atoms have a rotational motion perpendicular to their propagation with an associated circular polarization and angular momentum; the same work distinguishes this from “circular phonons” whose atomic rotation lies in a plane containing the propagation direction (Ueda et al., 2023). In $\alpha$-HgS, an additional distinction is made between rotating but nonpropagating modes and “truly chiral phonons,” namely propagating circularly rotating phonons in a 3D chiral crystal (Ishito et al., 2021).

2. Microscopic theory: angular momentum, magnetic moment, and phonon magnetons

For a circularly polarized mode, the angular momentum and magnetic moment per unit cell are written

$$\mathbf{L} = \mathbf{Q}\times\partial_t\mathbf{Q}, \qquad \mathbf{M} = \gamma \mathbf{L},$$

and, for the circular case,

$$\mathbf{L}=\Omega_0 Q^2\hat{z}\equiv L\hat{z}, \qquad \mathbf{M}=M\hat{z}.$$

The phonon gyromagnetic ratio is

$$\gamma=\sum_{i} \gamma_i \mathbf{q}_{i,1} \times \mathbf{q}_{i,2}, \qquad \gamma_i=\frac{e\mathbf{Z}_i^\ast}{2\mathcal{M}_i},$$

so the orbital magnetic moment depends on the phonon eigenvectors, Born effective charge tensors, and ionic masses rather than on frequency alone (Juraschek et al., 2018).

In the single-quantum language, the same formalism yields

$$L = N \hbar \equiv N l_\text{ph}, \qquad M = \gamma N \hbar \equiv N \mu_\text{ph},$$

so that each circular phonon quantum carries

$$l_\text{ph}=\hbar, \qquad \mu_\text{ph}=\gamma\hbar.$$

The quantity $$\Omega_1=\Omega_2\equiv\Omega_0,\qquad Q_1=Q_2\equiv Q,\qquad \varphi=\pi/2,$$0 is called the phonon magneton (Juraschek et al., 2018).

A second widely used description quantifies circular polarization through a mode “spin”

$$\Omega_1=\Omega_2\equiv\Omega_0,\qquad Q_1=Q_2\equiv Q,\qquad \varphi=\pi/2,$$1

with $$\Omega_1=\Omega_2\equiv\Omega_0,\qquad Q_1=Q_2\equiv Q,\qquad \varphi=\pi/2,$$2 the spin-1 matrices in Cartesian basis. The angular momentum per phonon is then effectively

$$\Omega_1=\Omega_2\equiv\Omega_0,\qquad Q_1=Q_2\equiv Q,\qquad \varphi=\pi/2,$$3

so $$\Omega_1=\Omega_2\equiv\Omega_0,\qquad Q_1=Q_2\equiv Q,\qquad \varphi=\pi/2,$$4 serves as a dimensionless handedness measure: positive and negative values correspond to opposite handedness, while zero corresponds to an achiral, linearly polarized mode (Pols et al., 2024).

A third, fully mode-theoretic definition writes the phonon angular momentum as the expectation value of rotation generators,

$$\Omega_1=\Omega_2\equiv\Omega_0,\qquad Q_1=Q_2\equiv Q,\qquad \varphi=\pi/2,$$5

which makes explicit that nonzero relative phase between orthogonal displacement components produces circular or elliptical motion and hence angular momentum (Zhang et al., 28 Mar 2025).

Recent work also shows that the conventional framework is incomplete. “Anomalous magnetic moments” identifies three cases beyond the standard picture: rotationless axial phonons, diverging gyromagnetic ratios, and noncollinear phonon magnetic moments (Chaudhary et al., 27 Apr 2025). In the first case, the phonon can have no real angular momentum from $$\Omega_1=\Omega_2\equiv\Omega_0,\qquad Q_1=Q_2\equiv Q,\qquad \varphi=\pi/2,$$6 yet still exhibit an effective magnetic response through pseudo-angular-momentum-mediated coupling (Chaudhary et al., 27 Apr 2025).

3. Symmetry, pseudo-angular momentum, and criteria for chirality

Pseudo-angular momentum is a symmetry label rather than an expectation value. At a rotation-invariant momentum,

$$\Omega_1=\Omega_2\equiv\Omega_0,\qquad Q_1=Q_2\equiv Q,\qquad \varphi=\pi/2,$$7

and $$\Omega_1=\Omega_2\equiv\Omega_0,\qquad Q_1=Q_2\equiv Q,\qquad \varphi=\pi/2,$$8 is the phonon pseudo-angular momentum (Zhang et al., 28 Mar 2025). In nonsymmorphic systems, the corresponding screw-rotation generalization yields momentum-dependent, non-quantized PAM. In Te, for example, along screw-preserving lines one has

$$\Omega_1=\Omega_2\equiv\Omega_0,\qquad Q_1=Q_2\equiv Q,\qquad \varphi=\pi/2,$$9

with the non-quantized part arising from the orbital Bloch-phase contribution $K/K'$0 (Zhang et al., 2021).

The modern symmetry-based literature stresses that nonzero PAM, nonzero real angular momentum, helicity, circular atomic motion, and helicity-changing Raman channels are not generally equivalent (Zhang et al., 28 Mar 2025). Several constraints follow directly. If the little group contains only a single $K/K'$1-fold rotation, a nondegenerate phonon can carry angular momentum only parallel to that axis. If it contains a single mirror, the angular momentum of a nondegenerate phonon must be perpendicular to the mirror plane. For $K/K'$2, these constraints conflict, so all nondegenerate phonons have zero angular momentum. Under time-reversal symmetry, every nondegenerate phonon at a time-reversal-invariant momentum has zero angular momentum. Under $K/K'$3, any nondegenerate phonon mode has zero angular momentum throughout the entire Brillouin zone (Zhang et al., 28 Mar 2025).

This is why the literature now treats several older heuristics as insufficient. Identifying chiral phonons solely through nonzero PAM or through circular-polarization inversion in Raman scattering is described as inadequate (Zhang et al., 28 Mar 2025). Black phosphorus provides the clearest counterexample in that work: a mode can flip circular polarization in Raman because of mirror symmetry yet still have zero phonon angular momentum everywhere in the Brillouin zone (Zhang et al., 28 Mar 2025).

The stricter notion of “true chirality” adds propagation. In $K/K'$4-HgS, the split $K/K'$5 optical branches along $K/K'$6 are treated as propagating circularly polarized phonons with finite group velocity, and the paper classifies them as “truly chiral phonons” in the sense that the two enantiomeric phonon states are interconverted by spatial inversion but not by time reversal combined with a suitable rotation (Ishito et al., 2021).

4. Generation, transport, and experimental detection

For coherent optical generation, a standard route is resonant excitation of an infrared-active mode with a mid-infrared pulse. In the harmonic approximation, the driven coordinate obeys

$K/K'$7

with $K/K'$8, $K/K'$9, and $\mathbf{k}$0, where the mode effective charge

$\mathbf{k}$1

determines how strongly the phonon is driven by light (Juraschek et al., 2018). The same framework connects the single-phonon magneton $\mathbf{k}$2 to a macroscopic coherent moment $\mathbf{k}$3 (Juraschek et al., 2018).

Thermal driving provides a different route. In chiral 2D halide perovskites, the nonequilibrium phonon angular momentum density induced by a temperature gradient is written

$\mathbf{k}$4

with a diagonal response tensor $\mathbf{k}$5 and the largest response along one in-plane axis because that direction hosts the most favorable combination of chirality and group velocity (Pols et al., 2024).

Detection has diversified across momentum, polarization, and nonequilibrium channels. Circularly polarized Raman scattering in $\mathbf{k}$6-quartz and tellurium resolves the split finite-$\mathbf{k}$7 branches of doubly degenerate modes and shows that the helicity assignment reverses between enantiomers (Oishi et al., 2022, Ishito et al., 2022). In quartz, circularly polarized O $\mathbf{k}$8-edge RIXS directly detects finite-momentum chiral phonons, and the dichroic signal changes sign between left and right quartz, establishing a momentum-resolved probe of lattice chirality (Ueda et al., 2023). In WC, meV-resolution IXS reveals anomalous geometry-dependent intensity asymmetries for circularly polarized LA and TA phonons near $\mathbf{k}$9, leading the authors to argue that the usual phonon scattering function must be revised for chiral phonons (Cai et al., 2021).

Several other experiments probe circular polarization indirectly through optical anisotropy. In Co$\alpha$0Sn$\alpha$1S$\alpha$2, infrared phonons below $\alpha$3 K show helicity-dependent Fano line shapes and sharp resonances in Kerr rotation and ellipticity, which are interpreted as phonon circular dichroism generated by coupling to a helicity-selective topological electronic continuum (Yang et al., 2024). In FePSe$\alpha$4, phase-resolved THz polarimetry reconstructs the trajectories of elliptically polarized magnon-phonon hybrids at the zone center, providing a route to nondegenerate rotating lattice modes without external magnetic field (Ning et al., 2024). For ionic bulk materials, direct magnetic detection remains a target; the macroscopic orbital moments predicted for coherently driven circular phonons are proposed to be observable indirectly by Faraday rotation or directly by NV-center magnetometry (Juraschek et al., 2018).

5. Materials platforms and representative phenomena

First-principles surveys of ionic solids identify clear design principles for large phonon magnetons: large Born effective charges, strong mass asymmetry, and mode eigenvectors with a large $\alpha$5 structure (Juraschek et al., 2018). The largest phonon magnetons in the 2018 survey occur in hydrides, especially CsH with $\alpha$6 and CuH with $\alpha$7, whereas typical values in most other binaries and perovskites are $\alpha$8–$\alpha$9 (Juraschek et al., 2018). Under a frequency-scaled mid-IR driving protocol, representative macroscopic moments reach the few-$$\mathbf{L} = \mathbf{Q}\times\partial_t\mathbf{Q}, \qquad \mathbf{M} = \gamma \mathbf{L},$$0 scale: GaN $$\mathbf{L} = \mathbf{Q}\times\partial_t\mathbf{Q}, \qquad \mathbf{M} = \gamma \mathbf{L},$$1, InN $$\mathbf{L} = \mathbf{Q}\times\partial_t\mathbf{Q}, \qquad \mathbf{M} = \gamma \mathbf{L},$$2, KTaO$$\mathbf{L} = \mathbf{Q}\times\partial_t\mathbf{Q}, \qquad \mathbf{M} = \gamma \mathbf{L},$$3 $$\mathbf{L} = \mathbf{Q}\times\partial_t\mathbf{Q}, \qquad \mathbf{M} = \gamma \mathbf{L},$$4, KNbO$$\mathbf{L} = \mathbf{Q}\times\partial_t\mathbf{Q}, \qquad \mathbf{M} = \gamma \mathbf{L},$$5 $$\mathbf{L} = \mathbf{Q}\times\partial_t\mathbf{Q}, \qquad \mathbf{M} = \gamma \mathbf{L},$$6, and SrTiO$$\mathbf{L} = \mathbf{Q}\times\partial_t\mathbf{Q}, \qquad \mathbf{M} = \gamma \mathbf{L},$$7 $$\mathbf{L} = \mathbf{Q}\times\partial_t\mathbf{Q}, \qquad \mathbf{M} = \gamma \mathbf{L},$$8, although the latter exceeds lattice-stability limits unless the field is reduced (Juraschek et al., 2018).

Chiral 2D halide perovskites provide a different regime: low-energy phonons of the inorganic Pb–I framework carry the dominant chirality, with $$\mathbf{L} = \mathbf{Q}\times\partial_t\mathbf{Q}, \qquad \mathbf{M} = \gamma \mathbf{L},$$9 below $$\mathbf{L}=\Omega_0 Q^2\hat{z}\equiv L\hat{z}, \qquad \mathbf{M}=M\hat{z}.$$0 meV, while higher-energy internal organic modes are nearly achiral (Pols et al., 2024). At $$\mathbf{L}=\Omega_0 Q^2\hat{z}\equiv L\hat{z}, \qquad \mathbf{M}=M\hat{z}.$$1 K, the response tensor components are reported as

$$\mathbf{L}=\Omega_0 Q^2\hat{z}\equiv L\hat{z}, \qquad \mathbf{M}=M\hat{z}.$$2

$$\mathbf{L}=\Omega_0 Q^2\hat{z}\equiv L\hat{z}, \qquad \mathbf{M}=M\hat{z}.$$3

$$\mathbf{L}=\Omega_0 Q^2\hat{z}\equiv L\hat{z}, \qquad \mathbf{M}=M\hat{z}.$$4

with the strongest response dominated by the three lowest-energy modes (Pols et al., 2024).

Bulk chiral crystals such as quartz, tellurium, and $$\mathbf{L}=\Omega_0 Q^2\hat{z}\equiv L\hat{z}, \qquad \mathbf{M}=M\hat{z}.$$5-HgS remain model systems for propagating or finite-$$\mathbf{L}=\Omega_0 Q^2\hat{z}\equiv L\hat{z}, \qquad \mathbf{M}=M\hat{z}.$$6 circular phonons. Quartz exhibits enantiomer-sensitive finite-$$\mathbf{L}=\Omega_0 Q^2\hat{z}\equiv L\hat{z}, \qquad \mathbf{M}=M\hat{z}.$$7 chiral phonons in RIXS and Raman (Ueda et al., 2023, Oishi et al., 2022). Tellurium and $$\mathbf{L}=\Omega_0 Q^2\hat{z}\equiv L\hat{z}, \qquad \mathbf{M}=M\hat{z}.$$8-HgS extend the same logic to nonsymmorphic or helical 3D chiral crystals in which circularly polarized Raman channels resolve opposite finite-$$\mathbf{L}=\Omega_0 Q^2\hat{z}\equiv L\hat{z}, \qquad \mathbf{M}=M\hat{z}.$$9 branches and permit direct handedness assignment of the crystal (Ishito et al., 2022, Ishito et al., 2021).

Several recent materials broaden the classification. In NiCo$$\gamma=\sum_{i} \gamma_i \mathbf{q}_{i,1} \times \mathbf{q}_{i,2}, \qquad \gamma_i=\frac{e\mathbf{Z}_i^\ast}{2\mathcal{M}_i},$$0TeO$$\gamma=\sum_{i} \gamma_i \mathbf{q}_{i,1} \times \mathbf{q}_{i,2}, \qquad \gamma_i=\frac{e\mathbf{Z}_i^\ast}{2\mathcal{M}_i},$$1, the $$\gamma=\sum_{i} \gamma_i \mathbf{q}_{i,1} \times \mathbf{q}_{i,2}, \qquad \gamma_i=\frac{e\mathbf{Z}_i^\ast}{2\mathcal{M}_i},$$2 Raman-active optical phonon shows perfect circular dichroism controlled by the ferroaxial combination $$\gamma=\sum_{i} \gamma_i \mathbf{q}_{i,1} \times \mathbf{q}_{i,2}, \qquad \gamma_i=\frac{e\mathbf{Z}_i^\ast}{2\mathcal{M}_i},$$3, and the authors therefore prefer the term “ferroaxial phonons” (Martinez et al., 2024). In MnTiO$$\gamma=\sum_{i} \gamma_i \mathbf{q}_{i,1} \times \mathbf{q}_{i,2}, \qquad \gamma_i=\frac{e\mathbf{Z}_i^\ast}{2\mathcal{M}_i},$$4, O $$\gamma=\sum_{i} \gamma_i \mathbf{q}_{i,1} \times \mathbf{q}_{i,2}, \qquad \gamma_i=\frac{e\mathbf{Z}_i^\ast}{2\mathcal{M}_i},$$5-edge RIXS identifies a $$\gamma=\sum_{i} \gamma_i \mathbf{q}_{i,1} \times \mathbf{q}_{i,2}, \qquad \gamma_i=\frac{e\mathbf{Z}_i^\ast}{2\mathcal{M}_i},$$6 meV circularly polarized $$\gamma=\sum_{i} \gamma_i \mathbf{q}_{i,1} \times \mathbf{q}_{i,2}, \qquad \gamma_i=\frac{e\mathbf{Z}_i^\ast}{2\mathcal{M}_i},$$7 phonon with non-reciprocal circular dichroism tied to the ferro-rotational order parameter $$\gamma=\sum_{i} \gamma_i \mathbf{q}_{i,1} \times \mathbf{q}_{i,2}, \qquad \gamma_i=\frac{e\mathbf{Z}_i^\ast}{2\mathcal{M}_i},$$8; these are termed “ferro-rotational phonons” (Huang et al., 26 Dec 2025). In graphyne, pristine stable allotropes have $$\gamma=\sum_{i} \gamma_i \mathbf{q}_{i,1} \times \mathbf{q}_{i,2}, \qquad \gamma_i=\frac{e\mathbf{Z}_i^\ast}{2\mathcal{M}_i},$$9 because $$L = N \hbar \equiv N l_\text{ph}, \qquad M = \gamma N \hbar \equiv N \mu_\text{ph},$$0 and $$L = N \hbar \equiv N l_\text{ph}, \qquad M = \gamma N \hbar \equiv N \mu_\text{ph},$$1 coexist, but atomic-selective substitutional doping with B, N, or BN breaks $$L = N \hbar \equiv N l_\text{ph}, \qquad M = \gamma N \hbar \equiv N \mu_\text{ph},$$2 and produces circularly polarized modes with nonzero $$L = N \hbar \equiv N l_\text{ph}, \qquad M = \gamma N \hbar \equiv N \mu_\text{ph},$$3, especially near $$L = N \hbar \equiv N l_\text{ph}, \qquad M = \gamma N \hbar \equiv N \mu_\text{ph},$$4 (Mishra et al., 14 Aug 2025).

6. Magnetism, topology, and terminological disputes

The direct magnetic consequences of circular phonons appear already in the phonon Zeeman effect. For $$L = N \hbar \equiv N l_\text{ph}, \qquad M = \gamma N \hbar \equiv N \mu_\text{ph},$$5,

$$L = N \hbar \equiv N l_\text{ph}, \qquad M = \gamma N \hbar \equiv N \mu_\text{ph},$$6

and the largest relative splittings reach the $$L = N \hbar \equiv N l_\text{ph}, \qquad M = \gamma N \hbar \equiv N \mu_\text{ph},$$7 level in a $$L = N \hbar \equiv N l_\text{ph}, \qquad M = \gamma N \hbar \equiv N \mu_\text{ph},$$8 T field (Juraschek et al., 2018). In SrTiO$$L = N \hbar \equiv N l_\text{ph}, \qquad M = \gamma N \hbar \equiv N \mu_\text{ph},$$9, however, the experimentally observed quasi-static Kerr signal under circular phonon pumping motivated a deeper reinterpretation: a local molecular-orbital model finds an atomic orbital contribution from chiral oxygen motion pumping Ti orbital angular momentum and points to an additional inter-atomic contribution from transient circulating currents, while also concluding that the simple local mechanism alone is too weak to explain the full experimentally inferred $$l_\text{ph}=\hbar, \qquad \mu_\text{ph}=\gamma\hbar.$$0-scale moment (Urazhdin, 2024).

The same theme appears in ultrafast magnetism. In Ni, ultrafast electron diffraction reveals a non-equilibrium population of anisotropic high-frequency phonons that appears as quickly as the magnetic order is lost; the anisotropy plane is perpendicular to the initial magnetization and the atomic oscillation amplitude is $$l_\text{ph}=\hbar, \qquad \mu_\text{ph}=\gamma\hbar.$$1 pm (Tauchert et al., 2021). That work interprets the missing angular momentum of femtosecond demagnetization as first entering polarized, effectively circularly polarized phonons before slower macroscopic rotation develops (Tauchert et al., 2021).

Circularly polarized phonons have also entered topological Floquet theory. On a honeycomb lattice, a zone-center circular optical phonon generates a Floquet correction

$$l_\text{ph}=\hbar, \qquad \mu_\text{ph}=\gamma\hbar.$$2

whose spin-independent part is a Haldane-type complex next-nearest-neighbor hopping; the resulting valley masses

$$l_\text{ph}=\hbar, \qquad \mu_\text{ph}=\gamma\hbar.$$3

drive a transition from a trivial insulator to a Chern insulator and induce orbital and spin magnetizations (Yao et al., 9 Jun 2026). In FePSe$$l_\text{ph}=\hbar, \qquad \mu_\text{ph}=\gamma\hbar.$$4, selective hybridization with magnons produces opposite-helicity elliptically polarized magnon polarons at the zone center, showing that intrinsic nondegenerate rotating lattice modes can also emerge from magnon-phonon coupling rather than from structural chirality (Ning et al., 2024).

These developments have sharpened several terminological disputes. “Magnetophononics and the Chiral Phonon Misnomer” argues that, in driven polar insulators, the dominant magnetic-like effect does not stem from the tiny Maxwell field of ions moving in circles but from a much larger electron-mediated non-Maxwellian field proportional to $$l_\text{ph}=\hbar, \qquad \mu_\text{ph}=\gamma\hbar.$$5, which can reach $$l_\text{ph}=\hbar, \qquad \mu_\text{ph}=\gamma\hbar.$$6–$$l_\text{ph}=\hbar, \qquad \mu_\text{ph}=\gamma\hbar.$$7 T; because this response depends on the square of the phonon displacements, the chirality transferred to the ions is said to play no role in magnetophononics (Merlin, 2024). A broader symmetry critique now states that identifying chiral phonons solely through nonzero PAM or through CPL polarization inversion is inadequate (Zhang et al., 28 Mar 2025). The resulting picture is more restrictive but also more precise: a phonon is circularly polarized in the physically meaningful sense when symmetry permits and the microscopic eigenmode indeed carries nonzero phonon angular momentum, while experimentally observed circular dichroism may, depending on context, reflect true circular lattice motion, ferroaxial selection rules, or coupling to electronic topology rather than one universal notion of chirality (Zhang et al., 28 Mar 2025, Martinez et al., 2024, Huang et al., 26 Dec 2025).