Ferro-Rotational Phonons

• Ferro-rotational phonons are quantized lattice vibrations with circular ionic displacements that impart a consistent phonon angular momentum.
• They emerge from intrinsic lattice symmetries, magnetoelastic couplings, and spin–phonon interactions in systems like ferromagnets and van der Waals materials.
• Their unique circular polarization facilitates coherent angular momentum transfer, topological phonon band formation, and novel nonreciprocal transport phenomena.

Ferro-rotational phonons are collective excitations in solids characterized by a uniform, macroscopically aligned angular momentum associated with circular (or rotational) ionic displacements. Unlike conventional phonons which typically entail linear or non-chiral atomic motion, ferro-rotational phonons possess a well-defined sense of rotation, resulting in a nonzero phonon angular momentum (PAM) per quantum. This order can emerge from intrinsic lattice symmetries, broken time-reversal symmetry (e.g., ferromagnetism), ferro-axial distortions, or strong spin–lattice entanglement, and is realized across a broad class of materials including magnetic insulators, van der Waals ferromagnets, ferro-axial crystals, charge-ordered charge density waves (CDWs), and interfacial systems with orbital angular-momentum quasiparticles.

1. Theoretical Foundations and Definitions

Ferro-rotational phonons are quantized lattice vibrations whose eigenmodes involve collective circular or elliptical trajectories of atoms, conferring well-defined angular momentum. In cubic, trigonal, or hexagonal lattices, the doubly-degenerate E-type optical or acoustic phonons can be recast in the circular basis: for atomic displacements $(u_x, u_y)$, the complex superpositions $u^\pm = u_x \pm i u_y$ yield right- and left-circularly polarized modes. Each carries PAM $l_{ph} = \pm \hbar$ (or multiples thereof, depending on ion count per unit cell) along the principal symmetry axis (Huang et al., 26 Dec 2025, Mekap et al., 22 Dec 2025, An et al., 2019, Solano-Carrillo, 2016).

Formally, the phonon angular-momentum operator is $L_z = u_x p_y - u_y p_x$, and in the second-quantized representation, the number operators for circular polarizations $b_{q,+}$, $b_{q,-}$ determine the total phonon "spin":

$$\hat{S}^3_{q,ph} = b^\dagger_{q,+}b_{q,+} - b^\dagger_{q,-}b_{q,-}$$

with eigenvalues $m = +1, -1$ per phonon quantum (Solano-Carrillo, 2016).

A ferro-rotational state is established when a majority of such phonons in the ground or a driven state occupy the same helicity, resulting in a macroscopic alignment of PAM—an analogy with spin alignment in conventional ferromagnets (Rückriegel et al., 2019, Mekap et al., 22 Dec 2025, Du et al., 2019).

2. Microscopic Mechanisms and Magnetoelastic Couplings

Ferro-rotational phonons arise via several microscopic couplings:

• Magnetoelastic Interaction: In ferromagnets or ferrimagnets, the uniform precession of the magnetization (Kittel mode) hybridizes with transverse shear phonons through the magnetoelastic Hamiltonian:

$$\mathcal{H}_{me} = B_1 \sum_\alpha \alpha_\alpha^2 \varepsilon_{\alpha\alpha} + B_2 \sum_{\alpha \neq \beta} \alpha_\alpha \alpha_\beta \varepsilon_{\alpha\beta}$$

where $\alpha_\alpha$ are normalized magnetization components and $\varepsilon_{\alpha\beta}$ the strain tensor (An et al., 2019, Rückriegel et al., 2019).

• Bond-Dependent Magnetoelastic Anisotropy: In low-symmetry or frustrated magnets (e.g., D$_3$ point group), off-diagonal magnetoelastic tensor elements yield an emergent gauge field in the phononic dynamical matrix, splitting circularly polarized modes and generating a net PAM (Ma et al., 2023).
• Spin-Phonon Hybridization via Internal Fields: In itinerant ferromagnets, the internal induction $B_i$ and hyperfine fields mediate coupling between electron spins and chiral phonons, leading to efficient spin–angular-momentum transfer (Solano-Carrillo, 2016).
• Interaction with Order Parameters (Ferro-axial and Quadrupolar Phases): In centrosymmetric ferro-axial crystals or systems with uniform quadrupole order (e.g., CDWs), collective rotations of molecular or cluster dipoles couple to circularly polarized phonons, producing ferro-rotational dynamics and associated order parameters, e.g., the toroidal moment $A_z = \sum_n \mathbf{r}_n \times \mathbf{p}_n$ (Huang et al., 26 Dec 2025, Luo et al., 2021).

3. Symmetry Analysis and Classification

The existence and observability of ferro-rotational phonons rely on symmetry considerations:

• Point-Group Selection Rules: In trigonal (P3m1, R3̄m), hexagonal (D$_{6h}$), and cubic lattices, E-type or $E_g$ phonons at $\Gamma$ enable circular combinations with defined $l_{ph}$ (Mekap et al., 22 Dec 2025, Du et al., 2019, Huang et al., 26 Dec 2025).
• Time-Reversal and Inversion: Breaking of time-reversal symmetry (by magnetization) lifts the degeneracy of right- and left-circular phonons, while axial ferro-rotational order can exist even in globally centrosymmetric crystals (Huang et al., 26 Dec 2025).
• Topological Character: The matrix structure of the phononic Hamiltonian under bond-anisotropy or magnetoelastic coupling can place the system in topological class D, permitting nonzero Chern numbers for phonon bands and topologically protected chiral phonon states (Ma et al., 2023).
• Ferro-rotational Quadrupole Modes: In CDWs such as 1T-TaS$_2$, the collective ferro-rotational order parameter belongs to the antisymmetric electric-quadrupole components, with associated amplitude (breathing) and rotation phonon modes (Luo et al., 2021).

4. Experimental Signatures and Material Realizations

Ferro-rotational phonons have been established across diverse experimental platforms:

• YIG/GGG/YIG Heterostructures: Strong magnetoelastic coupling in out-of-plane oriented bilayers produces standing transverse acoustic shear modes with circular polarization, mediating coherent angular momentum transfer over hundreds of microns—the phonon carries $\hbar$ PAM, and FMR absorption features periodic zero crossings and anti-crossings indicative of constructive/destructive interference mediated by these phonons (An et al., 2019).
• Van der Waals Magnets (Fe$_5$GeTe$_2$, Fe$_3$GeTe$_2$): Polarization-resolved Raman, including helicity-resolved measurements, reveals E-type modes that split into distinct RL, LR branches under ferromagnetic order, with direct observation of frequency splitting ($\Delta \omega \sim 1-2\,{\rm cm}^{-1}$) and a robust tilt in Raman patterns attributed to anisotropic electron-phonon coupling and chiral phonon formation (Mekap et al., 22 Dec 2025, Du et al., 2019).
• Ferro-axial Insulators (MnTiO$_3$): X-ray circular dichroism (XCD) via resonant inelastic X-ray scattering detects circularly polarized E$_g$ phonons, observing nonreciprocal dichroism that constrains the domain structure and directly visualizes the ferro-rotational order in the ground state (Huang et al., 26 Dec 2025).
• Ultrafast Nonlinear Spectroscopy in CDWs (1T-TaS$_2$): Time-resolved electric quadrupole RA-SHG reveals triplets of underdamped modes (breathing, rotation, and mixed) linked to ferro-rotational quadrupole excitations, separated from conventional phonons by symmetry of the nonlinear susceptibility tensor (Luo et al., 2021).
• Fröhlich Coupling to Rotational Modes at Interfaces: In MoS$_2$/SrTiO$_3$, the trion's orbital angular momentum selectively couples to substrate rotational optical (RO) phonons, giving strong anisotropic polaronic shifts in binding energy dependent on relative orientation (angle $\theta$ between trion and phonon rotation axes) (Trushin et al., 2019).

5. Quantitative Characterization

Key physical parameters and observables for ferro-rotational phonons include:

• Phonon Angular Momentum Content: Per mode, $$L_{ph}(\lambda) = \sum_s m_s (\dot u_{\lambda,s} \times u_{\lambda,s})$$, with circular polarization per branch $P_\lambda \sim \pm \hbar$ per unit cell (Mekap et al., 22 Dec 2025).
• Hybridization Strength: Magnetoelastic coupling constants (e.g., $\Omega/2\pi \approx 1.5$ MHz in YIG/GGG/YIG at 5.5 GHz) determine the linewidth and strong-coupling regime (cooperativity $\mathcal{C} \sim 3$) (An et al., 2019).
• Topological Invariants: Chern numbers $C_n$ for phononic bands signal robust chiral edge states and Hall responses (Ma et al., 2023).
• Electrical and Thermal Transport: Chiral phonon-mediated spin-flip scattering produces a linear-in-temperature resistivity $\rho(T) = \rho_0 + \gamma T$, with $\gamma$ found in Fe, Co, Ni to match measured values (Solano-Carrillo, 2016). Planar thermal Hall conductivity driven by Berry curvature in nontrivial phonon bands is obtained in monolayer systems (Ma et al., 2023).
• Spectroscopic Markers: Frequency splitting ($\Delta f$), Fano asymmetry parameters ($q$, $1/q$), circular dichroism contrasts $>$80%, and domain-dependent polarization reversal under XCD are direct experimental observables distinguishing ferro-rotational order (Huang et al., 26 Dec 2025, Mekap et al., 22 Dec 2025, Du et al., 2019).

6. Functional Consequences and Applications

Ferro-rotational phonons provide a bosonic angular-momentum transport channel distinct from magnonic or electronic conduction, with multiple potential implications:

• Coherent Angular Momentum Transfer: Ballistic propagation of circularly polarized phonons enables long-range, low-loss spin communication and magnon–phonon interfaces (An et al., 2019, Rückriegel et al., 2019).
• Topological Phononic Devices: Engineering bond-anisotropy or symmetry-breaking provides access to thermal Hall and chiral edge states for phononic logic and spintronic function (Ma et al., 2023).
• Nonreciprocal Transport and Diode Effects: Nonreciprocal dichroism and phononic nonreciprocity open possibilities for one-way energy flow and phononic isolation (Huang et al., 26 Dec 2025, Mekap et al., 22 Dec 2025).
• Ultrafast Manipulation of Order Parameters: Photoinduced enhancement or rotation of ferro-rotational phonons can access transient, nonthermal phases with new symmetry properties in CDWs (Luo et al., 2021).
• Anisotropic Polaronic Effects: Tunable polaronic shifts via rotational phonons at oxide interfaces suggest orientation-dependent energy engineering in 2D semiconductors (Trushin et al., 2019).

7. Future Directions and Outlook

The field of ferro-rotational phononics encompasses theoretical developments in angular momentum conservation, explicit Hamiltonian construction, and topological band theory, as well as advancing materials synthesis and spectroscopy:

• Novel Material Platforms: Exploration continues in van der Waals heterostructures, ferro-axial perovskites, engineered superlattices, and trion–phonon coupled interfaces (Huang et al., 26 Dec 2025, Mekap et al., 22 Dec 2025, Trushin et al., 2019).
• Topological Phononics and Device Integration: The topological robustness and thermal/optical activity of ferro-rotational phonons support technological applications in nonreciprocal devices, quantum transduction, and magnon–phonon hybrid logic (An et al., 2019, Ma et al., 2023).
• Probing and Control: Brillouin and Raman scattering, X-ray dichroism, and ultrafast nonlinear optical techniques offer direct means to interrogate and manipulate phononic angular momentum on ultrafast timescales and in domain-resolved contexts (Huang et al., 26 Dec 2025, Mekap et al., 22 Dec 2025, Luo et al., 2021).
• Theoretical Modeling: Ongoing advances in ab initio modeling of spin–phonon coupling, dynamical gauge fields in the phonon Hamiltonian, and Landau theory for ferro-rotational condensates are refining the understanding and design of these collective excitations (Ma et al., 2023, Huang et al., 26 Dec 2025).

Ferro-rotational phonons thus represent a fundamentally new paradigm in lattice, electronic, and spin dynamics, with strong implications for coherent control, topological transport, and strongly correlated electron–lattice–spin systems.

References:

(An et al., 2019, Solano-Carrillo, 2016, Mekap et al., 22 Dec 2025, Luo et al., 2021, Du et al., 2019, Rückriegel et al., 2019, Ma et al., 2023, Huang et al., 26 Dec 2025, Trushin et al., 2019)