Axial Ferro-Rotational Order

• Axial ferro-rotational order is a state defined by the uniform, axis-oriented alignment of microscopic rotations, resulting in an axial pseudo-vector order parameter unchanged by inversion and time reversal.
• The order manifests through chiral phonon modes carrying intrinsic angular momentum, which reveal nonreciprocal optical and thermal signatures in materials like MnTiO₃ and Fe₅GeTe₂.
• Microscopic mechanisms such as magnetoelastic coupling and topological effects enable control of these chiral modes, promising innovations in spintronics, nonreciprocal devices, and quantum transduction.

Axial ferro-rotational order refers to a symmetry-breaking state of matter characterized by a uniform, axis-oriented ordering of microscopic rotations, resulting in an axial (pseudo-)vector order parameter that is even under both spatial inversion and time reversal. In crystalline solids, such order arises not from polarization (as in ferroelectrics) or magnetization (as in ferromagnets), but from the alignment of rotational degrees of freedom associated with atomic loops, toroidal moments, or orbital currents, and is fundamentally associated with the concept of ferro-rotational (or “ferroaxial”) order. In contemporary condensed matter physics, the presence of axial ferro-rotational order is detected through the emergence of chiral or ferro-rotational phonon modes—collective lattice excitations where the vibrational pattern carries an intrinsic angular momentum or pseudo-angular momentum, often in the absence of broken time-reversal symmetry.

1. Symmetry, Order Parameter, and Group Theoretical Foundations

Axial ferro-rotational order is characterized by a uniform alignment of local rotational entities such as electric toroidal moments $$\boldsymbol{\mathcal A} = \sum_i \bm r_i \times \bm p_i$$, with $\bm r_i$ the position and $\bm p_i$ the local electric dipole moment or analogous local circulation (Huang et al., 26 Dec 2025). This axial vector is even under spatial inversion $\mathcal I$ and time reversal $\mathcal T$: $$\mathcal I:\boldsymbol{\mathcal A}\rightarrow\boldsymbol{\mathcal A}$$, $$\mathcal T:\boldsymbol{\mathcal A}\rightarrow\boldsymbol{\mathcal A}$$. Uniform nonzero $\mathcal{A}$ defines ferroaxial or axial ferro-rotational order.

Point group symmetry analysis is central: Axial ferro-rotational order is possible in point groups allowing a one-dimensional axial (rotational) pseudoscalar or pseudovector. In $C_{3i}$ (as in MnTiO$_3$) or $D_3,~S_6$ (as in $1T$-TaS$_2$ CCDW), irreducible representations such as $E_g$ or $A_{2u}$ support rotational phonon or electronic collective modes. The presence of degenerate multidimensional irreps—e.g., $E_g$ at $\Gamma$—permits recombination into circularly polarized partners with well-defined angular momentum $L_z$ about the high-symmetry axis, distinguishing ferro-rotational phonons from conventional shear or breathing modes (Huang et al., 26 Dec 2025, Luo et al., 2021).

2. Fundamental Phonon Modes and Their Angular Momentum

Chiral or ferro-rotational phonons constitute the dynamical realization of axial ferro-rotational order. In high-symmetry lattices (e.g., trigonal, hexagonal, or rhombohedral), the doubly degenerate in-plane $E$ or $E_g$ modes at the Brillouin zone center ($\Gamma$) can be recast as right- and left-circularly polarized vibrations—each associated with definite phonon pseudoangular momentum (PAM), $\ell_{\text{ph}}=\pm1$ or $\pm2$ (Mekap et al., 22 Dec 2025, Huang et al., 26 Dec 2025). For instance, in Fe$_5$GeTe$_2$ the $E$(1) and $E$(2) optical modes involve planar circular motions of Fe atoms, with the circular polarization (phonon angular momentum) computed as

$$P_\lambda(\mathbf{k}) = \sum_j \operatorname{Im}\big[u_{\mathbf{k},\lambda}(j) \times u^*_{\mathbf{k},\lambda}(j)\big]_z$$

where $u_{\mathbf{k},\lambda}(j)$ are complex vibrational eigenvectors (Mekap et al., 22 Dec 2025).

These ferro-rotational modes differ fundamentally from conventional (linearly polarized) phonons, as the two circular polarizations become energetically nondegenerate (lifting of chiral degeneracy) when time-reversal or inversion (or both) are broken, or, in some cases, by internal ferroaxial symmetry alone (Ma et al., 2023, Huang et al., 26 Dec 2025). The resulting circularly polarized phonon excitations carry quantized angular momentum per phonon, manifest in observable effects such as RL/LR splitting in Raman (Mekap et al., 22 Dec 2025), non-reciprocal X-ray circular dichroism (Huang et al., 26 Dec 2025), or velocity birefringence in acoustic resonators (Müller et al., 2023).

3. Microscopic Mechanisms: Magnetoelastic Coupling and Topological Effects

In ferro-rotational materials, the microscopic origin of chiral phonons is typically attributed to symmetry-allowed bond-dependent magnetoelastic coupling (MEC). For instance, in a triangular lattice ferromagnet with $D_3$ point symmetry, a bond-anisotropic exchange-magnetoelastic term,

$$H_{\mathrm{me}} = \sum_{\langle ij\rangle,\alpha\beta} K_{ij}^{\alpha\beta} S_i^\alpha S_j^\beta \left( \hat{\mathbf{R}}_{ij}^0\cdot \mathbf{u}_{ij} \right)$$

produces a $k$-dependent effective magnetic field in phonon equations of motion, breaking the left/right degeneracy of in-plane circular phonons (Ma et al., 2023). The phononic bands acquire nonzero Berry curvature, can attain quantized Chern numbers (typically class D in the tenfold-way), and exhibit thermally induced phonon angular momentum and planar thermal Hall conductivity.

In van der Waals ferromagnets and multiferroics, time-reversal or inversion symmetry breaking (including local, but not global, inversion breaking in “ferroaxial” scenarios) splits the PAM sublevels even in the absence of external fields. This can be further modulated by electron-phonon coupling, as evidenced by anisotropic Fano line shapes in Raman spectra and nontrivial polarization patterns (Mekap et al., 22 Dec 2025). In metallic systems, internal magnetization can couple to ionic orbital motion and phonon PAM via the internal induction and contact hyperfine field, yielding additional transport phenomena such as linear-in-T resistivity contributions (Solano-Carrillo, 2016).

4. Experimental Signatures and Detection

Axial ferro-rotational order manifests in several characteristic experimental signatures:

• Circular dichroism in inelastic X-ray or Raman scattering: Non-reciprocal X-ray circular dichroism (XCD) in MnTiO$_3$ provides direct evidence of ferro-rotational phonons, with measured dichroism $>80\%$ for specific phonon modes, sign reversal under beam direction reversal, and selection rules matching the $E_g$ irrep (Huang et al., 26 Dec 2025). In Fe$_5$GeTe$_2$, RL/LR Raman polarization channels distinguish split chiral $E$ phonons (Mekap et al., 22 Dec 2025).
• Ultrafast optical probes of multipolar order: In $1T$-TaS$_2$, time-resolved electric-quadrupole second harmonic generation (EQ RA-SHG) directly resolves breathing (symmetric amplitude, $A_{1g}$) and rotational (antisymmetric, $A_{2u}$) phonon modes, as well as sudden frequency shifts and magnitude changes upon photoinduced CDW melting (Luo et al., 2021).
• Phononic birefringence and magnon-phonon hybridization: Circularly polarized acoustic phonons exhibit velocity birefringence ($\Delta v/v\simeq10^{-5}$) in hybrid bulk acoustic resonator architectures, tunable via substrate symmetry and external magnetic field (Müller et al., 2023).
• Thermal and transport responses: Nonzero equilibrium phonon angular momentum and a finite planar thermal Hall conductivity directly probe ferro-rotational phonon-induced transport channels (Ma et al., 2023, Solano-Carrillo, 2016).

5. Coupling to Spin, Charge, and Other Degrees of Freedom

Ferro-rotational phonons provide new channels for coupling lattice and electronic/magnetic degrees of freedom. When chiral atomic rotations modulate spin-spin interactions (e.g., exchange and Dzyaloshinskii-Moriya), they act as “geometric pumps” of spin or magnon number, as shown in models where chiral phonons adiabatically modulate the magnon Hamiltonian, leading to geometric (Berry phase) pumping of the total magnon number with sign control by phonon chirality (Yao et al., 2023).

The presence of nonzero phonon angular momentum enables:

• Spin-lattice relaxation channels (linear-in-T contribution to resistivity, correct spin-lattice relaxation rates) (Solano-Carrillo, 2016)
• Magnon-phonon hybrid polarons with net angular momentum (coherent transfer in YIG/GGG/YIG trilayers, macroscopic quantum transduction) (An et al., 2019)
• Valley-polarized phonon currents and associated nonreciprocal transport in van der Waals and 2D materials (Mekap et al., 22 Dec 2025, Ma et al., 2023)

6. Material Realizations and Case Studies

Notable material classes exhibiting axial ferro-rotational order and associated chiral phonon phenomena include:

• MnTiO$_3$ (ilmenite): $C_{3i}$ point group, ferro-rotational phonon condensate evidenced by nonreciprocal XCD, uniform toroidal moment $\mathcal{A}\parallel c$ (Huang et al., 26 Dec 2025).
• Fe$_5$GeTe$_2$: Threefold rotation symmetry, direct Raman evidence for chiral $E$ phonons, room-temperature ferromagnetism (Mekap et al., 22 Dec 2025).
• $1T$-TaS$_2$ (CCDW phase): Ferro-rotational charge density modulation with triplet breathing/rotational modes resolved by ultrafast EQ RA-SHG (Luo et al., 2021).
• YIG/GGG/YIG trilayers: Long-range coherent angular momentum transfer via circularly polarized phonons (An et al., 2019).
• Co$_{25}$Fe$_{75}$ resonator hybrids: Phononic birefringence and Purcell-enhanced magnon-phonon coupling (Müller et al., 2023).

7. Theoretical Outlook and Applications

Axial ferro-rotational order and the associated physics of chiral phonons open a range of opportunities:

• Nonreciprocal phononic devices: Diodes, isolators, and filters based on nonreciprocal propagation of chiral phonons, achievable by symmetry engineering of ferroaxial order (Huang et al., 26 Dec 2025, Müller et al., 2023).
• Topological phononics: Quantized Berry curvature, Chern numbers, and topological edge states enabled by MEC-induced chiral phonon bands (class D). Tunable by in-plane magnetic field, with discrete topological transitions and sign switches in observable quantities (Ma et al., 2023).
• Spintronics and optomechanics: Chiral phonons enable angular momentum transfer between magnonic and phononic subsystems, offering routes for quantum transduction and novel pump-probe dynamics (An et al., 2019, Yao et al., 2023).
• Ultrafast manipulation of hidden multipolar orders: Time-resolved optical techniques can directly excite and image axial ferro-rotational dynamics, even in otherwise “hidden” order parameters inaccessible to conventional probes (Luo et al., 2021).

Further work is anticipated in mapping the momentum-resolved phonon angular momentum textures (via RIXS or inelastic neutron scattering), pump-probe control using circularly polarized THz pulses, and exploration of topological phonon edge phenomena in ferroaxial metamaterials (Huang et al., 26 Dec 2025, Ma et al., 2023).