Chiral Inertia in Advanced Systems

• Chiral inertia is the influence of structural chirality that couples rotational, translational, and dilatational dynamics in physical systems.
• It is modeled via enriched continuum mechanics, tensorial formulations in magnetism, and dynamic couplings in hydrodynamics and optics.
• Applications include mechanical metamaterials, spintronic devices, optomechanics, nuclear structure, and active matter, enabling novel device designs.

Chiral inertia refers to the influence of structural or interaction-based chirality on the inertial properties and dynamical response of physical systems. In contrast to conventional inertia—which describes resistance to acceleration arising from mass distribution—chiral inertia encompasses nontrivial couplings between rotational, translational, and other degrees of freedom that are intrinsically handed (non-centrosymmetric). This concept has become essential in the characterization and design of advanced mechanical, magnetic, optical, and quantum materials and systems, where chirality can dramatically alter transport, wave propagation, and dynamic stability.

1. Chiral Inertia in Continuum Elasticity and Lattice Mechanics

In classical elasticity, inertial effects are strictly governed by mass density and are insensitive to microstructural handedness. However, non-centrosymmetric or chiral solids exhibit additional couplings that demand an enriched continuum description.

For 2D isotropic chiral solids and lattices, standard micropolar (Cosserat) theories omit chirality in-plane because the traditional tensor structures do not detect the sign of handedness under mirror reflection. By reinterpreting isotropic tensors—specifically, integrating the 2D in-plane Levi-Civita tensor as a second-order object—one can construct an isotropic fourth-order tensor that retains chiral sensitivity. The constitutive law then includes a single chiral material parameter $A$, which quantifies the coupling between bulk dilatational strain and internal (microrotation) degrees of freedom:

$$\sigma_{\alpha\beta} = \lambda \delta_{\alpha\beta} \epsilon_{\gamma\gamma} + (\mu+\kappa)\epsilon_{\alpha\beta} + (\mu-\kappa)\epsilon_{\beta\alpha} + A(\delta_{\alpha\beta} \epsilon_{3\gamma\delta} + \delta_{\gamma\beta} \epsilon_{3\alpha\delta})\epsilon_{\gamma\delta}$$

A nonzero $A$ means that pure rotations cause volumetric changes, and vice versa, manifesting as rotation–dilatation coupling. In the context of a triangular chiral lattice, homogenization yields $A=V_{Es} n (12-\cos^2B) \sec B$ in terms of geometric and material parameters, where $B$ is a topology parameter and $n$ is ligament slenderness (Liu et al., 2012).

This framework predicts unique behaviors such as induced lateral displacement and rotation under uniaxial load, and the mixing of longitudinal and shear waves—phenomena absent in non-chiral models.

2. Chiral Inertia in Magnetism and Magnetic Spin Dynamics

Magnetic systems provide a rich platform for the emergence of chiral inertia, particularly at ultrafast timescales and in non-centrosymmetric environments.

Magnetic Skyrmions and Solitons

Topological magnetic textures, such as skyrmion bubbles, display inertial mass arising from their boundary dynamics. Their equations of motion feature a gyrotropic (Magnus) term and a true inertial term:

$$-{\cal M} \ddot{R} + {\cal G} \times \dot{R} - {\cal K} R = 0$$

The mass ${\cal M} = \pi \bar{r} \rho$, with $\rho = g^2/\kappa$, is rooted in domain wall physics, and results in atypical center-of-mass orbits and frequency splitting (Makhfudz et al., 2012). Furthermore, edge excitations display chiral dynamics, where edge waves propagate with different velocities in opposite directions due to Berry phase effects, producing non-reciprocal dynamic responses.

Tensorial Magnetic Inertia

The inertial Landau–Lifshitz–Gilbert (ILLG) equation generalizes magnetic inertia as a tensor:

$$\Delta_{ij} = \eta \delta_{ij} + \mathcal{I}_{ij} + \epsilon_{ijk} C_k$$

where $\eta$ is scalar inertia, $\mathcal{I}_{ij}$ is symmetric and anisotropic, and $\epsilon_{ijk} C_k$ is the antisymmetric (chiral) part (Ghosh et al., 28 Aug 2024). The chiral component fundamentally alters the damping of nutational (THz) magnetization modes—broadening and downshifting resonance peaks, while leaving GHz precession modes unchanged. In two-sublattice ferromagnets, chiral inertia introduces non-reciprocal spin-wave dispersion $\omega(k) \neq \omega(-k)$, providing a mechanism for magnonic nonreciprocity independent of Dzyaloshinskii–Moriya interaction (Ghosh et al., 7 Sep 2025).

Domain Walls and Droplet Solitons

In chiral magnetic domain walls, inertia is tunable: acceleration time depends on spin Hall torque while deceleration time is set by DM interaction. This decouples dynamic time scales and enables controlled overshoot mitigation and low-energy device applications (Torrejon et al., 2015). In magnetic droplet solitons, chiral inertia manifests as sideband excitations resulting from the interplay between Oersted field-induced forces and spin-torque, leading to localized chiral oscillation modes (Mohseni et al., 2019).

3. Chiral Inertia in Hydrodynamics, Particle Dynamics, and Nonequilibrium Systems

Chiral inertia also arises in fluid, active, and granular systems via symmetry-breaking couplings between translation, rotation, and environmental interaction.

Chiral Angular Momentum Transfer

In viscous fluids, periodic but zero-average translation and rotation of a chiral object can yield persistent angular momentum transfer (mean torque) to the container. The nonzero time-averaged torque depends on the existence of a nonvanishing pseudoscalar constructed from the phase correlation of oscillatory displacement and rotation—mirroring chiral symmetry breaking in low–Reynolds–number flows (Moffatt et al., 2019).

Self-Propelled Active Matter

For active self-aligning agents with both translational and rotational inertia, the sign of self-alignment determines the emergence of chiral inertial dynamics. For negative self-alignment, inertia triggers a pitchfork bifurcation, producing spontaneous symmetry breaking into left- and right-handed (orbiting) motion. The presence of a torque bias selects one chirality, and environmental effects (e.g. wall collisions) introduce singular oscillatory responses that cannot be captured without considering inertial terms (Fersula et al., 31 Jan 2024). In classical chiral particle models, inertial effects alter the statistics of circles/swimming and can induce superdiffusive behavior when mass or friction evolves nontrivially in time (Sprenger et al., 2021).

Turbulent Multiparticle Systems

In turbulent flows seeded with heavy, chiral inertial particles, the particles' rotational bias induces net fluid vorticity for moderate turbulence. However, as turbulence intensity increases, background fluctuations dominate and chirality ceases to affect large-scale features of the flow (Piumini et al., 5 Apr 2024). Distinctive collision dynamics—long-lasting entanglement in chiral particles vs impulsive rebounds in spheres—point to chiral inertia affecting microscale but not macroscale statistics when turbulence is strong.

4. Chiral Inertia in Optical, Nuclear, and Exotic Systems

Optical Forces and Torques

Chirality in dipolar optical scatterers introduces cross-couplings between linear and angular momentum, captured by an imaginary chiral polarizability $\chi$. The optical force and torque thus acquire ‘crossed’ chiral terms:

$${\bf F}_\chi^{diss} = (\omega I_0/k^2)\Im[\chi]\hat{z},\quad {\bf N}_\chi = (k I_0/\omega)\Im[\chi]\hat{z}$$

Depending on the sign and magnitude of $\Im[\chi]$, these terms can reverse the direction of force or torque, giving rise to pulling forces and left-handed torques. In the multipolar regime, symmetry breaking from chirality and higher-order modes can lead to sign changes not predicted by the dipolar model (Canaguier-Durand et al., 2015).

Nuclear Chiral Bands

In triaxially deformed nuclei, chiral doublet bands arise from the tunneling (reorientation) between left- and right-handed angular momentum geometries. Chiral inertia, in this context, is inferred from the small energy splittings ($\Delta E$), smooth energy staggering $S(I)$, and similar kinematic and dynamic moments of inertia between partner bands. The degree of degeneracy reflects the barrier and "inertia" associated with tunneling in the chiral space (Xiong et al., 2018).

Quantum and Relativistic Media

In rotating, chirally imbalanced Fermi gases, chiral vortical conductivities of the vector and axial vector currents couple to the moment of inertia. Importantly, only the spin angular momentum contributes to the total moment of inertia in the free gas ($I_{\text{total}} = I_{\text{spin}}$, $I_{\text{orbital}}=0$), and the anomalous transport response is dictated by the chemical potentials and temperature (Ahadi et al., 26 Feb 2025). In many-body systems, rotational motion can restore chiral symmetry (phase transition), with the chirally restored phase displaying a higher rotational inertia (Chernodub et al., 2016).

5. Mathematical Formalism and Theoretical Approaches

Chiral inertia is formalized by introducing antisymmetric, pseudoscalar, or tensorial coupling constants into the governing continuum or discrete equations.

• In elasticity: the chiral modulus $A$ couples volumetric strain and microrotation in constitutive matrices, with matrix form off-diagonal blocks indicating chiral coupling (Liu et al., 2012).
• In magnetism: the inertia tensor $\Delta_{ij}$ is decomposed into isotropic, anisotropic, and antisymmetric (chiral) components, the latter yielding frequency- and polarization-dependent inertial responses in spin-wave or nutation dynamics (Ghosh et al., 28 Aug 2024, Ghosh et al., 7 Sep 2025).
• In hydrodynamics: mean angular momentum transfer is proportional to time-averaged pseudoscalars such as $\langle \omega(\tau)D(\tau)\rangle$ (Moffatt et al., 2019).
• In nuclear physics: the tunneling splitting $\Delta E$ and moments of inertia $\mathcal{J}^{(1)}, \mathcal{J}^{(2)}$ serve as proxies for chiral inertia.

A general feature is that chiral inertia often results in non-reciprocal (direction-dependent) wave propagation, mixed mode dispersion, spontaneous symmetry breaking, and handed mode selection—which are inaccessible in non-chiral analogs.

6. Applications, Significance, and Future Perspectives

Chiral inertia plays a crucial role in the design and operation of advanced functional materials and devices:

• Mechanical Metamaterials: Microstructural rotation–dilatation couplings, twist response, and size-dependent elastic behavior are directed by chiral inertia (Liu et al., 2012, Kadic et al., 2019).
• Magnetic and Spintronic Devices: Tunable inertial responses, nonreciprocal magnonic channels, and ultrafast THz spin dynamics are made possible by engineering scalar and chiral components of the inertia tensor (Torrejon et al., 2015, Ghosh et al., 28 Aug 2024, Ghosh et al., 7 Sep 2025).
• Optomechanics and Nanophotonics: Control of pulling forces and torques on chiral (nano)objects by adjusting optical fields and polarizability, exploited for sorting, manipulation, and the creation of left-handed devices (Canaguier-Durand et al., 2015).
• Nuclear Structure: Extraction of chiral inertia through doublet band splittings and EM transitions enhances understanding of quantum tunneling and collective motion in nuclei (Xiong et al., 2018).
• Active Matter and Robotics: Predictive control of chiral propulsion, directional dynamics in active swimmers, and inertial delays in robotics rely on understanding how chirality and inertia interplay (Fersula et al., 31 Jan 2024, Sprenger et al., 2021).
• Quantum and Relativistic Fluids: The identification that only spin contributes to chiral inertia in a free Fermi gas may inform studies of the quantum Barnett effect and anomalous transports in chiral media (Ahadi et al., 26 Feb 2025).

Future directions include refined experimental measurements of chiral inertia at nanoscopic and mesoscopic scales, multi-physics coupling in metamaterials, and the integration of chiral inertial design principles in quantum technologies, spintronic logic, and soft robotics. Quantitative modeling of the interplay between chiral inertia and dissipation, disorder, or topological protection remains an open and active field of research.