In Fig. 2 we illustrate the real and pseudo phonon angular momenta of four phonons in 2D hexagonal boron nitride that exhibit different features but are all axial. We are now left with the task of bringing the concepts of phonon chirality and axiality together, and determining when the presence of phonon angular momentum also breaks improper rotational symmetry.

Based on the two distinct concepts of chirality and axiality introduced in the previous sections, we propose that three classes of phonons can be distinguished: geometric chiral phonons, axial chiral phonons and axial achiral phonons (illustrated in Fig. 3).

In achiral crystals, the instantaneous atomic displacement pattern of certain phonons, described recently as geometric chiral phonons, breaks the improper rotational symmetry of the crystal structure. Here only the displacement, not the motion, of the atoms is relevant and no angular momentum is present, so J = 0. The atomic displacements induced by a geometric chiral phonon reduce the point-group symmetry of the system to one of the 11 chiral point groups, and each of the 21 achiral point groups possesses at least one irreducible representation that introduces such a symmetry lowering. It has been shown that geometric chiral phonons at the centre of the Brillouin zone induce nonhanded chirality, whereas geometric chiral phonons at the Brillouin zone boundary induce handed chirality. They can also create chiral phases through displacive phase transitions. This is similar to the soft-mode theory in ferroelectrics. The mirror image and the original can be transformed into one another by a reflection, (Fig. 3, left column). Recent works have predicted and demonstrated that geometric chiral phonons can be used to induce achiral-chiral phase transitions when excited with ultrashort laser pulses. In chiral crystals, improper rotational symmetry is already broken and geometric chiral phonons can no longer be distinguished from regular phonons, as any instantaneous atomic displacement pattern results in a crystal structure within a chiral point group. This is similar to the way Raman-active and infrared-active phonons can no longer be distinguished when inversion symmetry is broken in non-centrosymmetric crystals.

Improper rotational symmetry can also be broken by the circular polarization of axial chiral phonons. Helical phonons in 3D -- for which the scalar product of linear and angular momentum is non-zero, k ⋅ J ≠ 0, similar to circularly polarized light -- represent this case (Fig. 3, middle column). A reflection reverses the handedness of circular polarization with respect to the direction of motion, which flips the sign of the pseudoscalar, . In turn, time reversal cannot transform |L〉 and |R〉 into one another, because it flips both J and k, preserving the sign of the pseudoscalar, . This situation precisely resembles the case of motion-based true chirality that we discussed above. We note that helicity and chirality are different terms in particle physics and only coincide for massless particles, but the definition of motion-based chirality here does not require a relativistic treatment.

In contrast, cycloidal phonons propagating in the plane of circular polarization or long-wavelength phonons at the centre of the Brillouin zone (the Γ point) do not break improper rotational symmetry (Fig. 3, right column). Simple 180° rotations transform their mirror images into the original states, |L〉 = C|R〉, rendering both cases achiral. In both cases, the pseudoscalar k ⋅ J = 0, because the linear momentum is zero, k = 0 or because it is perpendicular to the angular momentum, k ⊥ J. Therefore, not all phonons carrying angular momentum are chiral, and we can call those cases axial achiral phonons.

In 2D systems, helical phonons are absent due to the lack of out-of-plane propagation, and the symmetry of axial phonons needs to be considered more carefully. In 2D space, there are no mirror operations along the out-of-plane axis, , rotations are strictly limited to the 2D plane, C ≡ C, and the angular momentum becomes a scalar, not vectorial, quantity. Accordingly, even cycloidal and circular Γ-point phonons are chiral, because they cannot be superposed onto their mirror images with the operations available in 2D space, |L〉 ≠ C|R〉 (Fig. 3, middle column). The two states can be transformed into one another by a time-reversal operation, , making them false chiral. Cycloidal phonons at the Brillouin zone corners (K and K' points) in hexagonal 2D materials were among the earliest identified chiral phonons. They appear in non-centrosymmetric materials and are non-degenerate, in contrast to circular Γ-point phonons that are typically degenerate. When applying the rules of 3D space to axial phonons in 2D materials, however, they behave just like their counterparts in bulk materials, meaning that cycloidal and Γ-point phonons are achiral (Fig. 3, right column).

We make this distinction between 2D and 3D space because phonons in 2D materials seem to obey rules from both dimensionalities. For example, the absence of longitudinal optical-transverse optical phonon splitting in monolayers of materials such as hexagonal boron nitride can only be explained by invoking the Coulomb interaction in 2D space in the screening process. In addition, the direction of angular momentum for axial 2D phonons is locked to the out-of-plane direction, which limits the phase space of phonon scattering processes in contrast to axial phonons in 3D isotropic materials, where it can continuously rotate. Note that we have only considered the linear and angular momentum of phonons and the dimensionality in the classification of axial phonons above, not the symmetry of the material itself. More specific cases may arise when doing so. For example, if mirror symmetry perpendicular to the plane of circular polarization is already broken in equilibrium, such as in ferroelectrics, chiral materials, Janus monolayers, or 2D materials with surface adsorbents, then even cycloidal and Γ-point phonons in 3D space will break improper rotational symmetry, resembling the scenario of false chirality from the rotating cone in Fig. 1.

We conclude that 3D axial phonons are true chiral when the scalar product of linear and angular momentum is finite, k ⋅ J ≠ 0 -- that is, when the phonons have a helical component. This pseudoscalar has the symmetry of an electric toroidal monopole, which has been suggested as a potential measure of chirality. At the same time, all circularly polarized phonons in 2D systems are false chiral when symmetry operations along the z axis are not applicable. For geometric chiral phonons, the specific change in crystal symmetry induced by the atomic displacement pattern must be considered, which can be handed or nonhanded in nature. With the above classifications, we have been able to provide a framework that encompasses all chiral and axial phonons described in the literature so far, and we will provide a variety of examples in the following.