Chiral Surface Phonons in Crystals

• Chiral surface phonons are surface-localized vibrational modes with circular or elliptical atomic orbits that produce nonzero angular momentum, even in centrosymmetric bulk crystals.
• First-principles DFT studies in NaCl slabs reveal distinct chiral branches with quantifiable in-plane magnetic moments arising from symmetry reduction at the surface.
• The universal emergence of chiral surface phonons opens new pathways for surface magnetism and phonon-mediated interactions in a broad range of crystalline materials.

Searching arXiv for the specified paper and closely related work on chiral and surface phonons. Search query: arXiv id (Pols et al., 7 Jun 2026) and topic "chiral surface phonons" Chiral surface phonons are surface-localized vibrational normal modes whose atomic displacements execute circular or elliptical orbits and therefore carry nonzero phonon angular momentum and helicity. In the formulation used for crystalline slabs, the mode angular momentum is written as $$\mathbf{J}_{\mathbf{q}\nu} = \sum_{\kappa}\hbar\langle \epsilon_{\mathbf{q}\nu,\kappa}|\hat{S}|\epsilon_{\mathbf{q}\nu,\kappa}\rangle$$, and chirality is diagnosed by a nonvanishing projection $\hat{\mathbf{q}\!\cdot\!\mathbf{J}_{\mathbf{q}\nu}\neq0$. The central result of "Chiral Surface Phonons" is that all surfaces of crystalline materials host surface phonons that are chiral, even when the bulk crystal is centrosymmetric, because the presence of a surface or interface locally breaks inversion and improper rotation symmetries (Pols et al., 7 Jun 2026).

1. Definition and scope

A phonon is called chiral when the atoms in a normal mode execute circular or elliptical orbits, thereby carrying nonzero phonon angular momentum. In the surface setting, the relevant degrees of freedom are the surface-localized eigenvectors, and chirality is tied to the handedness of their in-plane polarization pattern. A simple handedness parameter introduced for the in-plane displacement $\mathbf{u}(\mathbf{q})=(u_x,u_y)$ is

$$C(\mathbf{q}) \;=\; |u^{+}(\mathbf{q})|^2 \;-\; |u^{-}(\mathbf{q})|^2,$$

which is proportional to the expectation value of the circular-polarization operator

$$\hat{S} \;=\; |R\rangle\langle R| \;-\; |L\rangle\langle L|.$$

The microscopic phonon angular momentum per mode is

$$J^\alpha_{\mathbf{q}\nu} =\sum_\kappa\hbar\, \langle\epsilon_{\mathbf{q}\nu,\kappa}|\hat S^\alpha|\epsilon_{\mathbf{q}\nu,\kappa}\rangle,$$

and the associated helicity is $$\hat{\mathbf{q}\!\cdot\! \mathbf{J}_{\mathbf{q}\nu}}$$ (Pols et al., 7 Jun 2026).

The surface case is distinct from the better-known bulk case. Bulk chiral phonons require global inversion- or mirror-symmetry breaking, as in noncentrosymmetric or chiral crystals. By contrast, chiral surface phonons arise even in centrosymmetric bulk materials because the surface itself lowers the symmetry. This distinction is fundamental: it shifts chirality from a property restricted to special bulk crystal classes to a generic consequence of boundary formation. A common misconception is therefore that chiral phonons require a chiral bulk lattice. The surface result shows that this is not so (Pols et al., 7 Jun 2026).

2. Symmetry origin at crystalline boundaries

The symmetry argument is the conceptual core of the subject. In an ideal infinite rocksalt crystal with space group $Fm\bar3m$, inversion and all mirror planes are present, so bulk phonons at generic $\mathbf{q}$ have zero helicity. When a (001) surface is introduced, the symmetry is lowered to the layer group $P4/mmm$. In that reduction, inversion and some mirror operations whose mirror planes are perpendicular to the propagation direction are removed. As stated in the group-theory analysis, any surface or interface subgroup of an originally centrosymmetric space group will in general lack the improper operations that forcibly invert phonon helicity, $\hat{\mathbf{q}\!\cdot\!\mathbf{J}_{\mathbf{q}\nu}\neq0$0, so chiral solutions become symmetry-allowed (Pols et al., 7 Jun 2026).

The symmetry allowance is not uniform over the entire surface Brillouin zone. Along remnant mirror-invariant lines in the surface two-dimensional Brillouin zone, such as $\hat{\mathbf{q}\!\cdot\!\mathbf{J}_{\mathbf{q}\nu}\neq0$1–X or $\hat{\mathbf{q}\!\cdot\!\mathbf{J}_{\mathbf{q}\nu}\neq0$2–M, helicity must vanish because reflection flips $\hat{\mathbf{q}\!\cdot\!\mathbf{J}_{\mathbf{q}\nu}\neq0$3. Away from those lines, no symmetry forbids nonzero helicity. This produces a characteristic momentum-space structure: surface chirality is generic, but constrained on special high-symmetry lines. A plausible implication is that momentum-resolved probes should observe null helicity on mirror-invariant cuts and finite helicity off those cuts, rather than a uniform handedness everywhere in the surface Brillouin zone.

3. First-principles realization in rocksalt slabs

The principal first-principles model system is an 11-layer, (001)-oriented slab of NaCl, with vacuum $\hat{\mathbf{q}\!\cdot\!\mathbf{J}_{\mathbf{q}\nu}\neq0$4 Å to avoid spurious interactions. The slab space group is $\hat{\mathbf{q}\!\cdot\!\mathbf{J}_{\mathbf{q}\nu}\neq0$5. The same phenomenology is reported to extend to RbF and CsH (Pols et al., 7 Jun 2026).

The density-functional framework uses VASP with PAW and PBEsol, with $\hat{\mathbf{q}\!\cdot\!\mathbf{J}_{\mathbf{q}\nu}\neq0$6 eV. For bulk calculations, an $\hat{\mathbf{q}\!\cdot\!\mathbf{J}_{\mathbf{q}\nu}\neq0$7 $\hat{\mathbf{q}\!\cdot\!\mathbf{J}_{\mathbf{q}\nu}\neq0$8-mesh is used, and the cell is relaxed to $\hat{\mathbf{q}\!\cdot\!\mathbf{J}_{\mathbf{q}\nu}\neq0$9 meV and $\mathbf{u}(\mathbf{q})=(u_x,u_y)$0 meV/Å. For slabs, a $\mathbf{u}(\mathbf{q})=(u_x,u_y)$1 $\mathbf{u}(\mathbf{q})=(u_x,u_y)$2-mesh is used and only the atoms are relaxed. Phonons are obtained by finite displacements of $\mathbf{u}(\mathbf{q})=(u_x,u_y)$3 Å using phonopy, with non-analytical corrections for dipole–dipole interactions. Bulk interatomic force constants are derived from a $\mathbf{u}(\mathbf{q})=(u_x,u_y)$4 supercell and slab force constants from a $\mathbf{u}(\mathbf{q})=(u_x,u_y)$5 supercell. Surface-mode dispersions are plotted on a $\mathbf{u}(\mathbf{q})=(u_x,u_y)$6 $\mathbf{u}(\mathbf{q})=(u_x,u_y)$7-mesh (Pols et al., 7 Jun 2026).

These details matter because the effect being isolated is subtle: the chiral character is not introduced by ad hoc symmetry breaking in the electronic structure calculation, but emerges from a slab geometry whose reduced symmetry is treated explicitly. The methodology therefore ties the universality claim to a concrete DFT workflow rather than to a purely abstract symmetry argument alone.

4. Surface localization, mode structure, and handed motion

Surface character is quantified through the surface localization index

$\mathbf{u}(\mathbf{q})=(u_x,u_y)$8

with $\mathbf{u}(\mathbf{q})=(u_x,u_y)$9 signifying modes confined almost entirely to the outermost atomic layers (Pols et al., 7 Jun 2026).

In NaCl slabs, surface branches occur at the low-frequency end of each bulk subband. A highly localized mode appears near X at approximately $$C(\mathbf{q}) \;=\; |u^{+}(\mathbf{q})|^2 \;-\; |u^{-}(\mathbf{q})|^2,$$0 THz with transverse-optical character, and another occurs at approximately $$C(\mathbf{q}) \;=\; |u^{+}(\mathbf{q})|^2 \;-\; |u^{-}(\mathbf{q})|^2,$$1 THz, arising from the heavier Cl ions. The atomic displacement patterns distinguish the two branches sharply: at $$C(\mathbf{q}) \;=\; |u^{+}(\mathbf{q})|^2 \;-\; |u^{-}(\mathbf{q})|^2,$$2 THz, Na$$C(\mathbf{q}) \;=\; |u^{+}(\mathbf{q})|^2 \;-\; |u^{-}(\mathbf{q})|^2,$$3 ions on the top layer traverse right-handed circles while Cl ions are nearly stationary; at $$C(\mathbf{q}) \;=\; |u^{+}(\mathbf{q})|^2 \;-\; |u^{-}(\mathbf{q})|^2,$$4 THz, Cl$$C(\mathbf{q}) \;=\; |u^{+}(\mathbf{q})|^2 \;-\; |u^{-}(\mathbf{q})|^2,$$5 ions execute left-handed loops while Na ions are stationary. At the bottom surface, the handedness is strictly reversed, so the global slab motion remains overall achiral (Pols et al., 7 Jun 2026).

Mode	Frequency	Reported displacement pattern
Surface TO-like branch	$$C(\mathbf{q}) \;=\; |u^{+}(\mathbf{q})|^2 \;-\; |u^{-}(\mathbf{q})|^2,$$6 THz	Top-layer Na$$C(\mathbf{q}) \;=\; |u^{+}(\mathbf{q})|^2 \;-\; |u^{-}(\mathbf{q})|^2,$$7 ions traverse right-handed circles
Surface Cl-dominated branch	$$C(\mathbf{q}) \;=\; |u^{+}(\mathbf{q})|^2 \;-\; |u^{-}(\mathbf{q})|^2,$$8 THz	Cl$$C(\mathbf{q}) \;=\; |u^{+}(\mathbf{q})|^2 \;-\; |u^{-}(\mathbf{q})|^2,$$9 ions execute left-handed loops

The reversal between top and bottom surfaces is important for interpretation. In a symmetric slab, local surface chirality does not imply that the entire finite slab has a net global chirality. This directly addresses a second common misconception: the existence of chiral surface phonons does not require the slab as a whole to be structurally or dynamically achiral. Rather, opposite terminations can host opposite handedness while preserving an overall achiral global motion.

5. Phonon-induced magnetism at surfaces

Because circular ionic motion forms microscopic current loops, chiral surface phonons carry an orbital phonon magnetic moment in the dynamical multiferroicity picture: $$\hat{S} \;=\; |R\rangle\langle R| \;-\; |L\rangle\langle L|.$$0 For the NaCl slab, these magnetic moments are strictly in-plane because the ions rotate about in-plane axes. The $$\hat{S} \;=\; |R\rangle\langle R| \;-\; |L\rangle\langle L|.$$1 THz chiral surface mode yields the largest local $$\hat{S} \;=\; |R\rangle\langle R| \;-\; |L\rangle\langle L|.$$2, on the order of $$\hat{S} \;=\; |R\rangle\langle R| \;-\; |L\rangle\langle L|.$$3–$$\hat{S} \;=\; |R\rangle\langle R| \;-\; |L\rangle\langle L|.$$4 per unit cell as a lower-bound DFT estimate. The $$\hat{S} \;=\; |R\rangle\langle R| \;-\; |L\rangle\langle L|.$$5 THz mode gives a smaller $$\hat{S} \;=\; |R\rangle\langle R| \;-\; |L\rangle\langle L|.$$6 of the same in-plane orientation, despite opposite ionic charge circulation (Pols et al., 7 Jun 2026).

The resulting magnetization density is described as a pair of oppositely oriented magnetic sheets, one on each surface. In continuum form,

$$\hat{S} \;=\; |R\rangle\langle R| \;-\; |L\rangle\langle L|.$$7

localized within the outermost atomic planes (Pols et al., 7 Jun 2026). This identifies a route to surface magnetism in nominally nonmagnetic crystals. A plausible implication is that surface-sensitive magnetic probes may register phonon-induced contributions that are absent in corresponding bulk analyses.

6. Generality, experimental access, and relation to other chiral-phonon settings

The general claim is explicit: any crystal surface or interface breaks inversion and improper rotations locally, leading to universal chiral surface phonons in all materials (Pols et al., 7 Jun 2026). Within the rocksalt examples, similar behavior is found in RbF and CsH slabs; the acoustic–optical gap grows with greater mass difference, but chiral surface branches and in-plane $$\hat{S} \;=\; |R\rangle\langle R| \;-\; |L\rangle\langle L|.$$8 persist. The proposed probes are surface-sensitive inelastic scattering, including He-atom scattering and EELS, to map the chiral dispersion, and nanoscale magnetometry, including NV-center scanning and scanning SQUID, to detect the tiny in-plane magnetic sheets (Pols et al., 7 Jun 2026).

A useful comparison comes from "Coexistent Topological and Chiral Phonons in Chiral RhGe: An ab initio study," which treats a different regime of chiral phonon physics. In RhGe, the crystal itself is chiral, and the phonon spectrum hosts topological bulk degeneracies, nontrivial surface states, and momentum-resolved phonon angular momentum on surface arcs. On the (001) surface at $$\hat{S} \;=\; |R\rangle\langle R| \;-\; |L\rangle\langle L|.$$9 THz, representative arc modes have $$J^\alpha_{\mathbf{q}\nu} =\sum_\kappa\hbar\, \langle\epsilon_{\mathbf{q}\nu,\kappa}|\hat S^\alpha|\epsilon_{\mathbf{q}\nu,\kappa}\rangle,$$0 values of $$J^\alpha_{\mathbf{q}\nu} =\sum_\kappa\hbar\, \langle\epsilon_{\mathbf{q}\nu,\kappa}|\hat S^\alpha|\epsilon_{\mathbf{q}\nu,\kappa}\rangle,$$1, $$J^\alpha_{\mathbf{q}\nu} =\sum_\kappa\hbar\, \langle\epsilon_{\mathbf{q}\nu,\kappa}|\hat S^\alpha|\epsilon_{\mathbf{q}\nu,\kappa}\rangle,$$2, and $$J^\alpha_{\mathbf{q}\nu} =\sum_\kappa\hbar\, \langle\epsilon_{\mathbf{q}\nu,\kappa}|\hat S^\alpha|\epsilon_{\mathbf{q}\nu,\kappa}\rangle,$$3, with corresponding $$J^\alpha_{\mathbf{q}\nu} =\sum_\kappa\hbar\, \langle\epsilon_{\mathbf{q}\nu,\kappa}|\hat S^\alpha|\epsilon_{\mathbf{q}\nu,\kappa}\rangle,$$4 values of $$J^\alpha_{\mathbf{q}\nu} =\sum_\kappa\hbar\, \langle\epsilon_{\mathbf{q}\nu,\kappa}|\hat S^\alpha|\epsilon_{\mathbf{q}\nu,\kappa}\rangle,$$5, $$J^\alpha_{\mathbf{q}\nu} =\sum_\kappa\hbar\, \langle\epsilon_{\mathbf{q}\nu,\kappa}|\hat S^\alpha|\epsilon_{\mathbf{q}\nu,\kappa}\rangle,$$6, and $$J^\alpha_{\mathbf{q}\nu} =\sum_\kappa\hbar\, \langle\epsilon_{\mathbf{q}\nu,\kappa}|\hat S^\alpha|\epsilon_{\mathbf{q}\nu,\kappa}\rangle,$$7 $$J^\alpha_{\mathbf{q}\nu} =\sum_\kappa\hbar\, \langle\epsilon_{\mathbf{q}\nu,\kappa}|\hat S^\alpha|\epsilon_{\mathbf{q}\nu,\kappa}\rangle,$$8. In that system, modes with $$J^\alpha_{\mathbf{q}\nu} =\sum_\kappa\hbar\, \langle\epsilon_{\mathbf{q}\nu,\kappa}|\hat S^\alpha|\epsilon_{\mathbf{q}\nu,\kappa}\rangle,$$9 are right-circularly polarized and couple preferentially to right-circularly polarized light, while $$\hat{\mathbf{q}\!\cdot\! \mathbf{J}_{\mathbf{q}\nu}}$$0 modes couple to left-circularly polarized light (Reddy et al., 2024).

The contrast is instructive. In RhGe, coexistence of topological and chiral phonons is tied to a chiral bulk space group and double-Weyl phonon nodes. In chiral surface phonons of the NaCl-type slab, surfaces alone suffice. This suggests two complementary research directions rather than competing explanations: one in which chirality is inherited from bulk crystal class and intertwined with topological band structure, and another in which chirality is generated generically by boundary-induced symmetry reduction. The latter broadens the domain of chiral phonon phenomena to essentially all crystalline surfaces and interfaces (Pols et al., 7 Jun 2026).