Chiral phononic and electronic edge modes of EuPtSi

Systems with P2${1}$3 symmetry are characterized by the realization of chiral edge modes, propagating in one direction along closed loops around some high symmetry points of the Brillouin zone. We study the phononic and electronic properties of EuPtSi, which crystallizes with P2${1}$3 symmetry. EuPtSi is also characterized by intriguing magnetic properties, such as the realization of the skyrmion lattice. Here, using ab initio techniques, we study bulk and slab properties of EuPtSi. The bulk phononic and electronic band structures exhibit a spin-1 Weyl point and a charge-2 Dirac point at the $Γ$ and R points, respectively. Consequently, the surface states exhibit chiral edge modes. Such features are present in both the phononic and electronic surface spectra. The chiral phononic edge mode is associated with the vibration of the atoms in close vicinity, while the chiral electronic surface states correspond to carrier accumulation at the edge of chiral atomic chains.

💡 Research Summary

The paper presents a comprehensive first‑principles investigation of the chiral intermetallic compound EuPtSi, which crystallizes in the non‑centrosymmetric space group P2₁3 (No. 198). Using density‑functional theory (DFT) with the projector‑augmented wave method as implemented in VASP, the authors optimized the lattice constant (a = 6.429 Å) and internal atomic coordinates, finding excellent agreement with experimental data. Phonon calculations performed with Phonopy on a 2 × 2 × 2 supercell reveal no imaginary frequencies, confirming dynamical stability.

The phonon dispersion exhibits three distinct frequency regions: low‑frequency acoustic modes (<5 THz), a middle band around 6 THz, and high‑frequency optical modes near 12 THz. Heavy Eu and Pt atoms dominate the low‑frequency vibrations, while Si contributes mainly to the middle and high‑frequency branches. Crucially, symmetry analysis shows a three‑fold degenerate spin‑1 Weyl point at the Brillouin‑zone center (Γ) and a four‑fold charge‑2 Dirac point at the R point. Both points carry opposite Chern numbers (±2), ensuring that the total topological charge in the Brillouin zone sums to zero.

Electronic structure calculations were carried out both with Eu 4f electrons treated as core states and as valence states within a DFT+U (U = 6 eV) framework. The localized Eu 4f manifold lies well below the Fermi level (≈ −1.75 eV) and does not significantly hybridize with the conduction bands. In the absence of spin‑orbit coupling (SOC), the electronic bands also display a spin‑1 Weyl point at Γ and a charge‑2 Dirac point at R, mirroring the phononic situation. Inclusion of SOC splits the spin‑1 Weyl point into a doubly degenerate node and a four‑fold degenerate node with Chern number +4, while the R‑point evolves into a six‑fold fermion (Chern = −4) plus a trivial doublet. The resulting Fermi surface consists of deformed spherical pockets centered at Γ (electron‑like) and R (hole‑like), in line with de Haas–van Alphen measurements. Pt d‑states dominate the density of states near the Fermi level, with modest contributions from Eu d and Si p orbitals.

Surface states were investigated using maximally localized Wannier functions (Wannier90) and the iterative Green’s‑function method (Wannier‑Tools). For the (001) termination, the phononic surface spectral function shows chiral edge modes that form closed loops around the projected Γ̄ and M̄ points at frequencies corresponding to the bulk spin‑1 Weyl (≈ 12 THz) and charge‑2 Dirac (≈ 6 THz) points. These modes are localized vibrations of atoms near the surface and propagate unidirectionally along the loop, a hallmark of topological phonons. The electronic surface spectral function reveals analogous chiral edge states at energies near the bulk Weyl/Dirac points. These electronic edge modes manifest as charge accumulation along the edges of the chiral atomic chains that run parallel to the