TaSi: Chiral Topological Material

• TaSi is a non‑symmorphic chiral topological material characterized by multifold degeneracies in both its electronic and phonon spectra.
• Its electronic structure hosts a fourfold spin‑3/2 Rarita–Schwinger fermion, a sixfold double spin‑1 excitation, and Weyl nodes along high‑symmetry lines.
• Topological phonon modes with Chern numbers ±2 in distinct frequency windows yield chiral bosonic edge states with promising THz and transport applications.

Searching arXiv for papers on TaSi to ground the article in current literature. TaSi is a non‑symmorphic chiral topological material crystallizing in space group $P\,2_1\,3$ (No. 198), whose electronic and phononic structures have been analyzed by first‑principles calculations in the 2025 study "Electron and phonon topology in transition metal material TaSi" (Sen et al., 15 Jul 2025). In the presence of spin orbit coupling, the electronic bands host a fourfold spin‑$3/2$ Rarita–Schwinger fermion at $\Gamma$, a sixfold “double spin‑1” excitation at $R$, and spin‑$1/2$ Weyl nodes along $\Gamma$–$R$, while the phonon spectrum hosts chiral bosonic excitations with Chern numbers $C=\pm 2$. The concurrent presence of chiral fermions and chiral phonons places TaSi within the broader class of space‑group‑198 chiral materials while distinguishing it as a single system in which both electronic and bosonic multifold topology are present.

1. Crystal symmetry and structural setting

TaSi crystallizes in the non‑centrosymmetric, chiral cubic space group $P\,2_1\,3$ (No. 198), with point group $23$ ($3/2$0). This space group is non‑symmorphic, hosting three $3/2$1 screw axes and threefold rotations. The key symmetry elements are the $3/2$2 screws $3/2$3, $3/2$4, $3/2$5 and a threefold rotation $3/2$6. The primitive cell contains 8 atoms $3/2$7; the study does not report the lattice constants or Wyckoff positions (Sen et al., 15 Jul 2025).

The absence of inversion symmetry and the presence of these non‑symmorphic operations enforce multifold degeneracies at time‑reversal invariant momenta $3/2$8 and $3/2$9 and along $\Gamma$0–$\Gamma$1 lines, while the chiral nature of the lattice allows net Chern‑number‑carrying nodes at high‑symmetry points. In the spinless case, the symmetry analysis already fixes the qualitative band topology. Without SOC, at $\Gamma$2, commuting screw symmetries $\Gamma$3 and $\Gamma$4, with $\Gamma$5, together with a nontrivial action of $\Gamma$6 on simultaneous eigenstates, yield a protected threefold crossing. Without SOC at $\Gamma$7, $\Gamma$8 and $\Gamma$9 anticommute, $R$0, square to $R$1, and with $R$2 generate a protected fourfold “double spin‑$R$3” crossing.

With SOC, $R$4 changes the allowed degeneracy structure. The same non‑symmorphic and cubic symmetries then protect a fourfold spin‑$R$5 Rarita–Schwinger node at $R$6, a sixfold “double spin‑1” fermion at $R$7, and a spin‑$R$8 Weyl node along $R$9–$1/2$0. In TaSi, symmetry is therefore not a secondary feature of the analysis; it is the organizing principle that determines which multifold excitations can appear and where they occur in the Brillouin zone.

2. Electronic multifold topology

The SOC band structure contains three electronically distinct topological features near the Fermi level: a fourfold RSW node at $1/2$1, a sixfold node at $1/2$2, and eight Weyl nodes along $1/2$3–$1/2$4. The study identifies the $1/2$5 node at $1/2$6 relative to $1/2$7, with Chern number $1/2$8 and positive chirality; the $1/2$9 node at $\Gamma$0, also with $\Gamma$1 and positive chirality; and representative $\Gamma$2 crossings along $\Gamma$3–$\Gamma$4 at $\Gamma$5, each carrying $\Gamma$6 and negative chirality. In the first Brillouin zone there are two multifold nodes and eight $\Gamma$7 Weyl nodes, so the total chiral charge is

$\Gamma$8

consistent with the Nielsen–Ninomiya theorem (Sen et al., 15 Jul 2025).

Sector	Node	Key data
Electronic	$\Gamma$9 fourfold RSW	$R$0, $R$1, positive chirality
Electronic	$R$2 sixfold “double spin‑1”	$R$3, $R$4, positive chirality
Electronic	$R$5 along $R$6–$R$7	$R$8, each $R$9; eight nodes total
Phononic	$C=\pm 2$0 spin‑1 phonon	windows $C=\pm 2$1–$C=\pm 2$2 and $C=\pm 2$3–$C=\pm 2$4, $C=\pm 2$5
Phononic	$C=\pm 2$6 charge‑2 Dirac phonon	same windows, $C=\pm 2$7

The local dispersions are represented by standard $C=\pm 2$8 Hamiltonians adapted to the multifold setting. For a spin‑$C=\pm 2$9 Weyl node, the paper uses

$P\,2_1\,3$0

or equivalently $P\,2_1\,3$1, with $P\,2_1\,3$2 the chirality. For the RSW fermion,

$P\,2_1\,3$3

with $P\,2_1\,3$4 the spin‑$P\,2_1\,3$5 matrices. For the threefold spin‑1 fermion, the representative form is

$P\,2_1\,3$6

with $P\,2_1\,3$7 the spin‑1 matrices. These linearized models encode the multifold dispersions enabled by SG 198 non‑symmorphic symmetry.

The SOC case shows clean, linearly dispersing crossings near $P\,2_1\,3$8 and $P\,2_1\,3$9, with an isolated $23$0 along $23$1–$23$2. Because the nodes lie within approximately $23$3 of $23$4, the study identifies them as accessible to angle‑resolved photoemission spectroscopy. This energy placement is not itself a topological invariant, but it is decisive for whether the multifold structure can be interrogated directly in experiment.

3. Phonon topology and chiral bosonic excitations

The primitive cell contains 8 atoms, so the phonon spectrum consists of 24 branches: 3 acoustic and 21 optical. Density functional perturbation theory yields no imaginary modes, which confirms dynamical stability. Within this stable phonon spectrum, TaSi hosts topological phonon modes in two frequency windows, $23$5–$23$6 and $23$7–$23$8, with a spin‑1 Weyl phonon at $23$9 and a “charge‑2 Dirac” phonon at $3/2$00 (Sen et al., 15 Jul 2025).

The chiral phonons carry Chern numbers $3/2$01. At $3/2$02, the threefold phonon acts as a source or sink depending on branch orientation; at $3/2$03, the fourfold phonon carries Chern magnitude $3/2$04 with opposite sign relative to $3/2$05. The same non‑symmorphic screw axes and cubic rotations that protect the electronic multifold crossings also protect the phonon degeneracies, but now in spinless bosonic bands and without SOC. This symmetry parallel between electron and phonon sectors is one of the central structural features of TaSi.

Phonon Berry curvature is computed from the eigenvectors of the dynamical matrix. The study states that one may write the phonon Berry connection as

$3/2$06

with appropriately mass‑normalized polarization vectors $3/2$07, and the corresponding Berry curvature as

$3/2$08

The resulting Berry‑flux patterns show $3/2$09 and $3/2$10 acting as source and sink with $3/2$11, and the surface spectral functions exhibit two phonon “Fermi arcs” connecting $3/2$12 to $3/2$13 in each of the two highlighted frequency windows.

A plausible implication is that the phononic sector of TaSi is not merely an analog of the electronic sector but an additional topological channel with its own surface transport signatures in the THz regime. The paper frames this coexistence explicitly as a basis for coupled electron–phonon topological phenomena.

4. Berry curvature, Chern numbers, and surface states

The topological analysis is organized through the standard Berry‑phase objects used for both electronic and phononic bands. For electronic bands, the Berry connection is

$3/2$14

the Berry curvature is

$3/2$15

and the Chern number on a closed surface $3/2$16 is

$3/2$17

In TaSi, Berry curvature was computed using a Wannier tight‑binding model. The resulting distribution on the $3/2$18 plane shows monopole‑like sources at $3/2$19 and $3/2$20, each with $3/2$21, and sinks with $3/2$22 along $3/2$23–$3/2$24. The paper states that Chern numbers are inferred from Berry flux patterns and arc counting (Sen et al., 15 Jul 2025).

The surface manifestations of these bulk invariants are explicit. On the $3/2$25 electronic surface projection, four chiral Fermi arcs emanate from $3/2$26, consistent with the $3/2$27 RSW node. The chiral sense of these arcs is visible in the surface spectral function obtained by iterative Green’s functions. In the phonon sector, the $3/2$28 surface states are traced along $3/2$29, where $3/2$30 and $3/2$31 are the projections of bulk $3/2$32 and $3/2$33. In both topological phonon frequency windows, chiral surface modes emanate from $3/2$34 and connect to $3/2$35.

These surface states serve two roles. First, they provide a boundary‑space realization of the bulk Chern numbers. Second, they supply experimentally tractable observables: electronic Fermi arcs in photoemission and phonon arcs in momentum‑resolved probes. In this sense, TaSi exemplifies a system in which bulk multifold topology and boundary arc structure are tightly aligned across both fermionic and bosonic spectra.

5. Computational framework and experimental access

The electronic structure calculations use VASP with the projector augmented wave method and GGA‑PBE for exchange–correlation. SOC is explicitly included for the electronic band structures and topological analysis. The plane‑wave cutoff is $3/2$36 the default value, with a nominal default of $3/2$37; the $3/2$38-point mesh is $3/2$39 Monkhorst–Pack; structural relaxation is carried out until forces converge to $3/2$40 and the total energy threshold reaches $3/2$41. Maximally localized Wannier functions from Wannier90 are then used to construct tight‑binding models, and surface states and Fermi arcs are computed by iterative Green’s functions via WannierTools (Sen et al., 15 Jul 2025).

For phonons, the study uses DFPT via Phonopy with a $3/2$42 supercell and lattice constants larger than $3/2$43. Phonon topological analysis and surface spectral functions are obtained with PhonopyTB linked to WannierTools. This division of labor—DFT/MLWF for electrons and DFPT/dynamical‑matrix eigenvectors for phonons—matches the dual electronic and bosonic scope of the work.

The experimental program proposed in the paper is correspondingly multi‑channel. ARPES is identified as suitable for resolving the multifold nodes at $3/2$44, $3/2$45, and along $3/2$46–$3/2$47; circular dichroism in ARPES is proposed to probe chirality. Magneto‑transport under $3/2$48 is proposed to test the chiral anomaly, and optical experiments under circularly polarized light are proposed to test the circular photogalvanic effect. For phonons, inelastic neutron scattering and THz Raman/IR spectroscopy are identified as probes of the chiral phonon modes and phonon surface arcs in the $3/2$49–$3/2$50 and $3/2$51–$3/2$52 windows. Because the nodes lie within roughly $3/2$53 of $3/2$54, the study states that ARPES accessibility is favorable, although synthesis details are not provided.

6. Response phenomena, materials context, and possible tuning

The paper presents transport and response phenomena as expected implications of the electronic and phononic topology rather than as experimentally established properties. In the electronic sector, it states that in the presence of time‑reversal symmetry the net integrated Berry curvature vanishes, but large local Berry curvature near nodes can yield strong gyrotropic and optical responses; breaking TRS, for example by magnetic field, can induce an anomalous Hall effect. As a Weyl semimetal, TaSi is expected to show negative magnetoresistance under parallel $3/2$55 and $3/2$56 fields due to the chiral anomaly. For circularly polarized light, an isolated chiral node can generate a quantized circular photogalvanic effect with diagonal tensor component

$3/2$57

so the $3/2$58 RSW and sixfold nodes provide a route to enhanced photocurrent when a single node dominates optical transitions. Strong Berry curvature and lack of inversion are also identified as a basis for second‑order responses such as shift current and nonlinear Hall effects (Sen et al., 15 Jul 2025).

In the phononic sector, the paper states that chiral phonons with $3/2$59 can contribute to a phonon Hall effect in the presence of TRS breaking, via magnetic field or spin–phonon coupling, with $3/2$60 sensitive to Berry curvature in the two topological frequency windows. It also proposes nonreciprocal phonon edge modes as a route to directional heat flow and phonon waveguiding in the THz regime. Thermoelectric enhancement and unconventional superconductivity are discussed conceptually in terms of the interplay between topological electrons and phonons.

Within the SG 198 family, TaSi is compared with CoSi, RhSi, AlPt, and FeSi. It shares the characteristic multifold electronic structure of chiral topological materials in this space group—RSW fermions at $3/2$61, sixfold fermions at $3/2$62, and Weyl nodes along $3/2$63–$3/2$64—and its phonon topology parallels predictions and observations in the MSi family. The study notes, however, that the node energies in TaSi lie approximately $3/2$65–$3/2$66 from $3/2$67, whereas in CoSi and RhSi the nodes are often closer to $3/2$68. Its distinguishing feature is therefore not simply the presence of multifold fermions, but the concurrent presence of chiral fermions and chiral phonons in one material.

The paper does not report strain, pressure, doping, or alloying studies. It does state more generally that, in SG 198 systems, strain or hydrostatic pressure can shift node energies toward $3/2$69 and tune the velocities $3/2$70 in the $3/2$71 Hamiltonians, while isoelectronic alloying such as $3/2$72 may move the chemical potential without altering symmetry. This suggests a potential route toward bringing TaSi’s nodes closer to experimentally optimal energies, although that possibility remains prospective rather than demonstrated for TaSi itself.