Unconventional Chiral Fermions and Large Topological Fermi Arcs in RhSi

The theoretical proposal of chiral fermions in topological semimetals has led to a significant effort towards their experimental realization. In particular, the Fermi surfaces of chiral semimetals carry quantized Chern numbers, making them an attractive platform for the observation of exotic transport and optical phenomena. While the simplest example of a chiral fermion in condensed matter is a conventional $|C|=1$ Weyl fermion, recent theoretical works have proposed a number of unconventional chiral fermions beyond the standard model which are protected by unique combinations of topology and crystalline symmetries. However, materials candidates for experimentally probing the transport and response signatures of these unconventional fermions have thus far remained elusive. In this Letter, we propose the RhSi family in space group No. 198 as the ideal platform for the experimental examination of unconventional chiral fermions. We find that RhSi is a filling-enforced semimetal that features near its Fermi surface a chiral double sixfold-degenerate spin-1 Weyl node at $R$ and a previously uncharacterized fourfold-degenerate chiral fermion at $\mathrm{\Gamma }$. Each unconventional fermion displays Chern number $\pm 4$ at the Fermi level. We also show that RhSi displays the largest possible momentum separation of compensative chiral fermions, the largest proposed topologically nontrivial energy window, and the longest possible Fermi arcs on its surface. We conclude by proposing signatures of an exotic bulk photogalvanic response in RhSi.

Figure 1

Lattice and electronic structure of RhSi in SG 198. (a) Crystal structure of RhSi. Each unit cell contains four Rh and four Si atoms lying at Wyckoff positions with the minimum multiplicity of SG 198. (b) The cubic bulk BZ of RhSi. (c) Band structure of RhSi in the absence of spin-orbit coupling (SOC). The highest valance and lowest conduction bands are colored in blue and red, respectively. (d) Band structure in the presence of SOC. A chiral double spin-1 Weyl point sits $\sim 0.4\mathrm{eV}$ below the Fermi energy at $R$, and a previously uncharacterized fourfold-degenerate chiral fermion lies at the Fermi energy at $\mathrm{\Gamma }$.

Figure 2

Energy dispersions and chiral character of the fourfold- and sixfold-degenerate unconventional fermions in RhSi. (a)–(c) 3D energy dispersions of the degeneracies at $\mathrm{\Gamma }$, $R$, and $M$, respectively. (d)–(f) Band structures in the vicinities of $\mathrm{\Gamma }$, $R$, and $M$, respectively. Because of the local Kramers theorem enforced under the combined operation of $(s_{2x,y,z}\times \mathcal{T}{)}^2=-1$, bands along $k_{x,y,z}=\pi$ are twofold-degenerate (e),(f). The absence of rotoinversion symmetries in SG 198 allows for nodes at TRIMs to have nontrivial Chern numbers; nodes with multiple finite-$q$ gaps can exhibit different Chern numbers occupying bands up to each gap (Supplemental Material, Sec. F [39]) (d)–(f). At the Fermi energy, the fourfold-degenerate fermion at $\mathrm{\Gamma }$ has Chern number $+4$, and the double spin-1 Weyl at $R$ has Chern number $-4$. The quadratic fourfold-degenerate crossing at $M$ (f) also exhibits a Chern number, but the bands dispersing from it are almost entirely covered by the Fermi energy. The fourfold-degenerate unconventional fermion at $\mathrm{\Gamma }$ (d) can be considered the combination of two $J=1/2$ and two $J=3/2$ states pinned together by time-reversal and screw symmetries. The analogous angular momentum eigenvalues for each band can then be deduced by observing the band eigenvalues of $C_{3,111}$ and considering the symmetry-allowed term $\overrightarrow{k}·\overrightarrow{J}$ (Supplemental Material, Sec. C2 [39]). (g),(h) The Berry curvature $\overrightarrow{\mathrm{\Omega }}$ on the $k_x=k_y$ plane flows almost directly from $\mathrm{\Gamma }$ to $R$ with minimal out-of-plane deviations. Measuring the intensity of the $xy$ (g) and $z$ (h) components of $\overrightarrow{\mathrm{\Omega }}$, we verify that $\mathrm{\Gamma }$ and $R$ exhibit the local vector fields of $C=\pm 4$ hedgehog defects.

Figure 3

Surface state texture of RhSi. (a) The (001)-surface states of RhSi calculated using surface Green’s functions (Supplemental Material, Sec. A [39]). Four Fermi arcs radiate at $\overline{\mathrm{\Gamma }}$ from the projection of the bulk fourfold-degenerate fermion at $\mathrm{\Gamma }$, grouping into two time-reversed pairs and spiraling around the BZ (c) until they meet at $\overline{M}$ at the projection of the bulk double spin-1 Weyl at $R$. (b) The surface states demonstrate $\sim 80\%$ spin polarization (Supplemental Material, Sec. A [39]). (d) An allowed simplified Fermi arc connectivity. For both possible connectivities (c),(d), plotting the surface bands along a clockwise loop surrounding $\overline{\mathrm{\Gamma }}$ [red loop in (a), dashed loop in (d)], the surface bands (e) demonstrate a $C=+4$ spectral flow. (f) Conversely, taking a loop along the zone-spanning dashed line at $k_y=0.15$ results in a surface state texture with just $C=+2$ spectral flow, as only two Fermi arcs cross each half of the surface BZ.

Figure 4

Quantized circular photogalvanic effect (CPGE) of the fourfold-degenerate unconventional fermion in RhSi. (a),(b) The CPGE of a single conventional Weyl node as proposed in Ref. [33]. (b) Calculations of the CPGE for the fourfold unconventional fermion at $\mathrm{\Gamma }$ in RhSi, tuned to half filling. The photocurrent rate saturates at 4 times the value it did for the conventional Weyl in (a), as the Chern number in this gap is 4 times as large. (c) The more realistic case of a partial occupation of this fourfold fermion; multiple transitions contribute to the photocurrent. (d) Contributions to the traced photocurrent rate from each transition in (c), calculated from the fitted TB model (Supplemental Material, Sec. C 3 [39]). Trend lines in (d) are labeled by the color of their contributing transition in (c), with pink representing the overall photocurrent rate.