Chiral optical response of multifold fermions

Multifold fermions are generalizations of twofold degenerate Weyl fermions with three-, four-, six-, or eightfold degeneracies protected by crystal symmetries, of which only the last type is necessarily nonchiral. Their low-energy degrees of freedom can be described as emergent relativistic particles not present in the standard model of particle physics. We propose a range of experimental probes for multifold fermions in chiral symmetry groups based on the gyrotropic magnetic effect (GME) and the circular photogalvanic effect (CPGE). We find that, in contrast to Weyl fermions, multifold fermions can have zero Berry curvature yet a finite GME, leading to an enhanced response. The CPGE is quantized and independent of frequency provided that the frequency region at which it is probed defines closed optically activated momentum surfaces. We confirm the above properties by calculations in symmetry-restricted tight-binding models with realistic density functional theory parameters. We identify a range of previously unidentified ternary compounds able to exhibit chiral multifold fermions of all types (including a range of materials in the families AsBaPt and ${\mathrm{Gd}}_3{\mathrm{Cl}}_3\mathrm{C}$), and provide specific predictions for the known multifold material RhSi.

Figure 1

Chiral optical responses and multifold fermions. (Top) Schematics of the physical effects considered in this work and the response functions that describe them. (a) The gyrotropic magnetic effect (GME), a current response parallel to the direction of an applied low-frequency magnetic field. The GME is the low-frequency limit of natural optical activity, the rotation of the plane of linearly-polarized light upon transmission through a gyrotropic material. (b) The circular photogalvanic effect (CPGE) is the photo generation of a current which changes sign under a change in the polarization of the incident light, and which is proportional to the light's intensity. The CPGE is quantized to an integer (given by a combination of Chern numbers) multiple of ${\beta }_0=\frac{\pi e^3}{h^2}$ for chiral multifold fermions. (c) Schematic dispersion relations for chiral multifold fermions. The threefold node harbors effective spin-1 quasiparticles. The Chern numbers of the bands are $C=-2,0,2$ from low to high. The sixfold node is a symmetry-protected doubling of the spin-1 node. The first fourfold node is a symmetry-protected doubling of a standard spin-1/2 Weyl node, and the bands have Chern numbers $C=-1,-1,1,1$ from low to high. The second node realizes effective spin-3/2 quasiparticles. The Chern numbers of the bands are $C=-3,-1,1,3$ from low to high.

Figure 2

CPGE for an effective threefold fermion model. (a) A representative band structure of Eq. (3) for $\varphi =\pi /6+0.2\sim 0.72$, with the chemical potential depicted by the dashed horizontal line. The frequencies ${\omega }_i$ where the different resonant surfaces $S_{nm}$ change from active to inactive and from open to closed are depicted with a consistent color coding with that of (b) and (c). Analytical expressions for the spectrum and for ${\omega }_i$ can be found in Appendix pp3. (b) Evolution of the different ${\omega }_i$ as a function of $\varphi$. The colored blue region indicates the region where exact quantization holds due to the fact that $S_{12}$ and $S_{13}$ are closed. In the region ${\omega }_1<\omega <{\omega }_2,S_{12}$ is closed and a nonquantized CPGE plateau exists. The vertical gray line corresponds to $\varphi =\pi /6+0.2$, the value used to calculate the CPGE in (c). (c) As in (b), the shaded area denotes the region with exact quantization. Between ${\omega }_1<\omega <{\omega }_2$ the plateau is nonuniversal yet close to the quantized value $2{\beta }_0$ due to the small magnitude of the corrections. (d) CPGE for different values of $\varphi$ deviating from the $\text{SU}\left(2\right)$ invariant case $\varphi =\pi /6$. The small deviations (even at $\varphi =\pi /6)$ from $2{\beta }_0$ are due to numerical artifacts that decrease with increasing momentum resolution.

Figure 3

CPGE for an effective fourfold fermion model. (a) A representative band structure of Eq. (4) for $\chi =-0.36$ (see Appendix pp4 for analytic expressions for arbitrary $\chi )$. The relevant frequency scales ${\omega }_i$ (see Appendix pp4 for expressions) are depicted with a consistent color coding with that of (b) and (c). The chemical potential is indicated by a dashed horizontal line. (b) Evolution of the different ${\omega }_i$ as a function of $\chi$ bounding the different types of surfaces $S_{nm}$ (open, closed, inactive or active) of allowed optical transitions. The colored blue region indicates the region where exact quantization holds due to the fact that $S_{13},S_{14},S_{23}$, and $S_{24}$ are closed. As for the threefold case other regions have closed surfaces resulting in a nonquantized CPGE plateau. The vertical gray line corresponds to $\chi =-0.36$, the value used to calculate the CPGE in (c). (c) The shaded area denotes the region with exact quantization $\left(4{\beta }_0\right)$, yet a plateau close to $\left(3{\beta }_0\right)$ is seen for $\hslash \omega /\mu \sim 1$. The latter is only exactly quantized at the value of $\chi$ that realizes the spin-3/2 multifold case, $\chi =\arctan (-3)\approx -0.32$ [solid curve in (d)]. (d) The CPGE for different values of $\chi$.

Figure 4

Tight-binding band structures. (a) The band structure of the eight-band model used for RhSi, from Ref. [29], including spin-orbit coupling. The model without spin-orbit coupling is shown in Appendix pp6. (b) The band structure of a minimal model featuring the symmetries of space group 199, without spin-orbit coupling. As this is not fit to any particular material the energies are schematic, but the band connectivities and node structures are accurate.

Figure 5

Trace of the GME tensor for space groups 198 and 199. In the top row, the different nodes are reproduced from Fig. 4. Spin-orbit (SO) coupling is either present (blue) or absent (red). In the bottom row the trace of the GME tensor is calculated numerically for the cases with and without spin-orbit coupling (same color scheme), and analytically for the case without spin-orbit coupling (black dashed lines). The chemical potentials of nodes are indicated with arrows. (a) the $\mathrm{\Gamma }$ point for RhSi, space group 198. Without spin-orbit coupling the $\mathrm{\Gamma }$ point features a doubly degenerate isotropic threefold node at $-0.07\mathrm{eV}\left({\mu }_2\right)$. The gyrotropic response is linear in chemical potential at energies above the node, and scales as a square root plus a linear part below the node. With spin-orbit coupling the node splits into a fourfold node $\left({\mu }_1\right)$ and a standard Weyl $\left({\mu }_3\right)$. (b) The $R$ point in the same model. Without spin-orbit coupling all eight bands meet at a spin-degenerate double spin-1/2 node $\left({\mu }_5\right)$. Spin-orbit coupling splits this into a sixfold node $\left({\mu }_4\right)$ and two separated bands. All cases are isotropic. (c) An isolated threefold node appears in space group 199 at the $\mathrm{\Gamma }$ point. The node is not isotropic, and the quadratic corrections approximately cancel out leaving a linear variation of the GME trace as a function of chemical potential close to the node.

Figure 6

GME and CPGE dependence on atomic positions. (a) The trace of the GME tensor in our model of RhSi (black), shown with the trace of the GME tensor in a hypothetical material with $x=1/8$, with the same spectrum but different orbital embeddings (dashed blue). Note the significant deviations in the GME tensor for $\mu$ away from 0. (b) Trace of the CPGE tensor for the same two models. Here we see that the dependence on orbital embedding is strongest for large $\omega$, far from the quantized plateaus.

Figure 7

Trace of the CPGE tensor for space groups 198 and 199 as a function of frequency $\omega$ for the tight-binding models considered in the text. See Fig. 4 for the corresponding band structures. (a) SG198 at $\mathrm{\Gamma }\left(\mu =0.\right)$ (b) SG198 at R $\left(\mu =-0.5\mathrm{eV}\right)$ and (c) SG199 at $\mathrm{\Gamma }\left(\mu =-2.7\mathrm{eV}\right)$. For 198, the CPGE with and without spin-orbit (SO) coupling is plotted. The origin of the observed deviations from quantization is discussed in the main text and is attributed to a combination of finite momentum space resolution, quadratic corrections and the contributions of small CPGE active pockets. The latter are only allowed in the case with spin-orbit coupling.

Figure 8

Band structures of several candidates displaying multifold fermions. (a) ${\mathrm{Gd}}_3{\mathrm{Cl}}_3\mathrm{C}$ in SG $I{4}_{1}32$ (214) without spin-orbit (SO) coupling, featuring a threefold crossing at the $H$ point close to the Fermi level. (b) For ${\mathrm{Gd}}_3{\mathrm{Cl}}_3\mathrm{C}$ with spin-orbit coupling included, we observe the same threefold crossing at the $H$ point owing to weak spin-orbit coupling. (c) ${\mathrm{Gd}}_3{\mathrm{I}}_3\mathrm{Si}\left(I{4}_{1}32\right)$ with spin-orbit coupling included. (d) AsBaPt in $P{2}_{1}3$ (198) with spin-orbit coupling, featuring a fourfold crossing close to the Fermi level at the $\mathrm{\Gamma }$ point.

Figure 9

RhSi band structure without spin-orbit coupling. The real material features a small spin-orbit coupling as shown in Fig. 4 in the main text.