Optical method to detect the relationship between chirality of reciprocal space chiral multifold fermions and real space chiral crystals

The chirality of chiral multifold fermions in reciprocal space is related to the chirality of crystal lattice structures in real space. In this study, we propose a strategy to detect and identify multifold fermions of opposite chirality in nonmagnetic systems using second-order optical transports. Chiral crystals related with inversion operations cannot be made to overlap with each other via any experimental operation. Further, chiral multifold fermions within such crystals host opposite chiralities corresponding to a given $k$ point. A change in chirality is indicated by a corresponding change in the sign of the second-order charge current dominated by chiral fermions. This property can be exploited to study the relationship between chiralities in reciprocal and real spaces by utilizing bulk transport.

Figure 1

Inversion of chirality in a chiral crystal structure leads to the inversion of chirality in associated chiral multifold fermions. (a), (b) Crystal lattice structures of semimetals with opposite chiralities in the space group $P{2}_{1}3$ with chiral multifold fermions, considering RhSi as an example. (c), (d) Schematics of fourfold degenerate fermions with spin-3/2 excitation at the $\mathrm{\Gamma }$ point. (e), (f) Schematics of Double threefold degenerate fermions with spin-1 excitation at the $R$ point. The local effective Hamiltonians around the crossing points can be expressed by $H=-\mathit{k}·{\mathit{S}}_{4\times 4}, H=\mathit{k}·{\mathit{S}}_{4\times 4}, H=-\mathit{k}·{\mathit{S}}_{3\times 3}$, and $H=\mathit{k}·{\mathit{S}}_{3\times 3}$ for (c)–(f), respectively. We use $\hslash v_F=1$ and $\hslash v_F=2$ for $H_{3-\mathrm{fold}}^1$ and $H_{3-\mathrm{fold}}^2$, respectivley. The parameter $\varphi =\pi /2$ in $H_{3-\mathrm{fold}}$. We use $\hslash v_F=1$ and $x=arctan(-3)$ for $H_{4-\mathrm{fold}}$ Further details of the effective model can be found in Ref. [2, 7].

Figure 2

Energy dispersion of the tight binding model Hamiltonian along high symmetry lines.

Figure 3

Local distribution of ${\mathrm{\Omega }}_z, {\tilde{\chi }}_{yz}^{CPGE,x}(\mathit{k};0,\omega ,-\omega )$, and ${\tilde{\chi }}_{yz}^{LPGE,x}(\mathit{k};0,\omega ,-\omega )$ around $R$ points from numerical calculations based on a tight binding model of RhSi. (a) Inversion-operation-induced sign change for the local Berry curvature (${\mathrm{\Omega }}_z$) distribution around multifold chiral fermions. (b) Light excitation between the lower and upper cones of multifold fermions. The green rings represent the transition path corresponding to a given frequency. The light excitation from the hot ringlike distribution for (c) ${\tilde{\chi }}_{yz}^{CPGE,x}(\mathit{k};0,\omega ,-\omega )$ and (d) ${\tilde{\chi }}_{yz}^{LPGE,x}(\mathit{k};0,\omega ,-\omega )$. ${\tilde{\chi }}_{yz}^{CPGE,x}$ and ${\tilde{\chi }}_{yz}^{LPGE,x}$ change the sign via the inversion operation on the crystal structure. The plot lies on the $k_z=\pi$ plane around the point, $R$. The color bars are expressed in arbitrary units. The left panel of (a) and the upper panels of (b)–(d) correspond to the crystal structure depicted in Fig. 1. The right panel of (a) and lower panels of (b)–(d) correspond to the crystal structure depicted in Fig. 1.

Figure 4

Frequency-dependent second-order conductivity for (a) CPGE and (d) LPGE based on numerical calculations based on the tight-binding model of RhSi. The red and blue curves correspond to opposite chiralities of the chiral fermions. The signs of all three conductivities change with the switch in chirality of the crystal structure. Schematic of the experimental setup to identify multifold fermions in crystals with opposite chiralities by (b), (c) CPGE and (e), (f) LPGE. CPL in (b), (c) represents circularly polarized light, and LPL in (e), (f) represents linearly polarized light. The calculations were performed based on the tight-binding model reported in Ref. [3] with the parameters $v_1=1.0, v_p=-1.0, v_{r1}=0.0, v_{r2}=-0.01, v_{r3}=0.01, v_{s1}=-0.01, v_{s2}=0.0$, and $v_{s3}=0$ and an on-site term $v_0=0.01$ to translate the degeneracy at the point, $R$, to the Fermi level. ${\beta }_0=i\pi \frac{e^3}{h^2}$ in (a). (b) and (e) correspond to the crystal structure depicted in Fig. 1. (c) and (f) correspond to the crystal structure depicted in Fig. 1.

Figure 5

Frequency dependent second-order susceptibility ${\chi }_{yz}^{SHG,x}$ for the (a) real and (b) imaginary part, respectively. The red and blue curves correspond to the crystal structures depicted in Fig. 1 and Fig. 1, respectively. (c) The magnitude of ${\chi }_{yz}^{SHG,x}$.