Single pair of charge-2 high-fold fermions with surface type-II Van Hove singularities in ultralight chiral crystals

The realization of the single-pair fermions in electronic systems remains challenging in topology physics, especially for the systems with larger chiral charges $C$. In this work, based on the symmetry analysis, low-energy effective models, and first-principles calculations, we identify the single-pair charge-two chiral fermions in cubic lattices. We first derive the minimal lattice model that exhibits a single pair of Weyl points with the opposite chiral charges of $C=\pm 2$. Furthermore, we show the ultralight chiral crystal ${\mathrm{P}4}_{3}32$-type ${\mathrm{LiC}}_2$ and its mirror enantiomer as high-quality candidate materials, which exhibit large energy windows to surmount the interruption of irrelevant bands. Since two enantiomers are connected by the mirror symmetry, we observe the opposite chiral charges $C$ and the reversal of Fermi-arc connections, showing the correspondence of chirality in the momentum space and the real space. In addition, we also reveal type-II van Hove singularities on the helicoid surfaces, which may induce chirality-locked charge density waves on the crystal surface. Our work not only provides a promising platform for controlling the sign of chiral charge through structural chirality but also facilitates the exploration of electronic correlations on the surface of ultralight chiral crystals.

Figure 1

Four types of unconventional chiral fermions with the high-fold degeneracy. (a) Threefold spin-1 Weyl fermions. (b) Fourfold charge-2 Dirac fermions, which are equivalent to adding two spin-$\frac{1}{2}$ Weyl fermions with the same chirality. (c) Fourfold spin-$\frac{3}{2}$ Weyl fermions. (d) Sixfold double spin-1 Weyl fermions, which are a composition of two identical spin-1 Weyl fermions. The bands marked the corresponding chiral charges, respectively.

Figure 2

The four-band minimal lattice model for SG 212. (a) The lattice structure of the 4a ($\{\frac{1}{8},\frac{1}{8},\frac{1}{8}\}$) Wyckoff position. (b) The bulk BZ and projected on the (001) surface BZ. (c) The bulk band structure of hopping parameters: $t_0=0.145$ and $t_1=0.160$. Here, the unit of energy is in eV. The left and right insets represent the WCCs for the SWP and C-2 DP at the $\mathrm{\Gamma }$ and R points, where $\phi \in [0,\pi ]$ is the polar angle, and $\theta \in [0,2\pi ]$ corresponds to the evolution of the WCCs. The red and blue dots indicate the positive and negative chiral charges. (d) The Berry curvature distributions on the (001) surface BZ. The SWP with $C=+2$ as the “source” flows into the “sink” generated by the C-2 DP with $C=-2$.

Figure 3

The TSSs and Fermi arcs of the four-band lattice model for SG 212. (a) The LDOS along $\overline{\mathrm{X}}\text{−}\overline{\mathrm{\Gamma }}\text{−}\overline{\mathrm{M}}\text{−}\overline{\mathrm{X}}$ paths on the (001) surface BZ. The white dashed line represents the isoenergy contour $\mathrm{Ecut}=0$ eV. (b) The Fermi arcs at $\mathrm{Ecut}=0$ eV and the associated WPs with the red and blue dots. The double-helicoid surface arcs occur at $\overline{\mathrm{\Gamma }}$ and $\overline{\mathrm{M}}$, respectively. (c) and (d) The surface LDOSs along the two white clockwise loops (${\mathrm{loop}}_1$ and ${\mathrm{loop}}_2$) centered at $\overline{\mathrm{\Gamma }}$ and $\overline{\mathrm{M}}$ in (b). The white dashed lines are the reference lines.

Figure 4

The representative materials for a single pair of WPs with opposite chiral charges of $C=\pm 2$. (a) The views of the two enantiomers along the [111] direction, which are connected by the mirror symmetry. (b) The bulk band structures along the HSLs and the relevant SVIRs with the corresponding SGs ${\mathrm{P}4}_{3}32$ and ${\mathrm{P}4}_{1}32$. The band crossings near the Fermi level are marked ${\mathrm{WP}}_1$ and ${\mathrm{WP}}_2$, respectively. (c) and (d) The evolutions of WCCs for ${\mathrm{WP}}_1$ and ${\mathrm{WP}}_2$, respectively. The WPs of the two enantiomers are characterized by opposite chiral charges.

Figure 5

(a) The LDOSs of the two enantiomers along $\overline{\mathrm{X}}\text{−}\overline{\mathrm{\Gamma }}\text{−}\overline{\mathrm{M}}\text{−}\overline{\mathrm{X}}$ paths on the (001) surface BZ. The white dashed line represents different isoenergy contours. (b) Their Fermi arcs at $\mathrm{Ecut}=-0.1$ eV and positions of the WPs. The double-helicoid surface arcs occur at $\overline{\mathrm{\Gamma }}$ and $\overline{\mathrm{M}}$, respectively. (c) The surface LDOSs along the two white clockwise loops (${\mathrm{loop}}_3$ and ${\mathrm{loop}}_4$) centered at $\overline{\mathrm{\Gamma }}$ in (b). Two right-moving and left-moving chiral double-helicoid TSSs occur naturally. (d) Three isoenergy contours at three energies (0.20, 0.215, and 0.23 eV) indicated by three horizontal dashed lines in (a). Lifshitz transitions occur in the evolutions of their surface arc with the opposite position.