Topological Charges, Fermi Arcs, and Surface States of $K_4$ Crystal

TL;DR

This work analyzes the topological electronic structure of the mathematical $K_4$ lattice via a minimal spinless tight-binding model. It identifies bulk Weyl nodes at $\Gamma_{\mathrm{high}}$, $\mathrm{H_{low}}$, $\mathrm{P_{low}}$, and $\mathrm{P_{high}}$ with chiralities $\chi = -2, +2, -1, +1$ and energies $E = \gamma, -\gamma, \mp\sqrt{3}\gamma$, including a triple Dirac cone at the high-symmetry points. Slab calculations in the $(001)$ direction reveal Fermi arcs that connect projected bulk nodes on the surface Brillouin zone, with high-energy arcs linking $\overline{\Gamma}$ to $\overline{\mathrm{R}}$ and low-energy arcs linking $\overline{\Gamma}$ to $\overline{\mathrm{R}}$ via the $P$ points, balanced by symmetry. The results establish the $K_4$ lattice as a novel spinless Weyl semimetal with intrinsic topological surface states and motivate symmetry-based classifications and photonic-analog explorations of higher-chirality fermions.

Abstract

We investigate the topological electronic properties of the $K_4$ crystal by constructing a tight-binding model. The bulk band structure hosts Weyl nodes with higher and conventional chiralities ($χ= \pm 2$ and $χ= \pm 1$) located at high-symmetry points in the Brillouin zone. Through analytical evaluation of the Berry curvature, we identify the positions and chiralities of these Weyl nodes. Furthermore, slab calculations for the (001) surface reveal Fermi arcs that connect Weyl nodes of opposite chirality, including those linking $χ= \pm 2$ nodes with pairs of $χ= \mp 1$ nodes. These results demonstrate that the $K_4$ crystal is a spinless Weyl semimetal featuring topologically protected surface states originating from multiple types of Weyl nodes.

Figures (3)

• Figure 1: (a) Crystal structure of the $K_4$ crystal. Viewed along the (001) direction, it exhibits a pattern of tiled squares and octagons and contains multiple helical motifs. (b) Graph representation of the $K_4$ structure, showing the connectivity of the four sublattices. The corresponding graph is a complete graph with four vertices, each connected to all others by three edges. (c) Unit cell of the $K_4$ crystal. The gray cube represents the conventional cubic unit cell, while the red rhombohedron denotes the primitive unit cell of the body-centered cubic (bcc) lattice. The primitive cell contains four sublattices, labeled A-D, each connected by three nearest-neighbor bonds. At each site, three bonds extend isotropically in a single plane, similar to graphene. However, the bond plane at one site is rotated by an angle of $\theta \simeq 70.5^\circ$ ($\cos \theta = 1/3$) relative to that of a neighboring site. (d) First Brillouin zone (BZ) of the $K_4$ crystal, based on the bcc lattice. (e) Energy band structure of the $K_4$ crystal. At the $\Gamma$ point, two linearly dispersing (conical) bands and one flat band intersect at energy $\gamma$, referred to as the $\Gamma_{\mathrm{high}}$ point. At the H point, a similar crossing occurs at energy $-\gamma$, labeled as the $\mathrm{H_{low}}$ point. The four-band structure around $\mathrm{H_{low}}$ is the vertical inversion of that around $\Gamma_{\mathrm{high}}$. At the P points, two conical bands form vertically aligned degenerate points at energies $-\sqrt{3}\gamma$ and $+\sqrt{3}\gamma$, denoted as $\mathrm{P_{low}}$ and $\mathrm{P_{high}}$, respectively.
• Figure 2: Energy dispersion around Weyl points in $K_4$ crystal. At the $\Gamma_{\mathrm{high}}$ and $\mathrm{H_{low}}$ points, two conical dispersions and one flat band intersect at a single point, forming what is referred to as a triple Dirac cone. The topological charge is calculated for each band, and the sign of the charge associated with the lowest band determines the chirality of the degeneracy point. At the $\Gamma_{\mathrm{high}}$ point, the chirality is $\chi = -2$, while at the $\mathrm{H_{low}}$ point, it is $\chi = +2$. At the $\mathrm{P_{low}}$ and $\mathrm{P_{high}}$ point, two conical bands intersect, forming what are referred to as simple Dirac cones, each carrying opposite chirality: $\chi = -1$ at $\mathrm{P_{low}}$ and $\chi = +1$ at $\mathrm{P_{high}}$.
• Figure 3: (a) Slab structure of the $K_4$ crystal cleaved along (001). Two different types of surface structures: (left) Layer 1 (red) forms chains along the $x$-axis, and (right) Layer 2 (blue) along the $y$-axis. Top and perspective views are shown in the upper and lower panels, respectively. Thus, for the slab structures, there are totally four combinations of possible surface terminations. (b) 1st BZ for Bulk and slab strcutures. (left) 1st BZ of bulk structure, with the yellow square indicating the slab BZ obtained by projecting along the $k_z$-direction. (right) 1st BZ of slab structures. $\overline{\Gamma}$ corresponds to the projection of the bulk $\Gamma$ and H points, while $\overline{\mathrm{R}}$ correspond to projections of the bulk P and P' points. (c) Left panel schematically shows the bulk BZ, where $\Gamma_{\mathrm{high}}$ (blue sphere) carries chirality $\chi = -2$, and $\mathrm{P_{high}}$ and $\mathrm{P'_{high}}$ (red spheres) each carry $\chi = +1$. These Weyl nodes are enclosed by distinct momentum-space subsystems: $\Gamma_{\mathrm{high}}$ by a cylinder centered at $\overline{\Gamma}$ (blue), and $\mathrm{P_{high}}$ and $\mathrm{P'_{high}}$ by a cylinder centered at $\overline{\mathrm{R}}$ (red). Accordingly, the subsystem centered at $\overline{\Gamma}$ acquires a Chern number $C = -2$, while that centered at $\overline{\mathrm{R}}$ acquires $C = +2$. Right panel shows the energy band structures of slab at $\overline{\Gamma}$ and $\overline{\mathrm{R}}$, respectively. In both cases, the subsystems cut through Weyl cones, yielding effectively two-dimensional systems with energy gaps that host nontrivial Chern numbers. Consequently, topologically protected edge states traverse these gaps. Since the bulk gap closes only at the Weyl nodes, the surface states connect their projections, forming a Fermi arc (pink line) that extends from $\overline{\mathrm{R}}$ ($C = +2$) to $\overline{\Gamma}$ ($C = -2$). (d) Energy dispersion of slab structures obtained from the tight-binding model. Gray shading indicates the bulk projection, and red curves highlight the Fermi arcs. Weyl points are labeled by their chiralities. High-energy arcs connect the projected $\Gamma_{\mathrm{high}}$ ($\chi = -2$) to $\mathrm{P_{high}}$ and $\mathrm{P'_{high}}$ ($\chi = +1 \times 2$), while low-energy arcs connect the projected $\mathrm{H_{low}}$ ($\chi = +2$) to $\mathrm{P_{low}}$ and $\mathrm{P'_{low}}$ ($\chi = -1 \times 2$). The apparent chirality imbalance is resolved by accounting for the projection of symmetry-equivalent Weyl points in the bulk.