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

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.

💡 Research Summary

In this paper the authors present a comprehensive theoretical investigation of the electronic topology of the K4 crystal, a three‑dimensional realization of the complete graph K4. Starting from the crystal’s geometric description, they show that the structure can be viewed as a body‑centered cubic lattice (space group I4₁32, No. 214) with four sublattices (A–D) occupying the 8a Wyckoff position. Each atom forms three coplanar nearest‑neighbor bonds; the planes of neighboring sites are rotated by about 70.5° (cos θ = 1/3), giving the lattice a chiral, non‑centrosymmetric character.

To capture the essential electronic features the authors construct a minimal tight‑binding model with a single spinless s‑orbital on each site and only nearest‑neighbor hopping of amplitude γ. The resulting 4 × 4 Bloch Hamiltonian contains complex phase factors e^{−i k·τ_{ij}} that encode the geometry of the K4 graph. Diagonalization yields four bands whose high‑symmetry behavior is striking: at the Γ and H points the bands form a three‑fold crossing (two linearly dispersing Dirac cones plus a nearly flat band), often called a triple Dirac point; at the P and P′ points a conventional two‑fold Dirac crossing appears. Group‑theoretical analysis identifies the Γ and H points as belonging to the O(432) point group with representations T₂⊕A₁, while P and P′ belong to the T (23) group with E⊕E. By comparing with elementary band representations (EBRs) from the Bilbao Crystallographic Server the authors confirm that the s‑orbital model reproduces an atomic limit, i.e., the bands are fully compatible with symmetry‑allowed atomic orbitals.

The topological character is then extracted by evaluating the Berry curvature analytically around each degeneracy. Expanding the Hamiltonian to first order in q = k − k₀ and applying appropriate unitary transformations, the authors obtain effective low‑energy Hamiltonians of Weyl type. At Γ and H the triple Dirac point splits into two Weyl nodes of chirality χ = ±2; the flat band contributes a compensating charge, leaving a net monopole charge of magnitude two. At P and P′ the two‑fold crossings correspond to ordinary Weyl nodes with χ = ±1. Thus the K4 crystal hosts both higher‑order (double‑charge) Weyl points and conventional single‑charge Weyl points, a rare coexistence in a single material.

To demonstrate the bulk–boundary correspondence, slab calculations are performed for a (001) termination using a 30‑layer slab and surface Green’s‑function techniques. The surface spectral function reveals clear Fermi arcs that connect projections of Weyl nodes with opposite chirality. Importantly, the χ = ±2 nodes are linked not directly to each other but to pairs of χ = ∓1 nodes, forming Y‑shaped or Λ‑shaped arc networks. This multi‑node connectivity is a distinctive hallmark of the higher‑charge Weyl points and provides an unambiguous experimental signature for angle‑resolved photoemission spectroscopy (ARPES).

The authors also explore the robustness of these features beyond the single‑orbital model. By adding pₓ, p_y, p_z orbitals (s + p model) the same symmetry analysis holds, and the triple and double Dirac points persist as s‑p hybridized π‑bands. First‑principles calculations reported in the literature for carbon, boron, and phosphorus K4 analogues corroborate the tight‑binding predictions, indicating that the minimal model already captures the essential topology.

In summary, the work establishes the K4 crystal as a spinless Weyl semimetal that simultaneously hosts double‑charge (χ = ±2) and single‑charge (χ = ±1) Weyl nodes. The presence of higher‑order monopoles leads to unconventional surface Fermi‑arc patterns, enriching the phenomenology of Weyl physics. The authors suggest that the K4 lattice can be realized in sp²‑hybridized carbon, boron, phosphorus, or multi‑element compounds, opening avenues for experimental synthesis and for exploiting the unique transport, optical, and nonlinear responses associated with mixed‑chirality Weyl fermions.