Magnetic-Field Effects on the Electronic States of the K 4 Lattice

We investigate the electronic states of the K 4 lattice in a uniform magnetic field within a nearest-neighbor tight-binding model. The K 4 lattice is a noncentrosymmetric three-dimensional network with triple-Dirac cones at the Γ and H points. By fixing the Bloch momentum component parallel to the field, the Peierls-substituted tight-binding problem reduces to an effective two-dimensional problem and yields a Hofstadter spectrum whose self-similar subpatterns reflect the geometry of the lattice projected onto the plane perpendicular to the field. A square-like substructure emerges on the flux scale ϕ = 1/8 for [Formula: see text], and honeycomb- and triangular-like substructures emerge on the finer scale ϕ = 1/32 for [Formula: see text]. Here ϕ is the flux through the elementary plaquette of the projected lattice, and these scales correspond to one flux quantum per projected unit cell. Owing to the helical connectivity of the chiral K 4 network, the field-parallel momentum acts as an additional flux, which, for commensurate values of this momentum, shifts the spectrum along the flux axis and brings the second triple-Dirac cone at the H point into the magnetic spectrum. In the weak-field limit, we derive the Landau-level structure from a low-energy effective Hamiltonian around the Γ point. The three-component nature of the triple-Dirac cone gives rise to a cubic secular equation, whose solutions describe how a pseudospin-flat branch coexists with two relativistic Landau ladders and a single chiral n = 0 level dispersing linearly along the field, in agreement with the lattice calculation.