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 \rev{noncentrosymmetric three-dimensional network with triple-Dirac cones at the $\mathrm{\Gamma }$ 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 \rev{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 $\phi =1\mathrm{/}8$ for $\mathit{B}\parallel [100]$, and honeycomb- and triangular-like substructures emerge on the finer scale $\phi =1\mathrm{/}32$ for $\mathit{B}\parallel [111]$\rev{. Here $\phi$ 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 screw axes 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 $\mathrm{\Gamma }$ 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 \rev{and a single chiral $n=0$ level dispersing linearly along the field, in agreement with the lattice calculation}.