Understanding the three-dimensional quantum Hall effect in generic multi-Weyl semimetals

The quantum Hall effect in three-dimensional Weyl semimetal (WSM) receives significant attention for the emergence of the Fermi loop where the underlying two-dimensional Hall conductivity, namely, sheet Hall conductivity, shows quantized plateaus. In tilt multi-Weyl semimetals (mWSMs) lattice models, we systematically study Landau levels (LLs) and magneto-Hall conductivity both under the parallel and perpendicular magnetic field (referenced to the Weyl node's separation), i.e., $\mathbit{B}\ensuremath{\parallel}z$ and $\mathbit{B}\ensuremath{\parallel}x$, to explore the impact of tilting and nonlinearity in the dispersion. We make use of two (single) node low-energy models to qualitatively explain the emergence of mid-gap chiral (linear crossing of chiral) LLs on the lattice for $\mathbit{B}\ensuremath{\parallel}z$ $(\mathbit{B}\ensuremath{\parallel}x)$. Remarkably, we find that the sheet Hall conductivity becomes quantized for $\mathbit{B}\ensuremath{\parallel}z$ even when two Weyl nodes project onto a single Fermi point in two opposite surfaces, forming a Fermi loop with ${k}_{z}$ as the good quantum number. On the other hand, the Fermi loop, connecting two distinct Fermi points on two opposite surfaces, with ${k}_{x}$ being the good quantum number, causes the quantization in sheet Hall conductivity for $\mathbit{B}\ensuremath{\parallel}x$. The quantization is almost lost (perfectly remained) in the type-II phase for $\mathbit{B}\ensuremath{\parallel}x$ $(\mathbit{B}\ensuremath{\parallel}z)$. Interestingly, the jump profiles between the adjacent quantized plateaus change with the topological charge for both cases. The momentum-integrated three-dimensional Hall conductivity is not quantized; however, it bears the signature of chiral LLs resulting in the linear dependence on $\ensuremath{\mu}$ for small $\ensuremath{\mu}$. The linear zone (its slope) reduces (increases) as the tilt (topological charge) of the underlying WSM increases.