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., $\mathit{B}\parallel z$ and $\mathit{B}\parallel x$, 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 $\mathit{B}\parallel z$ $(\mathit{B}\parallel x)$. Remarkably, we find that the sheet Hall conductivity becomes quantized for $\mathit{B}\parallel z$ 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 $\mathit{B}\parallel x$. The quantization is almost lost (perfectly remained) in the type-II phase for $\mathit{B}\parallel x$ $(\mathit{B}\parallel z)$. 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 $\mu$ for small $\mu$. The linear zone (its slope) reduces (increases) as the tilt (topological charge) of the underlying WSM increases.

Figure 1

Profile of 2D Chern numbers, calculated using Eq. (6), as a function of $k_z$ for single (a), double (b), and triple (c) WSMs following Eqs. (1), (2), and (3), respectively. We consider $t_z=t=1$ and $m_z=0$ throughout the paper.

Figure 2

LL energies as a function of $k_z$ under $\mathit{B}\parallel z$, shown for a single WSM with $m=1$ [Eq. (15)], a double WSM with $m=2$ [Eq. (16)], and a triple WSM with $m=3$ [Eq. (17)] in (a1)–(a3), (b1)–(b3), and (c1)–(c3), respectively, keeping $k_x=0$ fixed. We consider $t_0=0$ for (a1)–(c1), 0.5 for (a2)–(c2), and 1.2 for (a3)–(c3). The mid-gap chiral LLs, traversing through the WNs at ${\mathit{k}}_p^{\pm }=(0,0,\pm \pi /2)$, exist both in type-I and type-II phases as depicted in red color. These numerical results on the lattice are qualitatively consistent with analytical LL energies calculated using Eq. (10). The insets depict the number of nondegenerate chiral Landau levels, proportional to the topological charge $m$, inside the bulk gap as designated by the grey shaded region.

Figure 3

The variation of LL energies, computed from Eqs. (15, 16, 17) as a function of $k_x$ for $\mathit{B}\parallel x$ while following the format given in Fig. 2. The mid-gap chiral LLs, depicted in red color, linearly cross each other at $k_x=0$ (displayed in the insets) only for the type-I phase, while for the type-II, these chiral modes become indistinguishable from the bulk LLs. The chiral LLs are always gapped except at $k_x=0$. The number of linear crossings associated with chiral LLs at $k_x=0$ can directly determine the topological charge of the underlying WSMs. Each of the bulk LLs is doubly degenerate for untilted single WSM. By contrast, for untilted double and triple WSMs, the bulk LLs only degenerate at $k_x=0$. Similar to Fig. 2, the bulk gap is represented by the grey shaded region.

Figure 4

Normalized 2D Hall conductivity ${\sigma }_{xy}^{2D}\left(k_z\right)/{\sigma }_0$ (shown in left axis), following Eq. (18), and PDOS $D_{2D}\left(k_z\right)$ (shown in right axis), following Eq. (19), as a function of $\mu$, keeping the tilt term fixed at $t_0=0$, for $m=1$ in (a), $m=2$ in (b), and $m=3$ in (c) when $\mathit{B}\parallel z$. The upper-right insets show the flat $k_x$-independent LLs at $k_z=\pi /2$. One can notice that ${\sigma }_{xy}^{2D}\left(k_z\right)$ jumps by unity to the next plateau at certain $\mu$ when there exists a flat LL with the energy given by $\mu$. The jump profiles for different values of $k_z$ are in accordance with the LL spectrum illustrated in Fig. 2. We repeat (a), (b) and (c) in (d), (e), and (f) with ${\sigma }_{xy}^{2D}(k_z=0.4\pi )$ for the tilt $t_0=0,0.5,\text{and}1.2$, respectively. The staircase-like structure continues to exist for type-II phases as well. We set $\eta =0.01$. The 2D sheet Hall conductivity is measured in the unit of $e^2/h$. We follow this convention throughout.

Figure 5

Normalized 3D magneto-Hall conductivities ${\sigma }_{xy}/{\sigma }_0$, defined in Eq. (18), for $\mathit{B}\parallel z$ as a function of $\mu$ for $m=1$ in (a), $m=2$ in (b), and $m=3$ in (c) while varying $t_0=0,0.5,\text{and}1.2$. The DOSs are computed from Eq. (19), in (d), (e), and (f) with the same parameter set as used for (a), (b), and (c). The flat regions in DOS, observed for type-I phase only, cause the $\mu$-linear regions in ${\sigma }_{xy}$ where the chiral LLs contribute maximally. We associate the peaks in the DOS with the change in the slope of ${\sigma }_{xy}$ for $t_0=0$ with the blue dashed lines to emphasize the connection between the above two quantities. The 3D Hall conductivity is measured in the unit of $(e^2/h)/L$. We follow this convention throughout.

Figure 6

Normalized 2D Hall conductivity ${\sigma }_{xy}^{2D}(k_z=\pi /2)/{\sigma }_0$ (shown in left axis), following Eq. (18), and PDOS $D_{2D}(k_z=\pi /2)$ (shown in right axis), using Eq. (19), as a function of $\mu$ for $m=1$ in (a), $m=2$ in (b), and $m=3$ in (c), by varying the strength of the magnetic field $B=B_0, 2B_0, 4B_0$, and $10B_0$ keeping $t_0=0.5$ fixed. We show (d1), (e1), and (f1), [(d2), (e2), and (f2)] for the 3D magneto-Hall conductivity ${\sigma }_{xy}$ [DOS $D\left(\mu \right)$] as a function of $\mu$ with the above set of parameters. The degeneracy of the LLs increases with increasing $B$. This is reflected in the increased width of the quantized plateau [$\mu$-linear region] shown in (a)–(c) [(d1)–(f1)].

Figure 7

Normalized 2D sheet Hall conductivity ${\sigma }_{yz}^{2D}\left(k_x\right)/{\sigma }_0$, computed from Eq. (18), and PDOS $D_{2D}\left(k_x\right)$, evaluated from Eq. (19), as a function of $\mu$ with tilt term set at $t_0=0$, for $m=1$ in (a), $m=2$ in (b), and $m=3$ in (c) when $\mathit{B}\parallel x$. The insets show the $k_z$-independent flat LLs at $k_x=0.15\pi$. One can notice that ${\sigma }_{yz}^{2D}\left(k_x\right)$ jumps to the next plateau at certain $\mu$ when there exists a flat LL within the energy window scanned by $\mu$. We find double jumps, i.e., quantization changes by two, in ${\sigma }_{yz}^{2D}(k_x=0)$ for all of the WSMs, while the triple WSM additionally exhibits a nonmonotonic profile following the LL spectrum illustrated in Fig. 3. We repeat (a), (b), and (c), respectively, in (d), (e), and (f) with ${\sigma }_{yz}^{2D}(k_x=0.15\pi )$ for $t_0=0,0.5,\text{and}1.2$. The clean staircase-like structure almost vanishes in type-II phase while it exists in type-I phase.

Figure 8

We repeat Fig. 5 for $\mathit{B}\parallel x$. The flat regions in DOS, observed for type-I phase only, similar to Fig. 5 results in the $\mu$-linear regions in ${\sigma }_{yz}$ with the maximum contribution coming from the mid-gap chiral LLs. Similar to Fig. 5, the slope change in ${\sigma }_{yz}$ is mediated by the peak in the DOS as designated by vertical blue dashed line for $t_0=0$.

Figure 9

Normalized 2D Hall conductivity ${\sigma }_{yz}^{2D}(k_x=3\pi /20)/{\sigma }_0$ and PDOS $D_{2D}(k_x=3\pi /20)$ as a function of $\mu$ for $m=1$ in (a), $m=2$ in (b), and $m=3$ in (c), by varying $B=B_0, 2B_0, 4B_0$, and $10B_0$ at $t_0=0.5$. We show (d1), (e1), and (f1), [(d2), (e2), and (f2)] for the 3D magneto-Hall conductivity ${\sigma }_{yz}$ [DOS $D\left(\mu \right)$] as a function of $\mu$ with the above set of parameters. The degeneracy of LLs increases with $B$. This is apparently reflected in the increased width of the quantized plateau shown in (a)–(c) as well as $\mu$-linear region shown in (d1)–(f1). A stronger enough magnetic field $B=10B_0$ can lead to a nonmonotonic profile of ${\sigma }_{yz}^{2D}(k_x=3\pi /20)$ and a deviation from $\mu$-linear behavior in ${\sigma }_{yz}$.