Weyl fermions with arbitrary monopoles in magnetic fields: Landau levels, longitudinal magnetotransport, and density-wave ordering

We theoretically address the effects of strong magnetic fields in three-dimensional Weyl semimetals (WSMs) built out of Weyl nodes with a monopole charge $n$. For $n=1$, 2, and 3 we realize single, double, and triple WSM, respectively, and the monopole charge $n$ determines the integer topological invariant of the WSM. Within the linearized continuum description, the quasiparticle spectrum is then composed of Landau levels (LLs), containing exactly $n$ number of chiral zeroth Landau levels (ZLLs), irrespective of the orientation of the magnetic field. In the presence of strong backscattering, for example (due to quenched disorder associated with random impurities), these systems generically give rise to longitudinal magnetotransport. Restricting ourselves to the quantum limit (and assuming only the subspace of the ZLLs to be partially filled) and mainly accounting for Gaussian impurities, we show that the longitudinal magnetoconductivity (LMC) in all members of the Weyl family displays a positive linear-$B$ scaling when the field is applied along the axis that separates the Weyl nodes. But, in double and triple WSM, LMC displays a smooth crossover to a nonlinear $B$ dependence as the field is tilted away from such a high-symmetry direction. In addition, due to the enhanced density of states, the LL quantization can trigger instabilities toward the formation of translational symmetry-breaking density-wave orderings for sufficiently weak interaction (BCS instability), which gaps out the ZLLs. Concomitantly as the temperature (magnetic field) is gradually decreased (increased) the LMC becomes negative. Thus WSMs with arbitrary monopole charge $\left(n\right)$ can host an intriguing interplay of LL quantization, longitudinal magnetotransport (a possible manifestation of one-dimensional chiral or axial anomaly), and density-wave ordering, when placed in a strong magnetic field.

Figure 1

(a) Spectrum of Weyl fermions along the $k_z$ direction in any Weyl semimetal. Dispersion in the $xy$ plane respectively for (b) single, (c) double, and (d) triple Weyl semimetals. Notice that $E\sim |{\mathit{k}}_{\perp }{|}^n$, where $n$ is the monopole charge and ${\mathit{k}}_{\perp }=\left(k_x,k_y\right)$, but $E\sim |k_z|$ in all Weyl semimetals.

Figure 2

Schematic representation of the applied electric $\left(\mathit{E}\right)$ and magnetic $\left(\mathit{B}\right)$ fields (always parallel). Quasiparticle dispersion along $z$ direction is $v_z{k}_z$, while in the $xy$ plane is proportional to $|k_{\perp }{|}^{n/2}$, where ${\mathit{k}}_{\perp }=(k_x,k_y)$. Here $\theta$ parametrizes the tilting of magnetic field away from the $z$ direction. Weyl nodes are separated along the $z$ direction.

Figure 3

Vector field plots for the Berry curvature for Weyl fermion, (a) when the Weyl nodes are monopole (source) and antimonopole (sink) of Berry curvature, with unit charge. In double and triple WSMs the Weyl nodes are source and sink of Berry curvature with charge two and three, respectively, and the vector field plots are shown in (b) and (c).

Figure 4

Landau level (LL) spectrum in single Weyl semimetal, obtained from Eq. (6) when $n=1$. For details see Appendix pp1. Here the zeroth LL (red curve) is nondegenerate, and the energy $\epsilon$ (in units of $2{\alpha }_1^2{\hslash }^2/{\ell }_B^2$) is measured about a reference or zero point energy ${\hslash }^2/\left(2m{\ell }_B^2\right)$. At half-filling the zeroth LL cuts the Fermi energy (dotted line) at $k_z=\pm \pi /\left(2a\right)$ (we set $a=1$ for convenience) [87]. Similar LL spectrum is found for double and triple Weyl semimetals, for which, however, the zeroth LL has two- and threefold degenerate, respectively.

Figure 5

(a) Landau level spectrum for double-Weyl semimetals when the magnetic field is $B=1T$ and the direction of applied field is characterized by (i) $\theta =3\pi /100$, (ii) $\theta =\pi /4$, (iii) $\theta =\pi /3$, and (iv) $\theta =\pi /2$ for $v_z=658.2$ meV nm and ${\alpha }_2=500 \mathrm{meV}{\mathrm{nm}}^2$. (b) Landau level spectrum for triple-Weyl semimetals for same orientations of the magnetic field for $v_z=658.2$ meV nm and ${\alpha }_3=1782.29\mathrm{meV}{\mathrm{nm}}^3$.

Figure 6

DOS for Landau levels in a double-Weyl semimetal, in units of $L/\left(2\pi \hslash v_z\right)$. The parameters are the same as those in Fig. 5. The black curves are the DOS when the applied magnetic field is along the $z$ direction $\left(\theta =0\right)$, see Eq. (14).

Figure 7

DOS for Landau levels in a triple-Weyl semimetal, in units of $L/\left(2\pi \hslash v_z\right)$. The parameters are the same as those in Fig. 5. The black curves are the DOS when the applied magnetic field is along the $z$ direction $\left(\theta =0\right)$, see Eq. (15).

Figure 8

A comparative plot of longitudinal magnetoconductivity (LMC) in single (black), double (red), and triple (blue) Weyl semimetals in the presence of only Gaussian impurities. The field is applied along the axis separating the Weyl nodes in pristine system (the $z$ direction). In the shaded region Landau levels are not sharp and one needs to account for weak localization effects. In this region the semiclassical theory for magnetotransport may also be applicable. Here ${\sigma }_0=e^2\hslash v_z^2/\left(2\pi n_i{U}_0^2\right)$, and for discussion see Eq. (30).

Figure 9

A plot of longitudinal magnetoconductivity (LMC) in double Weyl semimetal for various orientations of the applied magnetic field (captured by the parameter $\theta$). We here only account for Gaussian impurities. In the shaded region weak localization effect dominates the LMC, where our analysis is qualitatively inapplicable. Here ${\sigma }_0=e^2\hslash v_z^2/\left(2\pi n_i{U}_0^2\right)$, and for discussion see Eq. (30).

Figure 10

A plot of longitudinal magnetoconductivity (LMC) in triple Weyl semimetal for various orientations of the applied magnetic field (captured by the parameter $\theta$). We here only account for Gaussian impurities. In the shaded region weak localization effect dominates the LMC, where our analysis is qualitatively inapplicable. Here ${\sigma }_0=e^2\hslash v_z^2/\left(2\pi n_i{U}_0^2\right)$, and for discussion see Eq. (30).

Figure 11

A schematic variation of longitudinal magnetoconductivity (LMC) upon accounting for density-wave instabilities (either CDW or SDW) within the subspace of the zeroth Landau levels (ZLLs). (a) Qualitative variation of LMC is shown for any WSMs (constituted by Weyl nodes with monopole charge $n$) when the field is applied along the $z$ direction, and therefore $n$ number of ZLLs are exactly degenerate in a WSM. LMC becomes positive when $B>B^{*}$. Qualitative variation of LMC in (b) double and (c) triple Weyl semimetals when the field is tilted away from the $z$ direction (assuming the Weyl nodes are separated along the $z$ direction in the absence of magnetic field). Separate discontinuities in LMC arise since the $n$ ZLLs are no longer degenerate when $\theta \ne 0$. Each discontinuity is associated with the density-wave ordering in separate branches of the ZLLs, before LMC become negative when all ZLLs are gapped out.

Figure 12

A plot of longitudinal magnetoconductivity (LMC) in double Weyl semimetal for various orientations of the applied magnetic field (captured by the parameter $\theta$). We here only account for Coulomb impurities. In the shaded region weak localization effect dominates the LMC, where our analysis is qualitatively inapplicable.