Chiral anomaly, dimensional reduction, and magnetoresistivity of Weyl and Dirac semimetals

By making use of the Kubo formula, we calculate the conductivity of Dirac and Weyl semimetals in a magnetic field. We find that the longitudinal (along the direction of the magnetic field) magnetoresistivity is negative at sufficiently large magnetic fields for both Dirac and Weyl semimetals. The physical reason of this phenomenon is intimately connected with the dimensional spatial reduction $3\to 1$ in the dynamics of the lowest Landau level. The off-diagonal component of the transverse (with respect to the direction of the magnetic field) conductivity in Weyl semimetals contains an anomalous contribution directly proportional to the momentum-space separation between the Weyl nodes. This contribution comes exclusively from the lowest Landau level and, as expected, is independent of the temperature, chemical potential, and magnetic field. The other part of the off-diagonal conductivity is the same as in Dirac semimetals and is connected with a nonzero density of charge carriers. The signatures for experimental distinguishing Weyl semimetals from Dirac ones through the measurements of conductivity are discussed.

Figure 1

Longitudinal conductivity ${\sigma }_{33}$ at zero temperature as a function of the magnetic field. The solid line shows the complete result, the dashed line shows the contribution without the lowest Landau level, and the dotted line shows the topological contribution of the lowest Landau level alone. The quasiparticle width in higher Landau levels is $\Gamma =0.1\mu$ (left panels) and $\Gamma =0.2\mu$ (right panels). The LLL quasiparticle width is the same (upper panels) or half (lower panels) the width in higher Landau levels.

Figure 2

Longitudinal resistivity ${\rho }_{33}$ at zero temperature as a function of the magnetic field. The solid line shows the complete result, the dashed line shows the contribution without the lowest Landau level, and the dotted line shows the topological contribution of the lowest Landau level alone. The quasiparticle width is $\Gamma =0.1\mu$ (left panels) and $\Gamma =0.2\mu$ (right panels). The LLL quasiparticle width is the same (upper panels) or half (lower panels) the width in higher Landau levels.

Figure 3

Diagonal components of the transverse conductivity ${\sigma }_{11}={\sigma }_{22}$ at zero temperature as a function of the magnetic field. The quasiparticle width is $\Gamma =0.05\mu$ (black solid line), $\Gamma =0.1\mu$ (blue dashed line), and $\Gamma =0.2\mu$ (red dotted line). The sum over Landau levels includes $n_{\max }={10}^4$ levels.

Figure 4

Off-diagonal components of the transverse conductivity ${\sigma }_{12}=-{\sigma }_{21}$ as a function of the magnetic field for the vanishing chiral shift $\mathbf{b}=0$. The results are shown for $\Gamma =T=0$ (green thin solid line), $\Gamma \to T=0.05\mu$ (black solid line), $\Gamma \to T=0.1\mu$ (blue dashed line), and $\Gamma \to T=0.2\mu$ (red dotted line). If $b\ne 0$, the conductivity will simply shift by $e^{2}b/\left(2{\pi }^2\right)$.

Figure 5

Transverse components of resistivity ${\rho }_{11}$ and ${\rho }_{12}$ at zero temperature as functions of the magnetic field for $b=0$ (upper panels) and $b=0.3\mu$ (lower panels). The quasiparticle width is $\Gamma =0.05\mu$ (black solid line), $\Gamma =0.1\mu$ (blue dashed line), and $\Gamma =0.2\mu$ (red dotted line). The sum over Landau levels includes $n_{\max }={10}^4$ levels.