Axial anomaly and longitudinal magnetoresistance of a generic three-dimensional metal

We show that the emergence of the axial anomaly is a universal phenomenon for a generic three-dimensional metal in the presence of parallel electric ($E$) and magnetic ($B$) fields. In contrast to the expectations of the classical theory of magnetotransport, this intrinsically quantum mechanical phenomenon gives rise to the longitudinal magnetoresistance for any three-dimensional metal. However, the emergence of the axial anomaly does not guarantee the existence of negative longitudinal magnetoresistance. We show this through an explicit calculation of the longitudinal magnetoconductivity in the quantum limit using the Boltzmann equation, for both short-range neutral and long-range ionic impurity scattering processes. We demonstrate that the ionic scattering contributes a large positive magnetoconductivity $\propto B^2$ in the quantum limit, which can cause a strong negative magnetoresistance for any three-dimensional or quasi-two-dimensional metal. In contrast, the finite range neutral impurities and zero-range point impurities can lead to both positive and negative longitudinal magnetoresistance depending on the underlying band structure. In the presence of both neutral and ionic impurities, the longitudinal magnetoresistance of a generic metal in the quantum limit initially becomes negative, and ultimately becomes positive after passing through a minimum. We discuss in detail the qualitative agreement between our theory and recent observations of negative longitudinal magnetoresistance in Weyl semimetals TaAs and TaP, Dirac semimetals ${\text{Na}}_3\text{Bi}, {\text{Bi}}_{1-x}{\text{Sb}}_x$, and ${\text{ZrTe}}_5$, and quasi-two-dimensional metals ${\text{PdCoO}}_2$ and $\alpha -{\left(\text{BEDT-TTF}\right)}_2{\text{I}}_3$, which do not possess any bulk three-dimensional Dirac or Weyl quasiparticles.

Figure 1

An illustration of the LMR due to the combined effects of neutral and ionic impurities [Eq. (30)]. The quantum limit is achieved at a magnetic field strength $B=B_0$. The initial decrease of LMR for $B>B_0$ is governed by ionic scattering and attains its minimum value at a nonuniversal field strength $B^{*}$ [Eq. (31)]. For $B>B^{*}$, the sharp increase of LMR is controlled by the neutral impurities. The choice of $B^{*}=2B_0$ has been made only for an illustrative purpose. The actual value of the crossover field $B^{*}$ depends on many nonuniversal sample-dependent details such as the comparative amounts of ionic long-range and neutral short-range disorder, the carrier density, etc., and $B^{*}$ could, in principle, be large or small, thus suppressing the positive or negative LMR regimes in experiments.

Figure 2

The dispersing LLs of a three-dimensional electron gas, where ${\omega }_c=eB/m$ is the cyclotron frequency and $l_B=\sqrt{\hslash /eB}$ is the magnetic length. The LLs with $N=0,1,2$ and 3 are, respectively, represented with the colors red, black, blue, and green. The dashed brown lines correspond to two possible locations of the Fermi energy. When the Fermi energy lies entirely within the $N=0$ LLL, we mark it by QL to illustrate that we are in the quantum limit. In the quantum limit only the partially occupied LLL produces a set of Fermi points. In contrast, away from the quantum limit, several partially filled LLs can produce Fermi points. Each partially occupied LL gives rise to the axial anomaly and participates in the longitudinal magnetotransport.

Figure 3

LMC for an electron gas in semiconductors [Eq. (33)] in the quantum limit. (a) Gaussian potential scattering: LMC with a fixed carrier density of $n=6.75\times {10}^{26}{\text{m}}^{-3}$ as a function of ${(a/l_B)}^4$, where $a$ is the range of the scattering potential, and ${\sigma }^s\left(0\right)$ is the conductivity in the absence of a magnetic field due to short-range scattering (see Appendix). We find ${\sigma }^s\left(B\right)/{\sigma }^s\left(0\right)$ decreases as a function of $B$, which gives rise to a positive LMR. (b) Ionic scattering: LMC with a fixed carrier density of $n=6.75\times {10}^{26}{\text{m}}^{-3}$ as a function of ${(a_0/l_B)}^4$, where $a_0$ is the Bohr radius in the vacuum and ${\sigma }^c\left(0\right)$ is the conductivity in the absence of a magnetic field (see Appendix). We find ${\sigma }^c\left(B\right)/{\sigma }^c\left(0\right)\sim B^2$, which gives rise to a negative LMR.

Figure 4

The dispersive LLs for a quasi-two-dimensional metal [Eq. (42)], when the magnetic field is applied along the $c$ axis. (a) $4t_z>\hslash {\omega }_c$ and (b) $4t_z<\hslash {\omega }_c$. For a high carrier density corresponding to Eq. (40), the underlying Fermi surface is a corrugated cylinder and $\hslash {\omega }_c$ exceeds $4t_z$ before reaching the quantum limit. In this case, each LL is well separated as shown in panel (b) which gives rise to two Fermi points, and the system behaves effectively like the one in the quantum limit.

Figure 5

The dispersive LLs of a quasi-two dimensional metal with a high carrier density in a magnetic field (solid) applied at an angle $\theta$ with respect to the $c$ axis, as a function of $k_z^{\prime }{c}^{\prime }$ (defined in the text), for $t_z/\left(\hslash {\omega }_c\right)=2$. We are considering a large Fermi energy such that the Fermi level crosses only the LLs with large index $N$. For (a) $\theta =0$, there are a large number of Fermi points from multiple LLs. (b) For the same field strength and tilt angle $\theta =0.09\pi$, the LLs with high index $N$ become completely flat and do not produce any Fermi points. Therefore the effective hopping strength along the field direction, for a large LL index vanishes, showing the existence of the Yamaji angle, independent of the magnitude of $t_z/\left(\hslash {\omega }_c\right)$. (c) For a small deviation away from the Yamaji angle, at $\theta =0.08\pi$, there is only one partially filled LL producing two Fermi points, similar to the situation in the quantum limit.

Figure 6

The dispersing LLs of a three-dimensional WSM. We have chosen the following parameters $\mathrm{\Delta }/t_z=0.2, t_{\perp ,1}/t_z=20$, and $B=2$ T. The LLs with $N=0$, 1, 2, and 3 are, respectively, represented with the colors red, black, blue, and green.

Figure 7

The dispersing LLs of a DSM with two Dirac cones. We have chosen the following parameters $\mathrm{\Delta }/t_z=0.2, t_{\perp ,1}/t_z=20$, and $B=2$ T. The LLS with $N=0$, 1, 2, and 3 are respectively represented with the colors red, black, blue, and green. In contrast to a WSM, the $N\ne 0$ LLs are twofold degenerate, and there are two $N=0$ levels. In the quantum limit as denoted by the dashed brown line, the carriers in the two levels are respectively the electrons and the holes.

Figure 8

The dispersing LLs of a DSM with a single Dirac cone as determined from Eq. (73). We have chosen the following parameters $t_{\perp ,1}/t_z=20, t^{\prime }/t=0.4$, and $B=2$ T. The LLs with $N=0$, 1, 2, and 3 are, respectively, represented with the colors red, black, blue, and green. The LLs with $N\ne 0$ are two fold degenerate. In contrast, $N=0$ levels are nondegenerate.