Scaling of the chiral magnetic effect in quantum diffusive Weyl semimetals

We investigate the effect of short-range spin-independent disorder on the chiral magnetic effect (CME) in Weyl semimetals. Based on a minimum two-band model, the disorder effect is examined in the quantum diffusion limit by including the Drude correction and the correction due to the Cooperon channel. It is shown that the Drude correction renormalizes the CME coefficient by a factor to a finite value that is independent of the system size. Furthermore, due to an additional momentum expansion involved in deriving the CME coefficient, the contribution of Cooperon to the CME coefficient is governed by the quartic momentum term. As a result, in contrast to the weak localization and weak antilocalization effects observed in the measurement of conductivity of Dirac fermions, we find that in the limit of zero magnetic field, the CME coefficients of finite systems manifest the same scaling of localization even in three dimensions. Our results indicate that while the chiral magnetic current due to slowly oscillating magnetic fields can exist in clean systems, its observability will be limited by strong suppression due to short-range disorder in condensed matter.

Figure 1

Feynman diagrams for the disorder correction of the CME coefficient. (a) Each solid line represents the Green's function with first-order Born approximation. (b) Vertex correction of the CME coefficient. (c) Correction due to the ladder diagram for particle-particle channel with the maximally crossed diagrams. Leading corrections of the current-current correlation due to (d) the bare and (e) and (f) two dressed Hikami boxes from the maximally crossed diagrams. It turns out that two dressed Hikami boxes cancel each other out due to the antisymmetric combination of current-current correlation in the CME coefficient (see text for details).

Figure 2

Renormalized total CME coefficient due to the Drude correction. Here dashed lines reproduce the CME coefficient $\alpha$ of Ref. [17] in the clean limit. Open circles are the renormalized total CME coefficient with the Drude correction.

Figure 3

Correction of the CME coefficient due to the Cooperon channel, $\mathrm{\Delta }\alpha$ (in unit of $\frac{e^2{\lambda }_{so}}{\hslash }$), due to short-range disorder. (a) $\mathrm{sgn}\left(\alpha \right)\mathrm{\Delta }\alpha$ vs $t_z$ for different ${\gamma }_e$ in the intermediate region $l<\lambda <L$. (b) ${log}_{10}\left|\mathrm{\Delta }\alpha \right|$ vs disorder strength ${\gamma }_e$ for different $t_z$. Here ${\lambda }_{so}=1,T=0.001,{\epsilon }_F=0,l=100$, and the system size $L/a={10}^7$ with $a$ being the lattice constant.

Figure 4

(a) Transition behavior of ${log}_{10}\left|\mathrm{\Delta }\alpha \right|$ from the intermediate region, $l<\lambda <L$, to the clean limit $l<L\ll \lambda$. Here $t_z=0.1,{\lambda }_{so}=1,T=0.001,{\epsilon }_F=0$, and $l=100$. The clean limit, ${\gamma }_e={10}^{-6}$, exhibits the scaling behavior with $\mathrm{\Delta }\alpha \propto -(L/a)$. (b) ${log}_{10}\left|\mathrm{\Delta }\alpha \right|$ vs $logL/a$ for different $t_z$. It is seen that ${\beta }_{\mathrm{CME}}$ is always positive and approaches zero for larger systems. Here ${\gamma }_e=0.001,{\lambda }_{so}=1,T=0.001,{\epsilon }_F=0$, and $l=100$.