Electromagnetic response of interacting Weyl semimetals

We study the electromagnetic properties of Weyl semimetals with strong interactions. Aiming for a large-$N$ expansion, we induce strong interactions by coupling a Weyl fermion with a tunable coupling constant $g_f$ to a quantum critical system with a large number of order $O\left(N\right)$ fermionic and bosonic degrees of freedom. The critical fluctuations are described by a conformal field theory containing also fermionic composite operators with scaling dimension $\mathrm{\Delta }$. Employing the methods of the holographic correspondence, we then derive the effective theory of the Weyl fermions in the presence of external electric and magnetic fields in the large-$N$ limit. In particular, we determine their frequency and momentum-dependent anomalous magnetic moment. We also determine the conductivity of the Weyl semimetal including the vertex corrections consistent with the Ward identity. Finally, we connect our construction to the case of Coulomb interactions in Weyl semimetals by tuning the parameters $\mathrm{\Delta }\to 5/2$ and $g_f^2\to e/\sqrt{\hslash c{\epsilon }_0}$.

Figure 1

Our model describes a charge-neutral Weyl semimetal interacting at zero temperature with a quantum critical system containing fluctuations of both chiralities. The quantum critical region can be reached by tuning a parameter $p$, such as the pressure, to its critical value $p_c$. The strength of the interaction is parametrized by $g_f$ and the universality class of the quantum critical point determines the scaling dimension $\mathrm{\Delta }$ of its most relevant fermionic composite operator.

Figure 2

The semiholographic Witten diagrams representing the full propagator (left) and the full vertex (right) in the effective action for $\chi$. The effective action is given by a sum over all possible ways of connecting insertions of $\chi$ and $A$ fields through propagation both on the flat boundary $z=0$, and in the curved AdS bulk spacetime $z>0$.

Figure 3

Contributions to the current-current correlation function. Double lines denote $G\left(k\right)$. The bare vertex diagram $\left(I\right)$ is the fermionic contribution $〈J_{\chi }^{\mu }{J}_{\chi }^{\nu }〉$. The diagrams with one dressed vertex $\left(II\right)$ provide the interference term $\langle J_{\chi }^{\mu }{J}^{\nu }\rangle +\langle J_{\chi }^{\nu }{J}^{\mu }\rangle$. The diagram with two dressed vertices $\left(III\right)$ corrects the CFT current-current correlation function by $\langle J^{\mu }{J}^{\nu }\rangle -{\langle J^{\mu }{J}^{\nu }\rangle }_{\text{CFT}}$.

Figure 4

The conductivity prefactors as a function of the scaling dimension $\mathrm{\Delta }$. We have plotted the dimensionless functions ${\tilde{C}}^{II}\left(\mathrm{\Delta }\right)=C^{II}(\mathrm{\Delta },g_f)g_f^2{c}^{6-2\mathrm{\Delta }}\hslash /e^2$ and ${\tilde{C}}^{III}\left(\mathrm{\Delta }\right)=C^{III}(\mathrm{\Delta },g_f)\hslash c/e^2$.