Electromagnetic response of interacting Weyl semimetals

TL;DR

This work develops a semiholographic framework to study how strong interactions at a quantum critical point modify the electromagnetic response of a Weyl semimetal. By coupling a single Weyl cone to a strongly interacting CFT via an operator with scaling dimension $Δ$ and tuning the coupling $g_f$, the authors compute the self-energy $Σ(k)$ and a corrected EM vertex using AdS/CFT techniques, capturing both frequency/momentum dependent charge renormalization and a novel, tensorial anomalous magnetic moment $μ_m(k)$. They derive the full conductivity by evaluating current-current correlators with both bare and vertex-dressed contributions, verify Ward identities, and show that the Coulomb limit $Δ→5/2$ with $g_f^2→ e/\sqrt{ħ c ε_0}$ yields the expected logarithmic corrections. The framework offers a controlled path to explore non-Fermi-liquid Weyl liquids, finite-temperature extensions, and other EM responses in interacting Weyl semimetals.

Abstract

We study the electromagnetic properties of Weyl semimetals with strong interactions. Focusing on a single Weyl cone in the band structure, we induce strong interactions by coupling the Weyl fermion with a tunable coupling constant $g_f$ to a quantum critical system. The critical fluctuations are described by a conformal field theory containing fermionic operators with scaling dimension $Δ$. Employing the methods of the holographic correspondence, we then derive the effective theory of the Weyl fermion in the presence of external electric and magnetic fields. In particular, we determine the frequency and momentum-dependent anomalous magnetic moment of the Weyl fermions. 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 $Δ\rightarrow 5/2$ and $g_f^2 \rightarrow e/\sqrt{\hbar cε_0$.

Figures (4)

• 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 parameterized by $g_f$ and the universality class of the quantum critical point determines the scaling dimension $\Delta$ 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 $G(k)$. The bare vertex diagram $(I)$ is the fermionic contribution $\langle J^{\mu}_{\chi} J^{\nu}_{\chi} \rangle$. The diagrams with one dressed vertex $(II)$ provide the interference term $2 \langle J^{(\mu}_{\chi} J^{\nu)} \rangle$. The diagram with two dressed vertices $(III)$ 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 $\Delta$. We have plotted the dimensionless functions $\tilde{C}^{II}(\Delta)=C^{II}(\Delta,g_f) g_f^2 c^{6-2\Delta}\hbar/e^2$ and $\tilde{C}^{III}(\Delta)=C^{III}(\Delta,g_f) \hbar c /e^2$.