Internodal excitonic state in a Weyl semimetal in a strong magnetic field

The simplest Weyl semimetal with broken time-reversal symmetry consists of a pair of Weyl nodes located at wave vectors $\mathbf{K}_{\tau }=\tau \mathbf{b}$ in momentum space with $\tau =\pm 1$ the node index and chirality. The electronic dispersion near each node is linear. In a magnetic field $\mathbf{B}$ along $\mathbf{b}$, this dispersion is modified into a series of positive and negative energy Landau levels $n=\pm 1,\pm 2,\ldots ,$which disperse along the direction of the magnetic field, and a chiral Landau level, $n=0$, with a linear dispersion given by $e_{\tau ,n=0}\left( k_{z}\right) =-\tau \hslash v_{F}k_{z},$ where $k_{z}$ is the component of the wave vector $\mathbf{k}$ along the magnetic field and $v_{F}$ is the Fermi velocity. In the extreme quantum limit, the Fermi level is in the chiral levels near the Dirac point. When Coulomb interaction is considered, a Weyl semimetal may be unstable towards the formation of a condensate of internodal electron-hole pairs. In this article we use the full long-range Coulomb interaction and the self-consistent Hartree-Fock approximation to generate the condensed state. We study its stability with respect to a change in the Fermi velocity, doping and strength of the Coulomb interaction and also consider Weyl nodes with higher Chern number $C=2,3$. We derive the response functions of the excitonic state in the generalized random-phase approximation (GRPA). We show that, in the mean-field gap induced by the internodal coherence, there is, in the GRPA excitonic response, a series of bound electron-hole states with a binding energy that decreases until the Hartree-Fock energy gap is reached. In addition, there is a collective mode gapped at exactly the plasmon frequency: the gapless mode present in the proper excitonic response function is pushed to the plasmon frequency by the long-range Coulomb interaction.