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 ${\mathbf{K}}_{\ensuremath{\tau}}=\ensuremath{\tau}\mathbf{b}$ in momentum space with $\ensuremath{\tau}=\ifmmode\pm\else\textpm\fi{}1$ the node index and chirality. The electronic dispersion in a small wave-vector region near each node is linear and isotropic. In a magnetic field $\mathbf{B}=B\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{z}}$, this band structure is modified into a series of positive and negative energy Landau levels $n=\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,...$ which disperse along the direction of the magnetic field, and a chiral Landau level $n=0$, with a linear dispersion given by ${e}_{\ensuremath{\tau},n=0}({k}_{z})=\ensuremath{-}\ensuremath{\tau}\ensuremath{\hbar}{v}_{F}{k}_{z},$ where ${k}_{z}$ is the component of the electron wave vector $\mathbf{k}$ along the direction of the magnetic field and ${v}_{F}$ is the Fermi velocity. In the extreme quantum limit and for a small doping, the Fermi level is in the chiral levels near the Dirac point. It has been shown before that, when Coulomb interaction is considered, a Weyl semimetal may be unstable towards the formation of a condensate of internodal electron-hole pairs which gives rise in real space to an excitonic charge-density wave. This new state of matter is usually studied by using a short-range interaction between the electrons. In this paper 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 the situation where the Weyl nodes have a higher Chern number $C=2,3$ and more complex excitonic states are possible. We derive the response functions and collective excitations of the excitonic state working in the generalized random-phase approximation (GRPA). We show that, in the mean-field gap induced by the internodal coherence, there is, in the excitonic response function, a series of bound electron-hole states (excitons) with a binding energy that decreases until the renormalized Hartree-Fock energy gap is reached. In addition, there is a collective mode gapped at exactly the plasmon frequency. By contrast, the plasmon mode is the only excitation present in the density and current response functions. Despite the U(1) symmetry of the excitonic state, there is no gapless mode in the GRPA excitonic response. Indeed, the gapless mode present in the proper excitonic response function is pushed to the plasmon frequency by the long-range Coulomb interaction.

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