  The behavior of three-dimensional (3D) semimetals under strong magnetic
fields is a topic of recurring interest in condensed matter physics. Recently,
the advent of Weyl and Dirac semimetals has brought about an interesting
platform for potentially uncovering phases of matter that combine nontrivial
band topology and interactions. While electronic instabilities of such
semimetals at strong magnetic fields have been explored theoretically and
experimentally, the role of electron-phonon interactions therein has been
largely neglected. In this paper, we study the interplay of electron-electron
and electron-phonon interactions in a minimal two-node model of Weyl semimetal.
Using a Kadanoff-Wilson renormalization group approach, we analyze lattice
(Peierls) instabilities emerging from chiral and nonchiral Landau levels as a
function of the magnetic field. We consider both the adiabatic and the
nonadiabatic phonon regimes, in the presence or in the absence of improper
symmetries that relate Weyl nodes of opposite chirality. We find that (i) the
Cooper channel, often neglected in recent studies, can prevent purely
electronic instabilities while enabling lattice instabilities that are not
Bardeen-Cooper-Schrieffer-like; (ii) breaking the improper symmetry that
relates the two Weyl nodes suppresses the Cooper channel, thereby increasing
the critical temperature for the lattice instability; (iii) in the adiabatic
phonon regime, lattice instabilities can preempt purely electronic
instabilities; (iv) pseudoscalar phonons are more prone to undergo a Peierls
instability than scalar phonons. In short, our study emphasizes the importance
of taking electron-phonon interactions into account for a complete
understanding of interacting phases of matter in Dirac and Weyl semimetals at
high magnetic fields.
