Anisotropic fixed points in Dirac and Weyl semimetals

The effective low energy description of interacting Dirac and Weyl semimetals is that of massless quantum electrodynamics with several Lorentz breaking material parameters. We perform a renormalization group analysis of Coulomb interaction in anisotropic Dirac and Weyl semimetals and show that the anisotropy persists in the material systems at the infrared fixed point. In addition, a tilt of the fermion cones breaking inversion symmetry induces a magnetoelectric term in the electrodynamics of the material whose magnitude runs to match that of the electronic tilt at the fixed point.

Figure 1

One-loop divergent diagrams of QED. (a) Electron self-energy, (b) vertex, and (c) vacuum polarization.

Figure 2

Isotropic runnings of various parameters for the initial values $v_i=0.01,{\epsilon }_i=2,{\mu }_i=2$, with $i=1,2,3$.

Figure 3

(a) Runnings of the fermion and photon velocities in the $X$ direction in the anisotropic case with no tilt for the initial values are described in the text. Fermion and photon velocities converge to the same value at the infrared. The same happens in the $Y$ and $Z$ directions. (b) Comparative runnings of the Fermi velocities in the three directions. The asymptotic fixed point is anisotropic.

Figure 4

Running of the tilts and velocities for the near-isotropic ($v_1=v_2=0.01$; $v_3=0.03$) case with a fermion tilt $t=t_3=0.0075$. ${\epsilon }_1={\epsilon }_2=1;{\epsilon }_3=3;{\mu }_1={\mu }_2=4;{\mu }_3=1$. Panels (a) and (b) show the running of the fermion and photon velocities in the direction perpendicular and parallel to the tilt. (c) The initial values of the photon tilts ${\omega }_i$ are set to zero and run to a common value with the fermion tilt at the fixed point.