Coulomb Interaction Effect in Weyl Fermions with Tilted Energy Dispersion in Two Dimensions

Weyl fermions with tilted linear dispersions characterized by several different velocities appear in some systems including the quasi-two-dimensional organic semiconductor $\alpha \text{−}(\mathrm{BEDT}\text{−}\mathrm{TTF}{)}_2{\mathrm{I}}_3$ and three-dimensional ${\mathrm{WTe}}_2$. The Coulomb interaction between electrons modifies the velocities in an essential way in the low-energy limit, where the logarithmic corrections dominate. Taking into account the coupling to both the transverse and longitudinal electromagnetic fields, we derive the renormalization group equations for the velocities of the tilted Weyl fermions in two dimensions, and found that they increase as the energy decreases and eventually hit the speed of light $c$ to result in the Cherenkov radiation. Especially, the system restores the isotropic Weyl cone even when the bare Weyl cone is strongly tilted and the velocity of electrons becomes negative in certain directions.

Figure 1

Two types of the energy dispersions of the tilted WFs. (a) Type I, with $v_i>w_i$ for all $i$. The conduction and valence bands touch at a single point $k=0$. (b) Type II. There are electron and hole pockets, which are bounded by lines and touch at $k=0$. In this figure, we set $v_i<w_i$ for a single $i$.

Figure 2

RG flows of the velocity parameters. (a) Flow of $v_x$ and $v_y$ without tilting $w=0$ and (b) flow of $v$ and $w$ with $v\equiv v_x=v_y$. The shaded region corresponds to type II where $w>v_x$. From those two figures, we can identify that the red point at $v_x=v_y=1$ and $w=0$ is the only stable fixed point.

Figure 3

(a) Scale dependence of the velocities as functions of $l=(g^2/2\pi )\ln (\mathrm{\Lambda }/\kappa )$. The initial values at $l=0$ ($\kappa =\mathrm{\Lambda }$) are given by $v_{x0}=v_{y0}=0.25$ and $w_0=0.2$. (b) Schematic picture of the renormalized energy dispersion of a tilted WF of type I. The orange and blue shapes depict the conduction and valence bands, respectively, while the green cones are the light cones. In the red region of the conduction band and the corresponding part of the valence band, the electron velocity exceeds the speed of light $c$ and the Cherenkov radiation takes place. Electron propagation decays and the energy dispersion is ill-defined in those regions.

Figure 4

Effect of the electron-electron interaction for a tilted Weyl cone of type II. (a) Schematic picture of the energy dispersion. (b) Corresponding Fermi surfaces, which consist of the blue and orange curves representing those of the valence and conduction bands, respectively, and the black point at the Weyl point.