Renormalization of Coulomb interactions in a system of two-dimensional tilted Dirac fermions

We investigate the effects of long-ranged Coulomb interactions in a tilted Dirac semimetal in two dimensions by using the perturbative renormalization-group (RG) method. Depending on the magnitude of the tilting parameter, the undoped system can have either Fermi points (type I) or Fermi lines (type II). Previous studies usually performed the renormalization-group transformations by integrating out the modes with large momenta. This is problematic when the Fermi surface is open, like type-II Dirac fermions. In this work we study the effects of Coulomb interactions, following the spirit of Shankar [Rev. Mod. Phys. 66, 129 (1994)], by introducing a cutoff in the energy scale around the Fermi surface and integrating out the high-energy modes. For type-I Dirac fermions, our result is consistent with that of the previous work. On the other hand, we find that for type-II Dirac fermions, the magnitude of the tilting parameter increases monotonically with lowering energies. This implies the stability of type-II Dirac fermions in the presence of Coulomb interactions, in contrast with previous results. Furthermore, for type-II Dirac fermions, the velocities in different directions acquire different renormalization even if they have the same bare values. By taking into account the renormalization of the tilting parameter and the velocities due to the Coulomb interactions, we show that while the presence of a charged impurity leads only to charge redistribution around the impurity for type-I Dirac fermions, for type-II Dirac fermions, the impurity charge is completely screened, albeit with a very long screening length. The latter indicates that the temperature dependence of physical observables are essentially determined by the RG equations we derived. We illustrate this by calculating the temperature dependence of the compressibility and specific heat of the interacting tilted Dirac fermions.

Figure 1

Conical band structure ($\xi =1$) for type-I Dirac fermions (left) and type-II Dirac fermions (right). For type-II Dirac fermions, the blue lines on the $E=0$ plane indicate the Fermi surface.

Figure 2

The RG flow of the ratio $r_l=v_{1l}/v_{2l}$ with $r\left(0\right)=1$ and various values of $\left|w\right|$ (left) and with $\left|w\right|=1.5$ and various various values of $r$ (right) for type-II Dirac fermions. In both diagrams we have taken $v_2=1.3g^2/\left(4\pi \right)$.

Figure 3

The RG flow of $|w_l|$ with $r=1$ and various values of $\left|w\right|$ (left) and with $\left|w\right|=1.5$ and various values of $r$ (right) for type-II Dirac fermions. In both diagrams we have taken $v_2=1.3g^2/\left(4\pi \right)$.

Figure 4

${\kappa }^{-1}={(\partial \mu /\partial n)}_T$ at $\mu =0$, in units of ${\kappa }_0^{-1}$, as a function of $T/T_0$ for type-I Dirac fermions (left) and type-II Dirac fermions (right), where ${\kappa }_0$ is the compressibility at $T=T_0=D/k_B$. In both diagrams, we take $v_1=v_2=1.3g^2/\left(4\pi \right)$.

Figure 5

The solid line shows the isothermal compressibility ${\kappa }^{-1}$ at $\mu =0$, in units of ${\overline{\kappa }}^{-1}$, as a function of $\left|w\right|$ at $T/T_0=0.01$ for both types of Dirac fermions, where $\overline{\kappa }$ is the compressibility of untilted Dirac fermions at $T=T_0$. For reference, ${\kappa }^{-1}$ for noninteracting tilted Dirac fermions at the same temperature is shown by the dashed line. In both cases, we take $v_1=v_2=1.3g^2/\left(4\pi \right)$.

Figure 6

The solid line shows the specific heat $c\left(T\right)$, in units of $\overline{c}$, as a function of $\left|w\right|$ at $T/T_0=0.01$ for both types of Dirac fermions, where $\overline{c}$ is the specific heat of untilted Dirac fermions at $T=T_0$. For reference, $c\left(T\right)$ for noninteracting tilted Dirac fermions at the same temperature is shown by the dashed line. [Due to the divergence of $c\left(T\right)$ near $\left|w\right|=1$, we have rescaled the value for noninteracting Dirac fermions by a factor of 0.01.] In both cases, we take $v_1=v_2=1.3g^2/\left(4\pi \right)$.