Functional renormalization group approach to interacting three-dimensional Weyl semimetals

We investigate the effect of long-range Coulomb interaction on the quasiparticle properties and the dielectric function of clean three-dimensional Weyl semimetals at zero temperature using a functional renormalization group (FRG) approach. The Coulomb interaction is represented via a bosonic Hubbard-Stratonovich field which couples to the fermionic density. We derive truncated FRG flow equations for the fermionic and bosonic self-energies and for the three-legged vertices with two fermionic and one bosonic external legs. We consider two different cutoff schemes—cutoff in fermionic or bosonic propagators—in order to calculate the renormalized quasiparticle velocity and the dielectric function for an arbitrary number of Weyl nodes and the interaction strength. If we approximate the dielectric function by its static limit, our results for the velocity and the dielectric function are in good agreement with that of A. A. Abrikosov and S. D. Beneslavskiĭ [Sov. Phys. JETP 32, 699 (1971)] exhibiting slowly varying logarithmic momentum dependence for small momenta. We extend their result for an arbitrary number of Weyl nodes and finite frequency by evaluating the renormalized velocity in the presence of dynamic screening and calculate the wave function renormalization.

Figure 1

Graphical representation of the FRG flow equations for the fermionic (upper panel) and bosonic (lower panel) self-energies as given in Eqs. (15) and (16), respectively. The dot over the symbol denotes the derivative with respect to cutoff $\mathrm{\Lambda }$, the solid line with the arrow represents the fermionic propagator ($G_{\mathrm{\Lambda }}^{b{b}^{\prime }}$), the solid line with the arrow and slash symbolizes the single-scale propagator (${\dot{G}}_{\mathrm{\Lambda }}^{b{b}^{\prime }}$), and the wavy line denotes the bosonic propagator ($F_{\mathrm{\Lambda }}$). The triangles illustrate renormalized three-legged vertices with two fermionic and one bosonic leg (${\mathrm{\Gamma }}_{\mathrm{\Lambda }}^b$) which carry the same band label $b$; conduction (C) shown in red and valence (V) in blue.

Figure 2

Diagrammatic representation of the FRG flow equation for the three-legged vertex, Eq. (17), with two fermionic and one bosonic leg and carrying the same band label. The shaded circle with three wavy lines is the purely bosonic three-legged vertex. The meaning of the other symbols is the same as in Fig. 1.

Figure 3

Static renormalized velocity in the limit of vanishing cutoff as a function of quasimomentum $k$ for (a) $N_W=2$, (b) $N_W=12$, and (c) $N_W=24$ and $\alpha =0.2,0.5,1.0,2.0$. The broken black line is a fit given in the form of Eq. (55).

Figure 4

Static renormalized dielectric function in the limit of vanishing cutoff as a function of quasimomentum $q$ for (a) $N_W=2$, (b) $N_W=12$, and (c) $N_W=24$ and $\alpha =0.2,0.5,1.0,2.0$. The broken black line is a fit given in the form of Eq. (56).

Figure 5

The values for $\mathcal{A}(\alpha ,N_W)$ and $\mathcal{B}(\alpha ,N_W)$, from Eq. (55), are shown in the upper panel as $◯$ and $\square$, respectively, and obtained as the fitting parameters to the numerical solution of Eq. (43) which is shown in Fig. 3. In the lower panel, the values for $\mathcal{C}(\alpha ,N_W)$ and $\mathcal{D}(\alpha ,N_W)$, from Eq. (56), are shown as $△$ and $▿$, respectively, and obtained as the fitting parameters to the numerical solution of Eq. (44) which is shown in Fig. 4.

Figure 6

Upper panel: Graphical illustration of the truncated FRG flow equation for the fermionic self-energy, where the slashed wavy lines are the bosonic single-scale propagators. Lower panel: Exact skeleton equation for the bosonic self-energy. Black dots on the right-hand side denote the bare vertex. All other symbols are the same as in Fig. 1.

Figure 7

The integral $\mathcal{I}(\lambda ,\mathrm{\Omega })$ as a function of dimensionless momentum $\lambda =\frac{2{\mathrm{\Lambda }}_0}{q}$ and frequency $\mathrm{\Omega }=\frac{\overline{\omega }}{v_{\mathrm{\Lambda }}q}$, Eq. (68), showing logarithmic divergence in the limit of small momentum $q$ and frequency $\overline{\omega }$.

Figure 8

Numerical solution of the dynamical RG flow equation of the velocity, Eq. (76), for two sets of $\alpha$ and $N_W$. The static results from Eq. (53), for the same values of $\alpha$ and $N_W$, are also shown as broken lines.

Figure 9

RG flow of wave function renormalization, $Z_{\mathrm{\Lambda }}$, obtained for two sets of $\alpha$ and $N_W$.

Figure 10

Diagrammatic representation of the FRG flow equation for the three-legged vertex, Eq. (A1), with band-label changing two fermionic and one bosonic legs. The magenta triangle signifies the band-label changing three-legged vertex, ${\mathrm{\Gamma }}^{(2,1)}$. The factor $2\times$ denotes the permutation of the single-scale propagator and the meaning of the other symbols is the same as in Fig. 1.

Figure 11

Graphical representation of the FRG flow equation for the purely bosonic three-legged vertex, Eq. (21). The factor $3\times$ signifies the cyclic permutation of the vertices in the second and third row for the fixed position of the single-scale propagator. The meaning of the symbols is the same as in Figs. 1 and 2.