Many-body effects and ultraviolet renormalization in three-dimensional Dirac materials

We develop a theory for electron-electron interaction-induced many-body effects in three-dimensional Weyl or Dirac semimetals, including interaction corrections to the polarizability, electron self-energy, and vertex function, up to second order in the effective fine-structure constant of the Dirac material. These results are used to derive the higher-order ultraviolet renormalization of the Fermi velocity, effective coupling, and quasiparticle residue, revealing that the corrections to the renormalization group flows of both the velocity and coupling counteract the leading-order tendencies of velocity enhancement and coupling suppression at low energies. This in turn leads to the emergence of a critical coupling above which the interaction strength grows with decreasing energy scale. In addition, we identify a range of coupling strengths below the critical point in which the Fermi velocity varies nonmonotonically as the low-energy, noninteracting fixed point is approached. Furthermore, we find that while the higher-order correction to the flow of the coupling is generally small compared to the leading order, the corresponding correction to the velocity flow carries an additional factor of the Dirac cone flavor number (the multiplicity of electron species, e.g. ground-state valley degeneracy arising from the band structure) relative to the leading-order result. Thus, for materials with a larger multiplicity, the regime of velocity nonmonotonicity is reached for modest values of the coupling strength. This is in stark contrast to an approach based on a $\text{large}-N$ expansion or the random phase approximation (RPA), where higher-order corrections are strongly suppressed for larger values of the Dirac cone multiplicity. This suggests that perturbation theory in the coupling constant (i.e., the loop expansion) and the $\text{RPA}/\text{large}-N$ expansion are complementary in the sense that they are applicable in different parameter regimes of the theory. We show how our results for the ultraviolet renormalization of quasiparticle properties can be tested experimentally through measurements of quantities such as the optical conductivity or dielectric function (with carrier density or temperature acting as the scale being varied to induce the running coupling). Although experiments typically access the finite-density regime, we show that our zero-density results still capture clear many-body signatures that should be visible at higher temperatures even in real systems with disorder and finite doping.

Figure 1

Leading-order polarization bubble ${\mathrm{\Pi }}_B\left(q\right)$.

Figure 2

Self-energy (left) and vertex (right) corrections to the polarization ${\mathrm{\Pi }}_{SE}\left(q\right)$ and ${\mathrm{\Pi }}_V\left(q\right)$, respectively.

Figure 3

One-loop electron self-energy ${\mathrm{\Sigma }}_1\left(q\right)$.

Figure 4

Plot of the function $f\left(z\right)$ appearing in Eq. (34), giving the dependence of the finite term in ${\mathrm{\Pi }}_{SE}\left(q\right)$ as a function of $z=q_0/v_F\left|\vec{q}\right|$.

Figure 5

Plot of $\lambda \frac{d}{d\lambda }F(z,\lambda )$ for $z=5$, where $\lambda =\mathrm{\Lambda }/|\vec{q}|$ and $z=q_0/v_F\left|\vec{q}\right|$. At large $\lambda$, this curve becomes approximately linear; the slope of the line is exactly $-4$ and is twice the coefficient of the ${ln}^2\lambda$ term, while the constant term gives us the coefficient of the $ln\lambda$ term. The solid red line is a linear fit to the points at large $\lambda$.

Figure 6

Plot of the function $g\left(z\right)$ appearing in Eq. (71), giving the dependence of the finite part of ${\mathrm{\Pi }}_V\left(q\right)$ as a function of $z=q_0/v_F\left|\vec{q}\right|$.

Figure 7

Two-loop rainbow correction to the electron self-energy ${\mathrm{\Sigma }}_{2b}\left(q\right)$.

Figure 8

Two-loop vertex correction to the electron self-energy ${\mathrm{\Sigma }}_{2a}\left(q\right)$.

Figure 9

Two-loop bubble correction to the electron self-energy ${\mathrm{\Sigma }}_{2c}\left(q\right)$.

Figure 10

Diagram corresponding to the counterterm ${\mathrm{\Pi }}_{SE,CT}\left(q\right)$ for the two-loop self-energy polarization diagram.

Figure 11

(a) Coulomb counterterm contribution to the second-order self-energy counterterm diagrams ${\mathrm{\Sigma }}_{2ct}\left(q\right)$. This is the only nonvanishing contribution. (b) Self-energy counterterm contribution. This diagram vanishes for the same reason that the “rainbow” diagram is zero. (c) Vertex counterterm contribution. This diagram only contributes at orders higher than ${\alpha }^2$.

Figure 12

Numerical solution of the RG equations for $\alpha$ (dark gray, red online) and $v_F$ (light gray, blue online) for $\alpha \left({\mu }_0\right)=0.9$ and $N=12$, which are typical values for the pyrochlore iridates. Since $\alpha \left({\mu }_0\right)$ is far below ${\alpha }_c$, indicated by the red dashed line, we find that it increases for small $\mu$, then saturates to ${\alpha }_c$. $v_F$, on the other hand, decreases as $\mu$ increases up to about $\mu \approx 0.95{\mu }_0$, then starts increasing. This is better illustrated in Fig. 13.

Figure 13

Plot of $v_F$ alone, with the range for $\mu$ adjusted to better illustrate the nonmonotonic behavior of $v_F$.

Figure 14

Drude weight for extrinsic (red solid line) and intrinsic (green dashed line) Dirac materials. At high temperature, there is no difference between the intrinsic and the extrinsic Drude weight, thus manifestly showing that temperature can act as an appropriate running scale to study the intrinsic Dirac point physics.

Figure 15

Plasmon dispersion for extrinsic (red solid line) and intrinsic (green dashed line) systems with ${\mathrm{\Lambda }}_L=8{\varepsilon }_F$. The manifest superlinearity in the temperature-dependent plasmon dispersion of the doped system arises from the ultraviolet renormalization with temperature also acting as the cutoff for $T\ll T_F$, as expected.

Figure 16

Two-loop self-energy vertex diagram ${\mathcal{V}}_{SE}(p,q)$.

Figure 17

Two-loop vacuum polarization bubble vertex diagram ${\mathcal{V}}_B(p,q)$.

Figure 18

Two-loop parallel Coulomb line vertex diagram ${\mathcal{V}}_{PC}\left(q\right)$.

Figure 19

Two-loop crossed Coulomb line vertex diagram ${\mathcal{V}}_{CC}\left(q\right)$.

Figure 20

Two-loop vertex correction vertex diagram ${\mathcal{V}}_V\left(q\right)$.

Figure 21

Chemical potential vs temperature as obtained by solving Eq. (E10). Dashed lines indicate the low- and high-temperature limits (E11).