Interacting nodal semimetals with nonlinear bands

We investigate the quasiparticle and transport properties of a model describing interacting Dirac and Weyl semimetals in the presence of local Hubbard repulsion $U$, where we explicitly include a deviation from the linearity of the energy-momentum dispersion through an intermediate-energy scale $\mathrm{\Lambda }$. Our focus lies on the correlated phase of the semimetal. At the nodal point, the renormalization of spectral weight at a fixed temperature $T$ exhibits a weak dependence on $\mathrm{\Lambda }$ but is sensitive to the proximity to the Mott transition. Conversely, the scattering rate of quasiparticles and the resistivity display high-temperature exponents that crucially rely on $\mathrm{\Lambda }$, leading to a crossover towards a conventional Fermi-liquid behavior at finite $T$. Finally, by employing the Nernst-Einstein relation for conductivity, we identify a corresponding density crossover as a function of the chemical potential.

Figure 1

The free-particle energy dispersion (on the right) and the correspondent DOS in three dimensions (on the left) are shown. The DOS is taken quadratic at low energy $\left|\epsilon \right|<\mathrm{\Lambda }$ and constant at high energy $\mathrm{\Lambda }<\left|\epsilon \right|<D$. The cutoff $\mathrm{\Lambda }$ sets the scale where the deviation from the linearity of the bands becomes relevant.

Figure 2

sMIT at half-filling with $\mathrm{\Lambda }/D=1$ and $T/D=0.01$ illustrated with the evolution of the local DMFT spectral function. The transition from the semimetallic to the insulating state occurs at a specific critical value $U_{c_2}$ that in this specific case is $U_{c_2}/D=5.6$.

Figure 3

The low-frequency spectral function at half-filling for $U/D=4.2$ and $T/D=0.01$ in the strongly correlated semimetallic phase for different values of $\mathrm{\Lambda }/D$. Inset: the same quantity with the frequency axis scaled with quasiparticle $Z$. The Fermi level is indicated by the dashed line.

Figure 4

(a) The quasiparticle weight $Z$ at half-filling for $T/D=0.01$ is calculated at several values of $\mathrm{\Lambda }/D$ as a function of $U/U_{c2}$. The same figure also shows the value obtained for a flat DOS ($\mathrm{\Lambda }/D=0$). The black lines represent the second-order diagram calculations both for the Dirac model with $\mathrm{\Lambda }/D=1$ [9] and the flat one with $\mathrm{\Lambda }/D=0$ (Appendix pp7). In the inset, the values $U_{c1}/D$ and $U_{c2}/D$ as a function of $\mathrm{\Lambda }/D$ are shown for $T/D=0.01$. (b) Comparison between the renormalization factor $Z$ calculated with IPT and QMC. Horizontal error bars are estimated considering the uncertainty in calculating $U_{c2}/D$, and they are shown in order to specify that the two methods give different values beyond errors, but the resulting behavior is qualitatively similar.

Figure 5

The scattering rate $\mathrm{\Gamma }/D$ at half-filling for $T/D=0.01$ calculated at several values of $\mathrm{\Lambda }/D$ as a function of $U/U_{c2}$.

Figure 6

The central U-shape in the spectral function at half-filling for $U/D=3.0$ and $\mathrm{\Lambda }/D=0.2$ as a function of temperatures. In the inset, the momentum distribution as a function of $\epsilon \left(\mathit{k}\right)$. The black bold line is $T^{*}/D=0.1$. The vertical dashed line indicates the position of the Fermi level.

Figure 7

Conductivity (a), charge compressibility (b), and diffusivity (c) at $\mu =0$ as a function of the temperature at weak coupling ($U/D=0.5$). The same quantities [(d), (e), and (f), respectively] at $\mu =0$ as a function of the scaled temperature at weak coupling ($U/D=0.5$). The dashed gray lines are useful to separate the crossover central region from the NSM one at lower temperatures and the bad metal one at higher temperatures.

Figure 8

Conductivity (a), charge compressibility (b), and diffusivity (c) at $\mu =0$ as a function of the temperature at strong coupling near the sMIT. Arrows indicate the temperature at which the quasiparticle U-shape disappears ($T^{*}/D=0.1$). The dashed gray lines are useful to separate the transition central region from the NSM one at lower temperatures and the bad metal one at higher temperatures.

Figure 9

Conductivity (a), charge compressibility (b), and diffusivity (c) for $U/D=0.5$ and $\mathrm{\Lambda }/D=0.2$ as a function of the chemical potential. Different curves refer to different temperatures. Dashed gray lines indicate the width of the U-shaped structure in the spectral function.

Figure 10

Conductivity (a), charge compressibility (b), and diffusivity (c) for $U/D=3.0$ and $\mathrm{\Lambda }/D=0.2$ as a function of the chemical potential. Different curves refer to different temperatures. The green bold line ($T^{*}/D=0.1$) marks the temperature at which, at half-filling, the quasiparticle U-shape vanishes. Dashed gray lines indicate the width of the U-shaped structure in the spectral function.

Figure 11

The imaginary part of the self-energy at half-filling for $U/D=0.5$ and $T=0.02$ scaled with $S^9\left(\mathrm{\Lambda }\right)$ in order to obtain the same Fermi velocity of any $\mathrm{\Lambda }/D$ (a). The same quantity for $U/D=3.0$ (b).

Figure 12

${\mathrm{\Gamma }}_{\text{opt}}$ and $\mathrm{\Gamma }$ at half-filling for $T/D=0.02$ and $U/D=0.5$ and 3.0. The dashed green line represents the $\mathrm{\Lambda }/D$ dependence from the perturbation theory [Eq. (C1)].

Figure 13

Model of free-particles DOS compared with the DOS calculated from the tight-binding (TB) model of Ref. [14] with parameters $\lambda =0.2$. We fit our model to the curvature at the nodal point, which sets our scale at $\mathrm{\Lambda }/D=0.3$.

Figure 14

The quasiparticle damping at half-filling normalized with $U^2$ as a function of the temperature ($T=T/D$) for $\mathrm{\Lambda }=\mathrm{\Lambda }/D=1.0, 0.2$, 0.05 and several values of $U=U/D$ from the metallic side approaching the MIT.

Figure 15

The quasiparticle damping at half-filling normalized with $U_0^2$ at fixed $U=U/D$ and $\mathrm{\Lambda }=\mathrm{\Lambda }/D$ as a function of the temperature ($T/D_0$) and scaled to obtain a comparison between quantities with the same bare Fermi velocity. The subscript 0 indicates that the quantity is in units $D_0$ (see Appendix pp4). The color code is the same as in Fig. 14.

Figure 16

Free-particle DOS used in the IPT codes. Notice the different curvature around $\omega =0$, which implies different bare Fermi velocity. Inset: Scaled free-particle DOS, which shows the same bare Fermi velocity.

Figure 17

Optical conductivity at half-filling for $T/D=0.02$ and $U/D=0.5$ (a) and $U/D=3.0$ (b). In both panels, the dashed red curve indicates the intraband contribution to the optical spectra at $\mathrm{\Lambda }/D=1$. They are obtained by taking into account only the intraband contribution from the Kubo formula [Eq. (B1)].

Figure 18

The spectral function shown at various chemical potentials for $\mathrm{\Lambda }/D=0.2, U/D=3.0$, and $T=T^{*}(\mu =0.0)=0.1D$, where at half-filling the semimetal behavior (U-shape) disappears. The dashed gray line indicates the position of the Fermi level.

Figure 19

Filling as a function of the chemical potential for different temperatures at $\mathrm{\Lambda }/D=0.2$ and $U/D=0.5$ (a) and $U/D=3.0$ (b). Dashed gray lines indicate the width of the U-shaped structure in the spectral function.