Transport in the non-Fermi liquid phase of isotropic Luttinger semimetals

Luttinger semimetals have quadratic band crossings at the Brillouin-zone center in three spatial dimensions. Coulomb interactions in a model that describes these systems stabilize a nontrivial fixed point associated with a non-Fermi liquid state, also known as the Luttinger-Abrikosov-Beneslavskii phase. We calculate the optical conductivity $\sigma \left(\omega \right)$ and the dc conductivity ${\sigma }_{dc}\left(T\right)$ of this phase, by means of the Kubo formula and the Mori-Zwanzig memory matrix method, respectively. Interestingly, we find that $\sigma \left(\omega \right)$, as a function of the frequency $\omega$ of an applied ac electric field, is characterized by a small violation of the hyperscaling property in the clean limit, which is in contrast with the low-energy effective theories that possess Dirac quasiparticles in the excitation spectrum and obey hyperscaling. Furthermore, the effects of weak short-ranged disorder on the temperature dependence of ${\sigma }_{dc}\left(T\right)$ give rise to a stronger power-law suppression at low temperatures compared to the clean limit. Our findings demonstrate that these disordered systems are actually power-law insulators. Our theoretical results agree qualitatively with the data from recent experiments performed on Luttinger semimetal compounds like the pyrochlore iridates $\left[{\left({\mathrm{Y}}_{1-x}{\mathrm{Pr}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7\right]$.

Figure 1

The noninteracting dispersion ${\varepsilon }_{\mathbf{k}}$ of the isotropic Luttinger semimetal [see Eq. (2)] shows quadratic band touching at the Brillouin-zone center. Here, we choose $m=1$ and $m^{\prime }=0.5$. For visualization, ${\varepsilon }_{\mathbf{k}}$ is shown as a function of $k_x$ and $k_y$ (i.e., we set $k_z=0$).

Figure 2

The four-fermion vertex arising due to Coulomb interactions.

Figure 3

Feynman diagram for the contribution to the current-current correlation function at one-loop order.

Figure 4

Feynman diagrams for the contributions to the current-current correlation function at two-loop order. (a) and (b) represent the diagrams with self-energy corrections, while (c) corresponds to the diagram with vertex correction.

Figure 5

Plot of the current-momentum susceptibility ${\chi }_{J_z{P}_z}^{1\mathrm{loop}}\left(T\right)$ at one-loop order versus temperature $T$. Here, we have chosen the parameters $m=1$, $m^{\prime }=5$, $N_f=1$, and the UV cutoff ${\mathrm{\Lambda }}_0$ for the momentum integrals has been taken to the infinity limit (note that this result does not depend on the UV cutoff). The temperature dependence of this one-loop contribution is found to be $|{\chi }_{J_z{P}_z}^{1\mathrm{loop}}\left(T\right)|\sim 0.170T^{3/2}$.

Figure 6

Plot of the current-momentum susceptibility ${\chi }_{J_z{P}_z}^{2\mathrm{loop}}\left(T\right)$ at two-loop order vs temperature $T$. Here, we have chosen the parameters $m=1$, $m^{\prime }=5$, $e=0.1$, $c=1$, $N_f=1$, and ${\mathrm{\Lambda }}_0=150$. The temperature dependence of this two-loop contribution is found to be $|{\chi }_{J_z{P}_z}^{\left(2\mathrm{loop}\right)}\left(T\right)|\sim 0.005T^{1/2}$.

Figure 7

Feynman diagram for the calculation of the leading-order contribution $M_{P_z{P}_z}^{\left(0\right)}\left(T\right)$ to the memory matrix. The solid line represents the bare fermionic propagator, whereas the dashed line represents the impurity line that carries only internal momentum and external energy $\omega$.

Figure 8

Plot of $M_{P_z{P}_z}^{\left(0\right)}\left(T\right)$ vs temperature $T$. Here, we have chosen the parameters $W_0=1$, $N_f=1$, $m=1$, $m^{\prime }=5$, and ${\mathrm{\Lambda }}_0=150$. To obtain the memory matrix, we have performed the analytical continuation $i\omega \to \omega +i\delta$, where we have set $\delta ={10}^{-9}$. The curve corresponds to the fit given by $g\left(T\right)=a^{\prime }+b^{\prime }/T$, where the parameters $a^{\prime }\approx 0.059$ and $b^{\prime }\approx 211.25$ depend only on the UV cutoff ${\mathrm{\Lambda }}_0$.