Optical response of Luttinger semimetals in the normal and superconducting states

We investigate the optical response properties of three-dimensional Luttinger semimetals with the Fermi energy close to a quadratic band touching point. In particular, in order to address recent experiments on the spectroscopy of pyrochlore iridates and half-Heusler superconductors, we derive expressions for the optical conductivity in both the normal and general superconducting states in the linear response regime within the random phase approximation. The response functions can be decomposed into contributions from intraband and interband transitions, the latter comprising a genuine signature of the quadratic band touching point. We demonstrate the importance of interband transitions in the optical response in the normal state both in the homogeneous and quasistatic limit. Our analysis reveals a factorization property of the homogeneous conductivity in the spatially anisotropic case and the divergence of the conductivity for strong spatial anisotropy. In the quasistatic limit, the response is dominated by interband transitions and significantly different from systems with a single parabolic band. As an applications of the formalism in the superconducting state we compute the optical conductivity and superfluid density for the $s$-wave singlet superconducting case for both finite and vanishing chemical potential.

Figure 1

The contributions to the dielectric tensor ${\varepsilon }_{ij}(\omega ,\mathbf{p})$ can be split into three parts. The first two arise from intraband transitions within, respectively, the upper or lower band, and as such can be computed without knowledge of the other bands. In contrast, interband transitions or genuine QBT contributions are not captured by a single-band model. They encode, however, many important physical features of Luttinger semimetals. For instance, in the normal state they lead to a divergent contribution at low frequency as $E_{\mathrm{F}}\to 0$, or they contain the response from Bogoliubov Fermi surfaces in certain time-reversal symmetry breaking superconducting states—a feature entirely absent in single band systems.

Figure 2

The low-momentum regime with $p^2/\left(2m^{*}\right)\ll \omega$ naturally decomposes into two sectors depending on whether the pro-duct $vp$ with Fermi velocity $v\propto \sqrt{E_{\mathrm{F}}}$ is dominating or irrelevant compared to the remaining energy scales such as $\omega$ or $T$. For $vp\ll \omega$, which amounts to setting $p=0$ in practice, we obtain the homogeneous limit, where L and T response coincide. For $vp\gtrsim \omega$, on the other hand, frequencies are necessarily small compared to $\mu$ and hence this regime is labeled the quasistatic limit. The inherent momentum dependence of the response then implies that L and T contributions differ.

Figure 3

QBT contribution to the homogeneous dielectric function $\varepsilon \left(\omega \right)$. We show the real and imaginary part in the upper and lower plot, respectively, as a function of $\omega /E_{\mathrm{F}}$. Here we normalize the expressions by the zero temperature limit $\varepsilon \left(0\right)=\frac{e^2}{2{\pi }^2{\epsilon }_0}\sqrt{2m^{*}/E_{\mathrm{F}}}$. The distinct curves (from bottom to top along the zero frequency axis) correspond to $T/E_{\mathrm{F}}$ values of 0 (black), 0.1 (orange), 0.3 (blue), and 0.5 (magenta). At zero temperature we observe singular behavior at $\omega =2E_{\mathrm{F}}$, which extends to an anomalously large, $1/\sqrt{\omega }$-divergent contribution to both the real and imaginary parts of the optical response as $E_{\mathrm{F}}\to 0$. At nonzero temperature the functions remain regular.

Figure 4

The homogeneous optical response for nonvanishing spatial anisotropy $\delta$ gets renormalized by a prefactor $\lambda \left(\delta \right)/\sqrt{1-{\delta }^2}$ that diverges for strong anisotropy. This statement is true for both the intraband and interband contributions, at both zero and nonzero temperature, for $x=0$. For $\delta =0$ we have, of course, $\lambda \left(0\right)=1$. The solid line shows the function $\lambda \left(\delta \right)$ computed from the two-dimensional integral in Eq. (S.164) in the SM [47], whereas the dashed line corresponds to the expansion around $\delta =0$ to quadratic order from Eq. (54). The latter should be sufficient for all practical purposes.

Figure 5

Intraband contributions to the dielectric tensor in the quasistatic limit as a function of $\omega /vp$. Results are plotted in units of $\frac{n_0{e}^2}{{\epsilon }_0{m}^{*}}\frac{3}{p^2{v}^2}$, the solid lines constitute the longitudinal response, the dashed lines the transverse response. The zero temperature results, shown in black, display singular behavior at $\omega =vp$. In particular, the real longitudinal component diverges logarithmically at this point. At nonzero temperature the functions are regular, shown here for $T/E_{\mathrm{F}}=0.1$ (orange) and $T/E_{\mathrm{F}}=0.3$ (blue). We observe the leading contribution at small frequencies to be imaginary and transverse. For large $\omega /vp\gg 1$, longitudinal and transverse response converge to the homogeneous limit.

Figure 6

QBT contribution to the dielectric tensor in the quasistatic limit as a function of $\omega /vp$. Curves are normalized by $\frac{e^2}{4{\pi }^2{\epsilon }_0}\sqrt{2m^{*}/E_{\mathrm{F}}}$, and longitudinal (solid lines) and transverse (dashed lines) contributions are shown for $T/E_{\mathrm{F}}=0$ (black), $T/E_{\mathrm{F}}=0.1$ (orange), and $T/E_{\mathrm{F}}=0.3$ (blue). The interband contributions remain finite at $\omega =vp$, although showing nonanalytic behavior at zero temperature. For large $\omega /vp$ we recover the large additive contribution to the real part of $\varepsilon \left(\omega \right)$. For small $\omega /vp$, the longitudinal contribution settles at a real value which is half the homogeneous limit. The transverse component diverges in both the real and imaginary parts with the real part being most dominant. As a result, the limit $\omega /vp\to 0$ of ${\varepsilon }_{\mathrm{L},\mathrm{T}}(\omega ,p)$ is fully dominated by the QBT contribution, see the discussion in the main text.

Figure 7

Crossover from normal to superfluid behavior in the QBT contribution to the optical conductivity at $T=0$. The black solid line shows the result in the $s$-wave superconducting case with gap $\mathrm{\Delta }/E_{\mathrm{F}}=0.1$, whereas the black dashed line shows the corresponding normal state result. The orange long-dashed line corresponds to the low-frequency behavior $n^{\prime \left(\mathrm{QBT}\right)}{e}^2/\left(m^{*}\omega \right)$ with QBT contribution to the Drude weight from Eq. (88). We observe that ${\sigma }_2\left(\omega \right)$ changes sign and so connects the negative normal state limit for $\omega \gg \mathrm{\Delta }$ to the positive Drude like scaling at low frequencies $\omega \sim \mathrm{\Delta }$. The same behavior is found in the strong coupling case with $\mu =0$, see Fig. 8.

Figure 8

QBT contribution to the optical conductivity of a strong coupling superconductor with $\mu =0$. The solid lines in the upper and lower panel show the real and imaginary part, respectively, for $T=0$ (black), $T/\mathrm{\Delta }=1$ (orange), and $T/\mathrm{\Delta }=2$ (blue). We only plot the regular part of ${\sigma }_1\left(\omega \right)$, see Eq. (92). The dashed lines show the corresponding normal state limit $\sigma \left(\omega \right)=\frac{e^2}{4\pi }\sqrt{m^{*}\omega }(1-i)$ for $\mu =T=0$. The real part displays threshold behavior at $\omega =2\mathrm{\Delta }$, whereas the imaginary part changes sign at this frequency. As a result, the imaginary part is negative for large frequencies—in agreement with the negative normal state limit—and it is positive with Drude-like behavior as in Eq. (80) for small frequencies.