Optical conductivity signatures of open Dirac nodal lines

We investigate the optical conductivity and far-infrared magneto-optical response of ${\mathrm{BaNiS}}_2$, a simple square-lattice semimetal characterized by Dirac nodal lines that disperse exclusively along the out-of-plane direction. With the magnetic field aligned along the nodal line the in-plane Landau level spectra show a nearly $\sqrt{B}$ behavior, the hallmark of a conical-band dispersion with a small spin-orbit coupling gap. The optical conductivity exhibits an unusual temperature-independent isosbestic line, ending at a Van Hove singularity. First-principles calculations unambiguously assign the isosbestic line to transitions across Dirac nodal states. Our work suggests a universal topology of the electronic structure of Dirac nodal lines.

Figure 1

(a) Temperature-dependent in-plane reflectivity of ${\mathrm{BaNiS}}_2$ below 0.8 eV. Inset: Crystal structure of ${\mathrm{BaNiS}}_2$. (b) Temperature dependence of the in-plane real part of the optical conductivity. The gray areas show the SW transfer from high to low energy between 300 and 5 K. The inset shows the differential spectral weight $\mathrm{\Delta }\text{SW}\left(T\right)=\text{SW}\left(T\right)-{\text{SW}}_{300\text{K}}$, for different upper cutoff energies, as a function of temperature. (c) Color plot of the relative magnetoreflectance $R_B/R_0$. Lighter areas correspond to inter-LL excitations, which were assigned to transitions $-n\to n+1$ and $-n-1\to n$ and fitted using the LL spectrum in Eq. (1).

Figure 2

(a) First Brillouin zone and the HSE-calculated Fermi surface. Hole and electron pockets are indicated in yellow and purple, respectively. The red cylinder represents the constrained momentum space used to isolate the contribution of the nodal line. (b) HSE band structure, and DOS at zero and 300 K. (c) Low-energy optical conductivity calculated at different temperatures for the full IBZ (solid lines). Dashed lines show ${\sigma }_1\left(\omega \right)$ for transitions in the restricted $k$ space, defined by the shaded volume in (a). (d) Full optical conductivity at 100 K decomposed into different $k$-space contributions, for various cylinders in IBZ. Green, blue, and red lines shows ${\sigma }_1\left(\omega \right)$ for a cylinder centered at $k_{\text{Dirac}}, \mathrm{\Gamma }$, and $X$, respectively. The radius $R_{\text{sel}}$ of each cylinder is given in reciprocal lattice units.

Figure 3

(a) 3D view of the connection between the two DNLs dispersing along $k_z$ obtained from HSE calculations. (b) Comparison between the measured ${\sigma }_1\left(\omega \right)$ at 10 and 100K (thick lines), and the analytic form of Eq. (3) (dashed line) with an electric field perpendicular to the nodal-line dispersion $k_z$. The thin green line shows the $k$-space restricted theoretical ${\sigma }_1\left(\omega \right)$ at 100 K. (c) Dispersion of a 2D model on a square lattice. (d) ${\sigma }_1\left(\omega \right)$ of the model in (c) at different temperatures. The peak corresponds to interband transitions between two saddle points (VHSs) of the model. The black dashed line is the universal value of ${\sigma }_1\left(\omega \right)$ for a perfect conical model with four valleys and spin-1/2 fermions.