Time-dependent electric transport in nodal loop semimetals

Close to the Fermi energy, nodal loop semimetals have a torus-shaped, strongly anisotropic Fermi surface, which affects their transport properties. Here we investigate the non-equilibrium dynamics of nodal loop semimetals by going beyond linear response and determine the time evolution of the current after switching on a homogeneous electric field. The current grows monotonically with time for electric fields perpendicular to the nodal loop plane however it exhibits nonmonotonical behavior for field orientations aligned within the plane. After an initial nonuniversal growth $\sim Et$, the current first reaches a plateau $\sim E$. Then, for perpendicular directions, it increases while for in-plane directions it decreases with time to another plateau, still $\sim E$. These features arise from interband processes. For long times or strong electric fields, the current grows as $\sim E^{3/2}t$ or $\sim E^3{t}^2$ for perpendicular or parallel electric fields, respectively. This nonlinear response represents an intraband effect where the large number of excited quasiparticles respond to the electric field. Our analytical results are benchmarked by the numerical evaluation of the current from continuum and tight-binding models of nodal loop semimetals.

Figure 1

Schematic picture of the current in nodal loop semimetals when the electric field perpendicular to the nodal loop (blue line) and parallel to it (red line). The black dashed lines represent the timescales, which split up the time domain into four different region.

Figure 2

Visualization of the Landau-Zener dynamics when the $E$ points in the $z$ direction. The gap between the two bands is determined by ${\widehat{p}}_z$ initially. During the time evolution, the gap starts to decrease and when ${\widehat{p}}_z-\widehat{t}=0$ it closes at ${\widehat{p}}_{\perp }=\pm \sqrt{\widehat{\mathrm{\Delta }}}$, then start to increase. During the time evolution, the system is driven through a quantum critical point [41].

Figure 3

Transition probability for short (left) and long (right) times. The $n_{\mathbf{p}}\left(t\right)$ resembles closely to the transition probability obtained for graphene and Weyl semimetals [41, 42], i.e., dipolar for short times and cylindrical for long times, but shifted to ${\widehat{p}}_{\perp }=\pm \sqrt{\widehat{\mathrm{\Delta }}}$.

Figure 4

Temporal behavior of the current after switching on the electric field. The blue and black dashed lines represent the polarization current in Eq. (17) and Eq. (18), respectively. When $\widehat{t}\gg 1$ the conductive current becomes dominant, scaling as $\sim \widehat{t}$.

Figure 5

Schematic plot of the dispersion relation in the $\widehat{\mathrm{\Delta }}\to 0$ and $\widehat{\mathrm{\Delta }}\to \infty$ limit. For a given energy (green dashed line) slightly above the gap edge, there are twice as many empty states in the conduction band for $\widehat{\mathrm{\Delta }}$ large, therefore the current is 2 times larger.

Figure 6

Left panel: The time evolution of the dispersion relation is illustrated. The time-dependent vector potential shifts the nodal loop in the $x$ direction. Right panel: The time evolution of ${\widehat{p}}_x$ at different values of ${\widehat{\mathrm{\Delta }}}_{\text{eff}}$ and ${\widehat{p}}_z$. When ${\widehat{\mathrm{\Delta }}}_{\text{eff}}>0, {\widehat{p}}_z$ determines the gap, and ${\widehat{\mathrm{\Delta }}}_{\text{eff}}>0$ sets the location of the two minimum/maximum points. On the other hand, when ${\widehat{\mathrm{\Delta }}}_{\text{eff}}<0$ the gap starts to increase and only one minimum/maximum remains.

Figure 7

Transition probability for short (top) and long times when ${\mathrm{\Delta }}_{\text{eff}}>0$ (middle) and ${\mathrm{\Delta }}_{\text{eff}}<0$ (bottom). For short times, the numerical result agrees with the result from first-order perturbation theory. For long times, when ${\mathrm{\Delta }}_{\text{eff}}>0$, we have three different regions (separated with white lines) depending on how far the nodal loop is shifted from its original position. In this case, the modified DDP formula works well for the transition probability. In the last case, when ${\mathrm{\Delta }}_{\text{eff}}<0$ the simple DDP formula is reliable only in the adiabatic limit (${\widehat{p}}_z\to \infty$). However, the disagreement has only a minor effect on the current since the number of excited particles decreases exponentially as ${\mathrm{\Delta }}_{\text{eff}}\to -\infty$.

Figure 8

Short and long time behavior of the current after switching on the electric field in $x$ direction for large and small values of $\widehat{\mathrm{\Delta }}$. The blue dashed line represents the $\sim 1/t$ decay of the polarization current while the black dashed line shows the $t\to \infty$ limit of the polarization current from Eq. (29), which is $\sim {\sigma }_x^{0}E$. The insets show the long time behavior when the conductive current is dominant. For large $\widehat{\mathrm{\Delta }}$ a linear term in time can arise, due to the single channel transitions, but it is overwhelmed by the leading $\sim {\widehat{t}}^2$ term with increasing time.

Figure 9

The electric current from the tight-binding model for $\widehat{E}$ in the $z$ direction. Top panel: Short-time response with $\delta /\gamma =1.5$. The brown and light blue dashed lines represent the two time scales $1/\delta$ and $1/\mathrm{\Delta }=1/(2\gamma -\delta )$, respectively. The polarization current dominates, represented by the black dashed line. Bottom panel: Long-time response with with $\delta /\gamma =1.5$. The orange dashed line represents the $\sim E^{3/2}t$ prediction. For large electric fields and long times, Bloch oscillations start to kick in.

Figure 10

The electric current from the tight-binding model for $\widehat{E}$ pointing to the $x$ direction. Top panel: Short-time response for $\delta /\gamma =1.5$. The brown dashed line represents the border of the ultrashort time domain at $1/\delta$. The black dashed line represents the polarization current. Bottom panel: Long-time response with $\delta /\gamma =1.97$. The orange dashed line represents our prediction $\sim E^3{t}^2$. The additional constant term, $j_x^0$ is the dimensionless static current coming from the polarization current. For longer times, the Bloch oscillations would kick in.