Dynamical characterization of Weyl nodes in Floquet Weyl semimetal phases

Due to studies in nonequilibrium (periodically driven) topological matter, it is now understood that some topological invariants used to classify equilibrium states of matter do not suffice to describe their nonequilibrium counterparts. Indeed, in Floquet systems the additional gap arising from the periodicity of the quasienergy Brillouin zone often leads to unique topological phenomena without equilibrium analogs. In the context of Floquet Weyl semimetal, Weyl points may be induced at both quasienergy zero and $\pi /T$ ($T$ being the driving period) and these two types of Weyl points can be very close to each other in the momentum space. Because of their momentum-space proximity, the chirality of each individual Weyl point may become hard to characterize in both theory and experiments, thus making it challenging to determine the system's overall topology. In this work, inspired by the construction of dynamical winding numbers in Floquet Chern insulators, we propose a dynamical invariant capable of characterizing and distinguishing between Weyl points at different quasienergy values, thus advancing one step further in the topological characterization of Floquet Weyl semimetals. To demonstrate the usefulness of such a dynamical topological invariant, we consider a variant of the periodically kicked Harper model (the very first model in studies of Floquet topological phases) that exhibits many Weyl points, with the number of Weyl points rising unlimitedly with the strength of some system parameters. Furthermore, we investigate the two-terminal transport signature associated with the Weyl points. Theoretical findings of this work pave the way for experimentally probing the rich topological band structures of some seemingly simple Floquet semimetal systems.

Figure 1

The quasienergy spectrum of the toy model has been shown in (a) for fixed $k_z=0$. (b)–(d) The eigenphase band spectrum of the toy model for fixed $\varphi =\pi /2$, (c) $t=0$ and (d) $t=T$. Weyl nodes in Floquet bands (a) and phase band (b) and (c) crossing can be observed. Phase band crossing occur at single point in time domain which is captured by dynamical winding number.

Figure 2

Weyl nodes at zero (red) and $\pi$ (blue) quasienergy in the Brillouin zone are shown. We consider a torus [(a)–(d)] and spherical [(e) and (f)] geometry. The 2D torus surface is defined such that ${\delta }_x=[R+rsin\left(\theta \right)]sin\left(\varphi \right),{\delta }_y=rcos\left(\theta \right)$, and ${\delta }_z=[R+rsin\left(\theta \right)]cos\left(\varphi \right)$, where $\theta$ and $\varphi$ are polar and azimuthal angle $\theta ,\varphi \in [0,2\pi )$. For spherical geometry we consider $R=0$ and $\theta \in [0,\pi )$.

Figure 3

A slice of the system's quasienergy spectrum at parameter values $J=\lambda =1$ and $V=16$ and (a) and (d) ${\alpha }_{z_0}=\pi /2$, (b) and (e) ${\alpha }_{z_0}={cos}^{-1}(\pi /V)$, (c) ${\alpha }_{y_0}=\pi /2$, and (f) ${\alpha }_{y_0}=\pi /4$. (a)–(c) and (d)–(f) are obtained under periodic (open) boundary conditions respectively. Red (green) color represent the states localized at the left (right) edge of the open lattice in $x$ direction.

Figure 4

Schematic of a one-dimensional lattice in the $x$ direction with $N_x$ number of unit cells for our two-terminal conductance analysis. Each unit cell consists of two orbital degrees of freedom and green bonds represent intercell coupling. We applied absorbing boundary conditions in the form of zero-dimensional point contacts (black) at $n_x=1$ and $n_x=N_x$.

Figure 5

Two-terminal conductances $(G^0,G^{\pi })$ at zero (blue) and $\pi /T$ (red) quasienergy of a one-dimensional lattice in the $x$ direction with $N_x=101$ number of unit cells. We consider $J=\lambda =1,V=16, {\alpha }_y=\pm \pi /2$, (a) ${\alpha }_z=\pm {cos}^{-1}\left[\frac{2\ell \pi }{V}\right]$, and (b) ${\alpha }_z=\pm {cos}^{-1}\left[\frac{(2\ell +1)\pi }{V}\right]$. All points are obtained after averaging over 1000 disorder realizations.

Figure 6

Two distinct species of Weyl points (marked as $A$ and $B$) may arise in general time-periodic Weyl semimetal systems. Solid lines depict the dispersion within the main BZ of $(-\pi /T,\pi /T]$, with $A$ being the band touching point within the main BZ. Dashed lines represent the equivalent of solid lines by shifting $2\pi /T$, with $B$ a result of the crossing between different Floquet side bands. This feature is unique in Floquet systems because the quasienergy is only defined up to $2\pi /T$.

Figure 7

The numerical results for the slice Chern number and slice dynamical winding number in the extended KHM for system parameter values $J=\lambda =1,V=16$. It can be observed that there is no jump at the transition point which are given as ${\alpha }_y=\pm \pi /2$. Chern number and dynamical winding numbers show trivial behavior and these negligible values are due to some numerical error.