Theory of photoinduced Floquet Weyl semimetal phases

The Weyl semimetal exhibits various interesting physical phenomena because of the Weyl points, i.e., linear band crossings. We show by Floquet theory that a linearly polarized light applied to a band insulator can induce controllable Weyl points. In a tight-binding model, we classify different types of photoinduced Weyl points that lead to a rich phase diagram characterized by the Chern number defined on each momentum slice of the bulk states. Taking into account the nonequilibrium electron distribution, we calculate and explain the nonmonotonous anomalous Hall conductivity in terms of the light frequency controlled shift of Weyl points' position, which also allows us to examine the conductivity's dependence on the driving electric field.

Figure 1

Illustration of three typical cases of photoinduced band shifting and hybridization (drawn in the $k_y\text{−}{k}_z$ plane) as a function of momentum $k_x$. Without the driving term, we have the gapped orange upper and lower bands. Lower orange band absorbs one photon of frequency $\mathrm{\Omega }$ (red arrow) and gets shifted upward to the purple lower band. The two purple bands are the Floquet bands (after hybridization) formed between the foregoing ascended one-photon-dressed band and the original orange upper band. (Left) The gap is larger than the light frequency $\mathrm{\Omega }$ and hence no band overlap occurs. (Middle) The gap equals to the light frequency at a certain $k_x$ position and the green band touching gets reshaped linearly and remains gapless under driving by the optical perturbation. (Right) The gap is smaller than the light frequency. Anticrossing gaps open around the ring of band overlap due to the hybridization introduced by the driving term.

Figure 2

(a)–(c) $\mathrm{\Omega }=9$ periodic drive applied to the band insulator. Left: Five adjacent Floquet bands near the zero energy along special lines in $k_y\text{−}{k}_z$ plane depicted by the cyan triangle in (d). Middle: Linear dispersion along $k_x$ (towards the left) and $k_y$ (towards the right) axes close to the Weyl point. Right: Berry curvature of the green $(0,-)$ band over the first Brillouin zone in $k_y\text{−}{k}_z$ plane when $k_x$ lies slightly above the Weyl point.

Figure 3

(a) Classification of the photoinduced Weyl points in the 2D 1st BZ in the $k_y\text{−}{k}_z$ plane. Four high-symmetry lines along the $k_x$ direction in the full 3D 1st BZ are signified by colored dots, where photoinduced Weyl points (monopoles in momentum space) are possible to reside. Type-$A$ or type-$B$ monopole has topological charge $+1$. Type-$C$ monopole has topological charge $-1$ and appears always in pair as shown by the two green dots since $k_y\text{−}{k}_z$ are on the same footing in the model. (b) Typical example of Chern number and monopoles along $k_x$ axis. $\mathrm{\Omega }=8$ for the model initially possessing two Weyl points at $\vec{k}=(\pm \frac{\pi }{2},0,0)$.

Figure 4

Chern-number diagram as a function of $k_x\in [-\pi ,0]$ and driving frequency $\mathrm{\Omega }$ for the $(0,-)$ band. Left: Numerical result via solving the Floquet Hamiltonian for $\mathrm{\Omega }\in [4,30]$. Middle: Analytic result for a larger range $\mathrm{\Omega }\in [1,30]$. Right: Analytic result for the small frequency region $\mathrm{\Omega }\in [1,5]$.

Figure 5

Photoinduced anomalous Hall conductivity vs driving frequency $\mathrm{\Omega }$. Temperature $T$ is measured in units of hopping energy $t_x$ in the Hamiltonian.

Figure 6

Typical creation and annihilation of photoinduced Weyl points are shown by the Chern-number diagrams along $k_x$ direction. Upper (lower) diagram in each subfigure is plotted when the driving frequency $\mathrm{\Omega }$ is 3‰ below (above) the frequency ${\mathrm{\Omega }}_0$ at which pair creation and/or annihilation occur. Meanings of various markers are the same as those introduced in Fig. 3 and Sec. 3a.

Figure 7

Hall conductivity's dependence on the strength of electric field $V$. Translucent gray dots are equilibrium conductivities without light irradiation. Other dots are numerically calculated data while lines are obtained from curve fitting. Temperature $T$ is measured in units of hopping energy $t_x$ in the Hamiltonian.