Multifold Weyl semimetals under irradiation: The particularity of singlet Weyl points

We investigate the topological phases of Weyl semimetals possessing a pair of multifold ($N$-fold) Weyl points (WPs) irradiated by an off-resonant circularly polarized light in the frame of Floquet theory. The irradiation splits each $N$-fold WP into two singlet WPs and one residual WP of $N-2$ fold in $k$ space. When the irradiation intensity increases, the two singlet WP pairs move faster and are gapped out first; therefore, a phase transition from $N$-fold to $(N-2)$-fold Weyl semimetal occurs. The pair of remaining $(N-2)$-fold WPs are annihilated later and a sequential phase transition to insulator happens. The irradiation influences the system by means of two mechanisms—the time average of the perturbation and the emission and reabsorption of photons, with the latter usually regarded as responsible for the phase transitions of matter and the former sometimes omitted due to its trivial effect. However, we find that these phase transitions are caused by the former, not by the latter, and the latter only affects how the WPs in the same pair approach each other, except for the case of $N=1$ in which both mechanisms take effect. The most common Weyl semimetal, say $N=1$, has the most special response. The irradiation can induce not only the transition from Weyl semimetal to insulator, but also the inverse transition, while the latter cannot happen in Weyl semimetals with $N\ge 2$.

Figure 1

(a),(b) Sketches of Weyl point (WP) splitting and moving in $k$ space when the multifold ($N=4$) Weyl semimetal is irradiated by a circularly polarized light in a different direction. The split WP pairs draw trajectories in the $k_x\text{−}{k}_z$ plane (see Fig. 3) when the irradiation intensity increases. (c) Transitions between phases of matter.

Figure 2

Light illuminating along the $z$ direction for $N=1$. Upper panels: Weyl point connection length $W$ as a function of $A_0$ for different $\omega$ (in units of $k_w$) at $v=1$ in Eq. (15). Lower panels: phase boundaries for different $v$.

Figure 3

Light illuminating along the $x$ direction. Left panels: trajectories of Weyl points (WPs) moving in the $k_x\text{−}{k}_z$ plane when $A_0$ increases. The outer and inner circles are the node circles defined in Eq. (28) for $A_0=0$ and $A_0=0.6k_w$, respectively. The thin solid curves are the hyperbolic curves at $A_0=0.6k_w$. The red and blue filled dots mark the intersections between the node circles and hyperbolic curves, which are the locations of the WPs for $A_0=0.6k_w$. The empty dots are the locations at which the WP pairs are gapped out. Right panels: distance between the WP pair canceled out first as a function of $A_0$ for different $\omega$ (in units of $k_w$). In all panels, $v=1$ is adopted.

Figure 4

Phase diagrams for light illuminating along the $x$ direction. (a)–(c) Phase diagrams for Weyl semimetals of fold $N=1,2,3$. (d) Boundaries between the phase of multifold Weyl semimetal and the adjacent phase of matter for $N=1\sim 4$.