Fate of Weyl semimetals in the presence of incommensurate potentials

We investigate the effect of an incommensurate potential on Weyl semimetal, which is proposed to be realized in ultracold-atomic systems trapped in three-dimensional optical lattices. For the system without the Fermi arc, we find that Weyl points are robust against the incommensurate potential and the system enters into a metallic phase only when the incommensurate potential strength exceeds a critical value. We unveil the transition by analyzing the properties of wave functions and the density of states as a function of the incommensurate potential strength. We also study the system with Fermi arcs and find that the Fermi arcs are fragile against the incommensurate potential and can be destroyed by a weak incommensurate potential.

Figure 1

Dispersions of the $N\mathrm{th}$ and $(N+1)\mathrm{th}$ levels of the system with (a) $V=0$, (c) $V=1$, (e) $V=2.1$, and (g) $V=2.5$ and the corresponding dispersions vs $k_y$ with fixed $k_x=0$ in (b), (d), (f), and (h), respectively. Here we consider the system with $t_x=t_y=t_z=1$ and $N=300$ under OBC in the $z$ direction and PBC in the $x$ and $y$ directions.

Figure 2

(a) IPR and (b) MIPR as a function of $k_x$ and $k_y$ with fixed $V=1.9$. (c) IPR and (d) MIPR as a function of $k_x$ and $V$ with fixed $k_y=\frac{\pi }{2}$. (e) IPR and (f) MIPR as a function of $k_y$ and $V$ with fixed $k_x=0$. The lattice size is $N=300$ and $t_x=t_y=t_z=1$.

Figure 3

(a) $\rho \left(0\right)$ vs $V$ for different lattice size $N$ with fixed $\sigma =0.02$. (b) DOS with $N=300$ as a function of energy for various values of incommensurate potential strength $V$. Here, $t_x=t_y=t_z=1$.

Figure 4

(a) The projection of dispersions of $N\mathrm{th}$ and $(N+1)\mathrm{th}$ levels of the system with $V=2.5$ and $t_x=t_y=t_z=1$ on the $k_y-E$ plane. (b) The cross section of energy dispersion $E(k_x,k_y)$ at $E=0\pm 0.001$.

Figure 5

(a) Dispersions of the $N\mathrm{th}$ and $(N+1)\mathrm{th}$ levels of the system with $V=0$. (b)–(d) Dispersions of the $N\mathrm{th}$ and $(N+1)\mathrm{th}$ levels as a function of $k_{\left|\right|}$ with fixed $k_z=\frac{\pi }{2}$ for the system with (b) $V=0$, (c) $V=0.1$, and (d) $V=1$. Here we have taken $t_x=t_y=t_z=1$. The dotted lines of (b) and (c) are the referenced line of $E=0$.

Figure 6

Dispersions of the $N\mathrm{th}$ and $(N+1)\mathrm{th}$ levels of the system vs $k_y$ with fixed $k_x=0$. The blue dotted lines are the fitted straight lines using the data near the Weyl points. The parameters of (a) are the same as Fig. 1 and the parameters of (b) are the same as Fig. 1.