Simulating and exploring Weyl semimetal physics with cold atoms in a two-dimensional optical lattice

We propose a scheme to simulate and explore Weyl semimetal physics with ultracold fermionic atoms in a two-dimensional square optical lattice subjected to experimentally realizable spin-orbit coupling and an artificial dimension from an external parameter space, which may increase experimental feasibility compared with the cases in three-dimensional optical lattices. It is shown that this system with a tight-binding model is able to describe essentially three-dimensional Weyl semimetals with tunable Weyl points. The relevant topological properties are also addressed by means of the Chern number and the gapless edge states. Furthermore, we illustrate that the mimicked Weyl points can be experimentally detected by measuring the atomic transfer fractions in a Bloch-Zener oscillation, and the characteristic topological invariant can be measured with the particle pumping approach.

Figure 1

Color-coded extended 3D bulk band structure $E_{+}(k_x,\theta ,k_z)$ for the typical parameters $m_z$. (a) $m_z=1$ with two pairs of analogous Weyl points located at $(k_x,\theta ,k_z)=(0,\pi ,\pm \pi /3)$ and $(\pi ,0,\pm \pi /3)$; (b) $m_z=2$, critical case with two pairs of analogous Weyl points merging at $(0,\pi ,0)$ and $(\pi ,0,0)$, and another pair appear at $(0,0,\pm \pi )$; (c) $m_z=4$ with one pair of analogous Weyl points located at $(0,0,\pm \pi /2)$; and (d) $m_z=6$, critical case with the pair of analogous Weyl points merging at $(0,0,0)$. Other parameters in (a)–(d) are $t_0=1$ as the energy unit and $t_s=1$.

Figure 2

Energy spectrum of the reduced tight-binding chain of analogous Chern insulator with lattice sites $L_x=40$ under open boundary conditions. (a) Nontrivial case for $k_z=0$ and $m_z=4$; (b) critical case for $k_z=0.5\pi$ and $m_z=4$; (c) trivial case for $k_z=\pi$ and $m_z=4$; (d) nontrivial case for $k_z=0$ and $m_z=3$; (e) nontrivial case for $k_z=0.5\pi$ and $m_z=3$. The inset figure in (a) shows the density distribution of two typical edge modes. Other parameters in (a)–(e) are $t_0=1$ as the energy unit and $t_s=1$.

Figure 3

Energy spectrum of the edge states in the mimicked WSM phase for (a) $m_z=4$ and (b) $m_z=3$. The green dashed lines denote straight Fermi lines formed by gapless zero-energy edge modes, which are analogous to Fermi arcs connecting the pair of the mimicked Weyl points ${\mathit{W}}_{\pm }$. They are located at $(\theta ,k_z)=(0,\pm \pi /2)$ in (a) and $(\theta ,k_z)=(0,\pm 2\pi /3)$ in (b), respectively. The regimes of left-edge and right-edge modes in the parameter space denoted by blue “L” and red “R” are $\{E<0,\theta <0\}\bigcup \{E>0,\theta >0\}$ and $\{E>0,\theta <0\}\bigcup \{E<0,\theta >0\}$, respectively. The edge modes have similar profiles with those shown in Fig. 2. Other parameters in (a) and (b) are $t_0=1$ as the energy unit and $t_s=1$.

Figure 4

The transfer fractions for tunable parameters. The quasimomentum distribution ${\xi }_x\left(k_z\right)$ for varying parameter $\theta$ with (a) $m_z=3$ and (b) $m_z=4$; the quasimomentum distribution ${\xi }_z\left(k_x\right)$ for varying parameter $\theta$ with (c) $m_z=3$ and (d) $m_z=4$; (e) ${\xi }_x\left(k_z\right)$ for different parameter $m_z$ with $\theta =0$; (f) ${\xi }_x\left(k_z\right)$ for different parameter $\theta$ with $m_z$ and $t_s=0.5$. The white dashed line in (e) shows the expected $k_z$ positions of the mimicked Weyl points ${\mathit{W}}_{\pm }$ for varying $m_z$, which agree well with the maximum transfer positions of ${\xi }_x\left(k_z\right)$. The maximum ${\xi }_x(k_z,\theta )$ in (a) and (b) also corresponds to the $k_z$ and $\theta$ positions, as shown in Fig. 3. The position of maximum dip inside the ring profile of ${\xi }_z(k_x,\theta )$ in (c) and (d) indicates $k_x=\theta =0$ for the Weyl points. Other parameters in (a)–(f) are $t_0=1$ as the energy unit, $F=1$, and $t_s=1$ except that $t_s=0.5$ in (f).

Figure 5

The HWF centers in a tight-binding chain of length $L_x=100$ at half-filling as a function of the adiabatic pumping parameter $\theta$ for different $k_z$. (a) The profile $\langle n_x(\theta ,k_z)\rangle$ for the parameter $m_z=4$, where $\langle n_x\left(\theta \right)\rangle$ (do not) shows a jump of one unit cell for $k_z$ (outside) within the region $(-0.5\pi ,0.5\pi )$, with typical examples shown in (b). (c) The profile for $m_z=6.1$ shows no jump of $\langle n_x\left(\theta \right)\rangle$ for all $k_z$, with typical examples shown in (d). The corresponding $k_z$-dependent Chern number $C_{k_z}$ is also plotted both in (a) and (c), with the white dashed lines denoting the critical value between the trivial ($C_{k_z}=0$) and nontrivial ($C_{k_z}=-1$) cases. Other parameters in (a)–(d) are $t_0=1$ as the energy unit and $t_s=1$.

Figure 6

The maximum variation of the HWF-center shift $\delta n_x$. (a) $\delta n_x$ as a function of $k_z$ for $m_z=4$. (b) $\delta n_x$ as a function of $m_z$ for $k_z=0$. Other parameters in (a) and (b) are $L_x=100, t_0=1$ as the energy unit, and $t_s=1$.