Weyl semimetal phases and implementation in degenerate optical cavities

We propose a scheme for simulating the behaviors of Weyl semimetals by using a two-dimensional array of degenerate optical cavities. To simplify such a three-dimensional system to a two-dimension system, the orbital angular momentum of the light is used to support an extra synthetic dimension. We find that this system is quite suitable for the purpose of investigating the features of the Weyl point by taking advantage of the input-output relation of the optical cavity system. We show that the transport properties of our system are determined by the number of sites along the constrained direction and the spin texture of the system makes the transport momentum dependent and spin dependent.

Figure 1

Schematics on simulating the Weyl semimetal phase based on degenerate optical cavities. (a) Realization of the Weyl semimetal phase using two-dimensional degenerate cavities. Red circles refers to degenerated optical cavities. (b) The hopping between the neighboring sites and their corresponding spin-dependent hopping term. (c) The spin-dependent hopping term along the $z$ direction emulated using the OAM of light. (d) The physical implementation of the degenerate cavity. SLM, WP, and BS represent spatial light modulator, wave plate, and beam splitter, respectively. The black arrows stand for the pump and detection processes.

Figure 2

Distribution of the Weyl point and the band structure. (a) The phase where the Weyl point exits with the varying of $t_z^{\prime }$. (b) The distribution of the Weyl point in the momentum space. The topological charges of each Weyl point are shown in the figure. The dots with +2 or −2 are the projection of Weyl points in the $k_x-k_z$ plane. The green lines are the theoretically obtained Fermi arcs. (c) The band structure when the Weyl point exits with $k_z=\pm k_0$.

Figure 3

Transmission rates along the $z$ direction for the bulk system at (a) $k_x=0$, (b) $k_x=2\pi /3$, and (c) $k_x=4\pi /3$. The results with only “spin-up” or “spin-down” input light are marked by the up and down arrows in panel (a) and shown in the insets in panels (b) and (c). (d) Transmission rates with $t_{xy}=1.2$ with the other parameters being the same as those in panel (a). (e) and (f) $t_z^{\prime }=-0.59$ with the other parameters being the same as those in panels (a) and (b), respectively. (g) Transmission rate in the momentum space with the $y$ direction being the constrained direction. Here, $N_y=3$ for blue curves and $N_y=5$ for red curves. (h) Transmission rate with the spin of the input light being up. (i) Transmission rate with the ‘spin of the input light being down. Here $N_y=4$ or 6. The inset figure in panel (h) is the case with $k_x$ being 1.2 times larger. In the calculation, $N_z=100, N_x=3, k_0=\pi /2, {\gamma }_l^{\text{P}}={\gamma }_l^{\text{D}}=0.1=10{\gamma }_l^{\mathrm{dis}}$, and $t_{xy}=t_y=t_z=1$.

Figure 4

Energy splitting induced by wave-function overlap between the two edge modes (see dots) in the case with even $N_y$. The lines represent the splitting determined by Eq. (20).

Figure 5

Schematics on the model Hamiltonian, Eq. (A1). $a_l, b_l, c_l$, and $d_l$ are the amplitudes of lights at the corresponding places shown in the figure at site $l$. The other elements are the same as those in Fig. 1.

Figure 6

Schematics on the input light (orange lines). The WPs are used to tune the relative phases of the cavities to simulate the particle momentum. The other elements are the same as those in Fig. 1.