Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas

Three-dimensional topological Weyl semimetals can generally support a zero-dimensional Weyl point characterized by a quantized Chern number or a one-dimensional Weyl nodal ring characterized by a quantized Berry phase in the momentum space. Here, in a dissipative system with particle gain and loss, we discover a new type of topological ring, dubbed a Weyl exceptional ring consisting of exceptional points at which two eigenstates coalesce. Such a Weyl exceptional ring is characterized by both a quantized Chern number and a quantized Berry phase, which are defined via the Riemann surface. We propose an experimental scheme to realize and measure the Weyl exceptional ring in a dissipative cold atomic gas trapped in an optical lattice.

Figure 1

Energy spectra and the Riemann surface of the toy model in Eq. (1). Spectra with respect to $k_x$ and $k_y$ for $k_z=0$ in (a) (real parts) and (b) (imaginary parts). Real (c) and imaginary parts (d) of the Riemann surface as a function of $k_y$ and $k_z$ for $k_x=0$. In (c) and (d), the color represents the strength of $\theta \text{mod}4\pi$, and the red tube arrow shows a path from $\theta =0$ to $\theta =4\pi$.

Figure 2

(a) A surface enclosing a Weyl exceptional ring and (b) a surface located inside the ring. (c) Lattice structure in the $(x,y)$ plane. (d) Schematic of trapped atoms being kicked out by a resonant optical beam (denoted by the grey arrow).

Figure 3

(a) Schematic of Weyl exceptional rings denoted by closed red, green, and cyan lines for the system described by the Hamiltonian in Eq. (8). The dashed box depicts the first Brillouin zone. (b) Real, (c) imaginary, and (d) absolute values of the eigenenergy with respect to $k_z{a}_z$ for $k_y=0$ and $\gamma =0.7J$ when the open boundary condition is imposed along the $x$ direction. The red lines are the surface states. In (d), additional surface states for $\gamma =0,0.35J,0.86J$ are plotted as blue, green, and yellow lines, respectively. Note that only the parts with zero absolute energy are associated with surface states.