Double exceptional links in a three-dimensional dissipative cold atomic gas

We explore the topological properties of non-Hermitian nodal-link semimetals with dissipative cold atoms in a three-dimensional optical lattice. We construct a two-band continuum model in three dimensions with a spin-dependent gain and loss, where the exceptional points in the energy spectrum can comprise a double Hopf link. The topology of the bulk band is characterized by a winding number defined for a one-dimensional loop in the momentum space and a topological transition of the nodal structures emerges as the change of the non-Hermiticity strength. A non-Bloch theory is built to describe the corresponding lattice model which has anomalous bulk-boundary correspondence. Furthermore, we propose that the model can be realized by using ultracold fermionic atoms in an optical lattice, and the exceptional nodal links as well as the topological properties can be detected by measuring the atomic spin textures.

Figure 1

(a)-(d) The exceptional degeneracies in the energy spectra of the model described by Eq. (2) with (a) $m=1, {\gamma }_z=0.8$; (b) $m=1, {\gamma }_z=1.2$; (c) $m=-0.25, {\gamma }_z=0.15$; (d) $m=-0.25, {\gamma }_z=0.3$. (e)-(f) The exceptional degeneracies with an additional symmetry-breaking non-Hermitian perturbation $i{\gamma }_y{\sigma }_y$ term; parameters are adopted for (e) $m=1, {\gamma }_z=0.8, {\gamma }_y=0.5$ and (f) $m=1, {\gamma }_z=0.8, {\gamma }_y=1$. Thin lines ${\ell }_1$ and thick lines ${\ell }_2$ are two solutions given by Eq. (4).

Figure 2

The winding numbers $w(k_x,k_y)$ for (a) $m=1, {\gamma }_z=0.8$; (b) $m=1, {\gamma }_z=1.2$; and (c) $m=-0.25, {\gamma }_z=0.15$.

Figure 3

The energy spectra for (a,b) $m=1, \gamma =0.8, k_y=0.5$ and (c,d) $m=1, \gamma =1.2, k_y=0.5$ when open boundary condition is imposed along the $z$ direction. The energy unit is set to be 1.

Figure 4

(a)–(c) The energy spectra for (a) $m=1, \gamma =0.8, k_y=0$, (b) $m=1, \gamma =0.8, k_y=0.5$, and (c) $m=1, \gamma =1.2, k_y=0.5$ when periodic and open boundary condition is imposed along the $z$ direction. (d) The wave function distribution for both the zero mode and bulk state for systems with parameters in (a) and (b). (e) The wave function distribution for topological zero modes in the Hermitian limit ${\gamma }_z=0$ when open boundary condition is imposed along both $x$ and $z$ directions, and for (i) $m=1, k_y=0$, and (ii) $m=1, k_y=0$, (iii) $m=1, k_y=0.5$. (f) The wave function distribution for topological zero modes (and bulk state) corresponding to (e) for ${\gamma }_z=1.2$ (when $k_y=0$, there is no skin effect).

Figure 5

The generalized Brillouin zone for (a) $k_x=\pi /4, k_y=0, {\gamma }_z=0.8, m_0=1$ and (b) $k_x=\pi /4, k_y=0.5, {\gamma }_z=0.8, m_0=1$.

Figure 6

(a) The exceptional nodal rings in the spectrum of Hamiltonian (19) with $m=-0.5$ and $g=0.25$; (b) $\mathrm{Re}\left(\varphi \right)$ in Eq. (6) with parameters in (a) and with (i) $k_y=1$, (ii) $k_y=0$, and (iii) $k_y=-1$; (c) $\mathrm{Re}\left({\eta }_{xz}\right)$ defined from long-time averaged spin textures with parameters in (a) and with (i) $k_y=1$, (ii) $k_y=0$, and (iii) $k_y=-1$. The evolution time is $T=80$.

Figure 7

Schematics of the laser configurations to realize the Hamiltonian (19). Polarizations of each beam are shown in brackets. Rabi frequencies for each beam are ${\mathrm{\Omega }}_1^x={\mathrm{\Omega }}_0{e}^{ikx}, {\mathrm{\Omega }}_2^{x\left(z\right)}=\frac{t_{x\left(z\right)}}{4}{\mathrm{\Omega }}_{x\left(z\right)}{e}^{-ikx\left(z\right)}, {\tilde{\mathrm{\Omega }}}_2^{x\left(z\right)}=-\frac{t_{x\left(z\right)}}{4}{\mathrm{\Omega }}_0{e}^{-ikx\left(z\right)}, {\overline{\mathrm{\Omega }}}_2^z\left({\sigma }^{\pm }\right)=\pm \frac{1}{2}{\mathrm{\Omega }}_{z\pm }{e}^{ikz}, {\mathrm{\Omega }}_1^y\left({\sigma }^{\pm }\right)=\pm im{\mathrm{\Omega }}_{y1}{e}^{iky}, {\mathrm{\Omega }}_2^y\left({\sigma }^{\pm }\right)=\pm \frac{t_y}{4}{\mathrm{\Omega }}_{y1}{e}^{iky}$.

Figure 8

Snapshots of the density distribution at different evolution times (denoted by $\tau$) for (a) a smaller lattice and (b) a bigger lattice. The initial state takes the Gaussian form for (a) $\psi (\tau =0)=\mathcal{N}exp[-{(x-15)}^2/10-z^2/5]{(1,1)}^T$ and (b) $\psi (\tau =0)=\mathcal{N}exp[-{(x-5)}^2/3-z^2/0.5]{(1,1)}^T$, where $\mathcal{N}$ is the normalization factor. The intensity profile of an evolved state $\psi \left(\tau \right)$ is normalized.

Figure 9

(a) Wannier-Stark function centered at site 0 and site 1. (b) Wannier-Stark function centered at site 0 and site 2. The parameters are chosen as $V_0=2.3E_r$ and linear tilt ${\mathrm{\Delta }}_x/2\pi =200\mathrm{Hz}$.