Higher-Order Weyl-Exceptional-Ring Semimetals

For first-order topological semimetals, non-Hermitian perturbations can drive the Weyl nodes into Weyl exceptional rings having multiple topological structures and no Hermitian counterparts. Recently, it was discovered that higher-order Weyl semimetals, as a novel class of higher-order topological phases, can uniquely exhibit coexisting surface and hinge Fermi arcs. However, non-Hermitian higher-order topological semimetals have not yet been explored. Here, we identify a new type of topological semimetal, i.e., a higher-order topological semimetal with Weyl exceptional rings. In such a semimetal, these rings are characterized by both a spectral winding number and a Chern number. Moreover, the higher-order Weyl-exceptional-ring semimetal supports both surface and hinge Fermi-arc states, which are bounded by the projection of the Weyl exceptional rings onto the surface and hinge, respectively. Noticeably, the dissipative terms can cause the coupling of two exceptional rings with opposite topological charges, so as to induce topological phase transitions. Our studies open new avenues for exploring novel higher-order topological semimetals in non-Hermitian systems.

Figure 1

(a) Weyl exceptional rings (blue curves) obtained by solving Eqs. (S7) and (S8) in Ref. [120]. They are formed by a two-band coalescence at the energy $E=0$. The bulk bands of the non-Hermitian Hamiltonian $\mathcal{H}(\mathbf{k})$ in Eq. (1) exhibit four Weyl exceptional rings along the $k_z$ direction. The parameters used here are $m_0=1.5$, $m_1=-1$, $v_z=0.8$, $\gamma =0.8$, and ${\mathrm{\Delta }}_0=0.8$. The red loop denotes a closed path $\mathcal{L}$ encircling a Weyl exceptional ring. (b) Real vs imaginary parts of eigenenergies along the path $\mathcal{L}$. The red dots denote exceptional points (EPs). (c) A Weyl exceptional ring is enclosed by a closed surface of a cuboid.

Figure 2

(a) Real, (b) imaginary, and (c) absolute values of the surface band structure along the $k_z$ direction, when the open boundary condition is imposed along the $x$ direction with 200 sites for $k_y=0$. Yellow, green, and red lines in (c) denote surface states for $\gamma =0.4$, $\gamma =0.6$, and $\gamma =0.8$, respectively. Note that only the modes with zero absolute energy are surface states. (d) Real, (e) imaginary, and (f) absolute values of the band structure of a finite-sized system with $60\times 60$ unit cells in the $x\text{−}y$ plane. The red lines represent hinge states. Probability density distributions $|{{\psi }_n(x,y,z)|}^2$ of midgap modes for the eigenenergies (g) $E=4.6\times {10}^{-6}(1+i)$ and (h) $E=0.0072+0.0072i$ with open boundaries along the $x$, $y$, and $z$ directions. The number of unit cells is $20\times 20\times 30$. The parameters used here are $m_0=1.5$, $m_1=-1$, $v_z=0.8$, $\gamma =0.8$, and ${\mathrm{\Delta }}_0=0.8$.

Figure 3

$|{\lambda }_{1,R}|$ and $|{\lambda }_{2,R}|$ vs $k_z$, according to Eq. (8), with $m_0=1.5$, $m_1=-1$, $v_z=0.8$, and $\gamma =0.4$. The dashed black line denotes $|\lambda |=1$, which is a guide for the eyes.

Figure 4

(a) Weyl exceptional rings (blue curves) for $\gamma =1.1$. The bulk bands of the non-Hermitian Hamiltonian exhibit eight Weyl exceptional rings. (b) Real and (c) imaginary parts of the band structure of a finite-sized system with $80\times 80$ unit cells in the $x\text{−}y$ plane. The red lines represent hinge states. The parameters used here are $m_0=1.5$, $m_1=-1$, $v_z=0.8$, and ${\mathrm{\Delta }}_0=0.8$.