Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems

We analyze chiral topological edge modes in a non-Hermitian variant of the 2D Dirac equation. Such modes appear at interfaces between media with different “masses” and/or signs of the “non-Hermitian charge.” The existence of these edge modes is intimately related to exceptional points of the bulk Hamiltonians, i.e., degeneracies in the bulk spectra of the media. We find that the topological edge modes can be divided into three families (“Hermitian-like,” “non-Hermitian,” and “mixed”); these are characterized by two winding numbers, describing two distinct kinds of half-integer charges carried by the exceptional points. We show that all the above types of topological edge modes can be realized in honeycomb lattices of ring resonators with asymmetric or gain-loss couplings.

Figure 1

Real and imaginary parts of the complex spectrum (2) of the Hamiltonian (1) with exceptional points at ${\mathbf{k}}_{\mathrm{EP}}^{\pm }=(0,\pm |m|)$.

Figure 2

Schematic diagrams indicating the zero-energy chiral edge modes (6) and (7) at the interface $x=0$ between two media with different masses $m_1>0$ and $m_2$ and the same non-Hermitian charges $s_1=s_2=1$. Exceptional points ${\mathbf{k}}_{\mathrm{EP}}$ are indicated for the two media, with chiralities ${\chi }_{\mathrm{EP}}$ marked by the black $+$ and $-$ signs. The edge modes with positive and negative chiralities ${\chi }_{\mathrm{edge}}$ are marked by red and blue colors.

Figure 3

Phase diagrams for the chiral edge modes (6) and (7) at the interface between two media with (a) equal non-Hermitian charges $s_1=s_2=1$ and (b) opposite charges $s_1=-s_2=1$. The mass in medium 1 is fixed as $m_1>0$. The arrows above (a) indicate the cases shown in Fig. 2. The numbers in parentheses indicate the differences of winding numbers (9) and (11) between the two media: $(\mathrm{\Delta }{w}_1,\mathrm{\Delta }{w}_2)$. Striped, empty, and dotted zones indicate gapped, ungapped, and mixed cases of the bulk spectra $\mathrm{Re}({\lambda }^{\pm })$ in the two media.