Edge states at the interface of non-Hermitian systems

Topological edge states appear at the interface of two topologically distinct Hermitian insulators. We study the extension of this idea to non-Hermitian systems. We consider $\mathcal{P}T$-symmetric and topologically distinct non-Hermitian insulators with real spectra and study topological edge states at the interface of them. We show that $\mathcal{P}T$ symmetry is spontaneously broken at the interface during the topological phase transition. Therefore, topological edge states with complex energy eigenvalues appear at the interface. We apply our idea to a complex extension of the Su-Schrieffer-Heeger model.

Figure 1

The energy bands for $\gamma =0.2$, where the curves in blue and orange represent the real and imaginary parts of the energy eigenvalues, respectively. For $\omega =1.3$ and $\omega =0.7$: The energy eigenvalues are real and they are gapped. However, the two cases are topologically distinct. Whenever a phase transition occurs, i.e., the band gap closes, the system enters a broken $\mathcal{PT}$-symmetric region since a complex spectrum appears. Therefore, at the interface of two systems with ${\displaystyle \omega =1.3}$ and ${\displaystyle \omega =0.7}$ (left plots), there exist complex energy interface states. Bottom right plot: The interval of $\omega$ for the appearance of complex energy eigenvalues.

Figure 2

Our structure is displayed for $N=6+6=12$ lattice sites. The gain and loss (shaded circles) are assumed to be located as in the figure. The non-Hermitian degree is chosen in such a way that both chains have real spectra. The first six sites are in the chain-I region, while the second six sites are in the chain-II region. Both chains are SSH systems so tunneling amplitudes alternate from site to site, but they are topologically distinct. The two chains are coupled with a tunneling amplitude ${\displaystyle \mathrm{\Delta }}$. Therefore, the topological phase transition occurs and topological edge states with complex energy appear at the interface. Note also that topological edge states occur at the open end of a topologically nontrivial chain too.

Figure 3

The real and imaginary parts of the energy eigenvalues for two topologically distinct complex SSH chains with $N=20+20=40$ and $\gamma =0.5$. The two chains are coupled together with ${\displaystyle \mathrm{\Delta }=0.1}$. The hopping amplitudes are parametrized as ${\displaystyle \nu =1-0.5cos\left(\mathrm{\Phi }\right)}$ and ${\displaystyle \omega =1+0.5cos\left(\mathrm{\Phi }\right)}$, where $\mathrm{\Phi }$ is an additional degree of freedom. The coupled system has topological zero-energy modes for all $\mathrm{\Phi }$, unless $\mathrm{\Phi }$ is not around either ${\displaystyle \pi /2}$ or ${\displaystyle 3\pi /2}$. We also plot the densities of topological edge states for $\mathrm{\Delta }=0$ and $\mathrm{\Delta }=0.1$. The stable edge states are depicted in orange color, while the edge state with complex energy are in blue color. As can be seen, the topological edge state with a complex energy eigenvalue appears at the interface when $\mathrm{\Delta }=0.1$.