Stable topological edge states in a non-Hermitian four-band model

We introduce a one-dimensional non-Hermitian four-band tight-binding lattice system. We find stable topological edge states protected by particle-hole and parity-time symmetries. We show that a topological phase appears in the system. We discuss that the system can also be used as a topological laser if the gain and loss positions in a unit cell are judiciously arranged.

Figure 1

The energy spectra for the Hamiltonian (4) at $\lambda =\mp 0.6$ and $\lambda =\mp 0.3$ in the upper panel and $\lambda =\mp 0.1$ and $\lambda =0$ in the lower panel. The real and imaginary parts of the energy spectrum at ${\gamma }_0=0.2$ are depicted in blue and orange colors, respectively. The spectrum is complex valued for all values of $\lambda$ since the system has no $\mathcal{PT}$ symmetry.

Figure 2

The energy spectra for the Hamiltonian (5) at $\lambda =\mp 0.6$ and $\lambda =\mp 0.45$ in the upper panel and $\lambda =\mp 0.2$ and $\lambda =0$ in the lower panel. The real and imaginary parts of the energy spectrum at ${\gamma }_0=0.2$ are depicted in blue and orange colors, respectively. The $\mathcal{PT}$ symmetry is spontaneously broken for the middle two bands and exceptional points occur at a nonzero value of $\lambda$. The top and the bottom bands retains the $\mathcal{PT}$ symmetry and the two bands are real-valued as long as the non-Hermitian degree does not exceed the critical value.

Figure 3

The energy spectra for the Hamiltonian (6) at $\lambda =\mp 0.6$ and $\lambda =\mp 0.3$ in the upper panel and $\lambda =\mp 0.2$ and $\lambda =0$ in the lower panel. The real and imaginary parts of the energy spectrum at ${\gamma }_0=0.2$ are depicted in blue and orange colors, respectively. As opposed to the case in the Fig. 2, the $\mathcal{PT}$ symmetry is spontaneously broken not for the middle two bands but for the top and the bottom bands.

Figure 4

The energy spectra as a function of $\mathrm{\Phi }$ at ${\gamma }_0=0.1$ and $\lambda =0.5$. The first figure (a) plots the real part of the energy spectrum for the Hamiltonian (6). Note that the real part of the energy eigenvalues are almost the same for two other complex Hamiltonians. (b), (c) and (d) show the imaginary parts of the energy eigenvalues for Hamiltonians (4), (5), and (6), respectively. All three non-Hermitian Hamiltonians have particle-hole symmetries at $\mathrm{\Phi }=0$. However, only the system in (b) has no $\mathcal{PT}$ symmetry, which leads to complex energy eigenvalues. The $\mathcal{PT}$ symmetry is spontaneously broken in (c) when the system has open edges. As a result, topological lasing can be studied for Hamiltonian (5), since all the bulk states have real-valued energy while topological edge states have a nonvanishing imaginary part at $\mathrm{\Phi }=0$. Finally, the $\mathcal{PT}$ symmetry exists for both periodic and lattice form for Hamiltonian (6) at $\mathrm{\Phi }=0$, so stable topological edge states can be realized.