Effective Hamiltonian for Photonic Topological Insulator with Non-Hermitian Domain Walls

The gain and loss in photonic lattices provide possibilities for many functional phenomena. In this Letter, we consider photonic topological insulators with different types of gain-loss domain walls, which will break the translational symmetry of the lattices. A method is proposed to construct effective Hamiltonians, which accurately describe states and the corresponding energies at the domain walls for different types of photonic topological insulators and domain walls with arbitrary shapes. We also consider domain-induced higher-order topological states in two-dimensional non-Hermitian Aubry-André-Harper lattices and use our method to explain such phenomena successfully. Our results reveal the physics in photonic topological insulators with gain-loss domain walls, which provides advanced pathways for manipulation of non-Hermitian topological states in photonic systems.

Figure 1

(a) Schematic diagrams of the 1D-AAH model with non-Hermitian domain walls. (b) Part of the $\varphi$ spectrum of non-Hermitian 1D-AAH chain described by Eq. (1). Domain-induced edge states are colored in red (blue), whose wave functions concentrate in the left(right) region of the domain wall. Inner plot: corresponding imaginary parts of the spectrum. (c) At $\gamma ={\gamma }_c$, the dispersion relations of two domain-induced bands have the square root form.

Figure 2

(a) Schematic plot of constructing $\phi$. (b) Comparison of eigenvalues of domain-induced edge states given by $H_{1\mathrm{D}}$ (solid lines) and $H_{\mathrm{eff}}$ with $\lambda =9$ (discrete circles). (c) Comparison of normalized field distributions of six representative domain-induced edge states. Only six sites around the domain wall are shown.

Figure 3

(a) Layout of a 2D-AAH model with a cross-shaped domain wall. (b) Real and imaginary $\varphi$ spectrum of Eq. (5) on the plane ${\varphi }_x={\varphi }_y$ in the neighborhood of ${\varphi }_x=\pi /3$. Only domain induced corner states are colored. Inner plot: The enlarged drawing of the gray dot-dash box.

Figure 4

(a) Schematic plot for constructing assistant states $\phi \mathrm{s}$ in the 2D case. (b) Relations between ${\kappa }_{ij}$ and $\lambda$. Some quantities have been shifted by a constant for a better distinction. (c) $\omega$’s at different ${\varphi }_x$’s. We always have ${\omega }_1={\omega }_3$. (d) Comparison of eigenvalues of four domain-induced corner states given by $H_{2\mathrm{D}}$ (solid lines) and $H_{2\mathrm{D}}^{\mathrm{eff}}$ with $\lambda =9.5$ (discrete circles). (e) Comparison of normalized field distributions of four chosen states around the intersection.

Figure 5

A multidomain wall configuration and typical density plots at each domain corner.