Fig. 2: Macroscopic model of non-Hermitian multiple domain walls (1D topological states).
From: Non-Hermitian reconstruction of photonic hierarchical topological states
a Schematic diagram of closed non-Hermitian multiple domain walls. The non-reciprocity of three domain walls can be described by (\({t}_{1}^{-},{t}_{1}^{+}\)), (\({t}_{2}^{-},{t}_{2}^{+}\)) and (\({t}_{3}^{-},{t}_{3}^{+}\)), respectively. b Conditions for valid solutions of this macroscopic model, which are categorized into three cases (I–III). g is defined as \(\frac{| {t}_{1}^{+}{t}_{2}^{+}{t}_{3}^{+}| }{| {t}_{1}^{-}{t}_{2}^{-}{t}_{3}^{-}| }\) and ∣z1(2)∣ represents the (second) largest absolute value of the product of complex wavevectors on three boundaries. c Spectra, zq,α and boundary field distributions for different cases. Here we have δt1 = δt3 = 0.1 and δt2 = 0.2 for case I, δt1 = −δt3 = 0.1 and δt2 = 0 for case II as well as δt1 = −0.1 and δt2 = δt3 = −0.2 for case III. Other parameters are t0 = 1.5 and t = 0.5. The TBM calculation is based on N = 50. The black dotted lines are closed loops for ideally ∣zq,α∣ = 1. The red, green and blue curves indicate allowed complex wavevectors on three boundaries, corresponding to α = 1, 2, 3, respectively. a1, a3 and a5 indicate 0D non-Hermitian topological states, which are not significantly reconstructed because the array is not small enough. a2 and a6 indicate 1D non-Hermitian topological states with maximum imaginary parts of eigenenergies. a4 indicate 1D non-Hermitian topological states with E = 1.5.
