Fig. 3: Verification of the non-Hermitian approximation at real and complex frequencies.
The spectral functions under non-Hermitian approximation (NHA), AS,nH(k, ω), and from the exact case, AS,eff(k, ω), respectively. The color bar ranges are [0.002, 2.98] for (a) and [0.003, 2.99] for (b). c A comparison between the real frequency density of states (DOS) with and without the NHA, i.e., ρS,nH(ω) (blue circles) and ρS,eff(ω) (green curve), respectively. d, e The complex frequency DOS with and without NHA, defined as \({D}_{{{{\rm{S}}}},{{{\rm{nH}}}}}(\omega \in {\mathbb{C}})\) and \({D}_{{{{\rm{S}}}},{{{\rm{eff}}}}}(\omega \in {\mathbb{C}})\) respectively, where the black circles indicate the poles of the complex frequency Green’s function. The poles in (d) form the spectra with a nonzero winding number. The non-Hermitian approximation remains accurate near the real axis but breaks down in the complex plane as \(| {{{\rm{Im}}}}\,\omega |\) increases. For (d) and (e), we choose ω = − 1 − ωii, with ωi ∈ [0.08, 0.16] to compute the winding number shown in Fig. 8 in the “Methods” section. Frequencies from Fig. 1c's non-Bloch region, i.e., ω3, ω4, ω5 = − 0.6 − 0.14i, − 0.6 − 0.16i, − 0.6 − 0.18i are marked in (d) as red, green, and pink dots, respectively. The color bar ranges are [−10, 10] for (d) and [−2.8, 2.8] for (e). Other parameters are \({t}_{1}=1.2,{t}_{2}=-1,\lambda =-1,{\mu }_{{{{\rm{S}}}}}=0.2,{t}_{x}=1,{t}_{y}=3.5,{\mu }_{{{{{\rm{S}}}}}^{{\prime} }}=0.3,{t}_{{{{\rm{A}}}}}=0.6,\gamma =0.1\), NS = 80 and Ny = 300.
