Fig. 6: Error analysis of the NHA: deviation from the exact result.
a Relative error of the imaginary part of the self-energy. The color bar ranges from −0.17 to 0.04. b Relative error of the density of states (DOS, green curve); the red dashed lines mark band edges at ω = ± 2.21, and the gray shaded areas indicate the band gaps ( ≈ 0.5). c Relative error of the spectral function at \({{{\rm{Im}}}}\,\omega =-0.08\). The color bar ranges from −0.65 to 0.4. d Validity breakdown diagram of the non-Hermitian approximation (NHA) versus \({{{\rm{Im}}}}\,\omega ,{t}_{A}\) and ty/tx. Boundary lines are drawn as yellow, blue, green, and red curves, with their corresponding validity regions filled with the respective colors. Here, \({t}_{1}=1.2,{t}_{2}=-1,\lambda =-1,{\mu }_{{{{\rm{S}}}}}=0.2,{t}_{x}=1,{\mu }_{{{{{\rm{S}}}}}^{{\prime} }}=0.3,\gamma =0.1\), NS = 80 and Ny = 300, \({E}_{\min }=-2.21,{E}_{\max }=2.21\). In (a)–(c) ty = 3.5, tA = 0.6.
