Fig. 1: The real and imaginary parts of the energy spectra under periodical boundary condition.

The blue and purple parts represent the eigenvalues E₊ and E₋ respectively, where \({E}_{\pm }={\varepsilon }_{0}\pm \sqrt{{{d}_{x}}^{2}+{{d}_{y}}^{2}}\), and the red lines denote the spectral degeneracy that E₊ = E₋ = ε₀. For simplicity and without loss of generality, the on-site energy ε₀ is set as zero, and the hopping parameters are set as t_1x = t_1y = t₁. a, b The energy spectra defined by \({E}_{\pm }=\pm \sqrt{{({t}_{0}+{t}_{1}\cos {k}_{x}+{t}_{1}\cos {k}_{y})}^{2}-{({\gamma }_{0}+{\gamma }_{2x}\cos {k}_{x})}^{2}}\) with hopping parameters t₀/t₁ = 1, γ₀/t₁ = 0.2, γ_2x/t₁ = 0.4 in (a), and t₀/t₁ = 1, γ₀/t₁ = 0.6, γ_2x/t₁ = 1.2 in (b), which can exhibit exceptional rings and hyperbolic lines respectively. c–f The energy spectra defined by \({E}_{\pm }=\pm \sqrt{{({t}_{0}+{t}_{1}\cos {k}_{x}+{t}_{1}\cos {k}_{y})}^{2}-{{\gamma }_{0}}^{2}}\). The parameter space is classified into phase I-IV by the distinct forms of the exceptional rings. c Phase I with t₀/t₁ = 1.85, γ₀/t₁ = 0.25. d Phase II with t₀/t₁ = 1.2, γ₀/t₁ = 2.5. e Phase III with t₀/t₁ = 0.5, γ₀/t₁ = 0.75. f Phase IV with t₀/t₁ = 0.75, γ₀/t₁ = 0.5.