Table 1 Illustration to the phase diagram of the tight-binding model. The parameter space (t₀, t₁, γ₀) is classified into phase I-IV according to the distinct locations and centers of the exceptional ring (ER) in the momentum space (k_x, k_y). Phase I and phase II both possess a single exceptional ring, but encircle (π, π) and (0, 0) respectively. Phase III and phase IV both possess two exceptional rings, where the former encircle (π, π) and (0, 0), while the latter encircle (π, π) and (π, π).

I

(t₀ + γ₀)/t₁ ∈ (2, ∞)&(t₀ − γ₀)/t₁ ∈ (0, 2)

\(\cos {k}_{x}+\cos {k}_{y}=-({t}_{0}-{\gamma }_{0})/{t}_{1}\)

(π, π)

II

(t₀ + γ₀)/t₁ ∈ (2, ∞)&(t₀ − γ₀)/t₁ ∈ (− 2, 0)

\(\cos {k}_{x}+\cos {k}_{y}=-({t}_{0}-{\gamma }_{0})/{t}_{1}\)

(0, 0)

III

(t₀ + γ₀)/t₁ ∈ (0, 2)&(t₀ − γ₀)/t₁ ∈ (− 2, 0)

\(\cos {k}_{x}+\cos {k}_{y}=-({t}_{0}+{\gamma }_{0})/{t}_{1}\)

(π, π)

\(\cos {k}_{x}+\cos {k}_{y}=-({t}_{0}-{\gamma }_{0})/{t}_{1}\)

(0, 0)

IV

(t₀ + γ₀)/t₁ ∈ (0, 2)&(t₀ − γ₀)/t₁ ∈ (0, 2)

\(\cos {k}_{x}+\cos {k}_{y}=-({t}_{0}+{\gamma }_{0})/{t}_{1}\)

(π, π)

\(\cos {k}_{x}+\cos {k}_{y}=-({t}_{0}-{\gamma }_{0})/{t}_{1}\)

(π, π)