## 3 CPA-LPs and EPs Splitting
The eigenvalues and eigenvectors of \(S\)-matrix are given by \(\lambda_{1,2}=t\pm(r_{1}r_{2})^{1/2}\), and \((r_{1}^{1/2},\,r_{2}^{1/2})\), respectively. The two eigenvectors (\(\lambda_{1,2}\)) coalesce at \((r_{1}r_{2})^{1/2}=0\) signifying the EPs of the eigenvalues--or, to be more precise, the EPs originate from \(r_{1}\) or \(r_{2}=0\). As light impinges upon the structure from the right, Figure 3c provides the reflectance. One can see that there are also some CPA-LPs and EPs\({}_{2}\) in the parameter space. The position of the CPA-LPs in parameter space does not change for light incident from the left or right. But the EPs are located at the left side of the CPA-LPs. To demonstrate the relation between the CPA-LPs and EPs, we give the phase transition shown in Figure 3d as the gain-loss factor increases. The EPs\({}_{1}\) and EPs\({}_{2}\) split, and the interval of splitting EPs increases with the increase of the gain-loss factor. The CPA-LPs, which are proved to be the defective modes, are sandwiched between the EPs\({}_{1}\) and EPs\({}_{2}\). Figure 2: The calculated spectra for the anti-PT-symmetric PCs. The transmission spectrum \(T\) (**a**), the reflection spectrum \(R_{1}\) (**b**), and the phase of reflection coefficient (**c**) for light incident from the left, respectively. The reflection spectrum \(R_{2}\) (**d**), and the phase of reflection coefficient (**e**) for light incident from the right, respectively. The gain-loss factor \(q=2\). In the parameter space composed of angular frequency \(\omega\) and gain-loss factor \(q\), one of the CPA-LP emerge as the gain-loss factor approaches \(q_{\text{LP1}}=1.975\). To verify the CPA-LPs in the anti-PT-symmetric PCs, Figure 4a,b depicts the real part Re(\(\lambda_{1,2}\)) and imaginary part Im(\(\lambda_{1,2}\)) of the eigenvalues \(\lambda_{1,2}\) of \(S\)-matrix. It shows that the eigenvalues degenerate at the CPA-LPs as the angular frequency changes. It is the unique property of CPA-LPs usually utilized for confirming CPA-LP in non-Hermitian systems. In the vicinity of the CPA-LP, the curves of the imaginary part are crossing, while the curves of the real part are anti-crossing [52]. The CPA-LPs are connected with the surface mode localized at the gain-loss boundary. Figure 4c gives the electric field intensity distribution of the CPA-LP\({}_{1}\) at (\(q_{\text{LP1}}=1.975\), \(\omega_{\text{LP1}}=2.3\times 10^{13}\) rad/s). It demonstrates that most of the light field power is located in the center layers and is strongly restrained at the interfaces of dielectrics. Along the z-axis, the field exponentially decays as it departs from the center. As light is incident from the left, Figure 5a illustrates the reflection coefficient phase, \(\varphi_{r1}\), in parametric space, where the angular frequency and gain-loss gain can be modified in the vicinity of the CPA-LP\({}_{1}\). The phase dislocates around the CPA-LP\({}_{1}\), which is unique for the CPA-LPs in non-Hermitian systems [64]. The singularity in phase indicates that the phase should abruptly change once the parameters undergo a little variation. Figure 4: (**a,b**) The real part and imaginary part of eigenvalues for scattering matrix varying with the angular frequency around the LP for \(q=q_{\text{LP}}\). (**c**) The electric field intensity (\(|E_{2}|^{2}\)) distribution at the CPA-LP. Figure 3: (**a–c**) The transmittance and reflectance varying with the gain–loss factor and angular frequency for light incident from the left and right, respectively. (**d**) Phase transition as the strength of gain–loss modulation increases from zero. Discrete splitting EPs are located at the blue dots. CPA-LPs are located at the purple dots. Figure 5b plots the coefficient phase, \(\varphi_{r1}\), versus the angular frequency for several different gain-loss factors. The sampling gain-loss factors are \(q=1.875\), \(1.975\), and \(2.075\), which distribute around the \(q_{\rm LP1}\), respectively. There is a hop point in the phase curve as \(q=1.875\), and the phase difference is \(-2\pi\).