## 1 Introduction  
In photonic crystals (PCs), one can manipulate photons in the same way as electrons in semiconductors. By modulating the refractive indices of dielectric materials, it forms periodic distribution in space and obtains special optical properties. PCs, which are microstructures of wavelength-scale, have photonic bandgaps, that is, light with frequency within the bandgaps cannot pass through the PCs [1; 2; 3; 4]. There are natural photonic crystals, such as opals and butterfly wings. What is more, we can achieve specific optical functions by synthesizing PCs. PCs can also realize localization of light field [5; 6; 7; 8], unidirectional transmission [9; 10], directional cloaking [11], solitons [12; 13; 14], and other functions [15; 16; 17; 18; 19]. Otherwise, PCs are used to control photon flows, which can realize the miniaturization and integration of photonic devices [20; 21; 22; 23; 24]. It is often necessary to modulate the refractive index of material in the study on PCs. In the past, it mainly focused on the modulation of the real part of the refractive index, while in recent years, the regulation of the imaginary part of the refractive index has gradually attracted people's attention. The imaginary part of the refractive index represents the loss or gain of material. Loss seems to be an unfavorable factor, which causes energy attenuation, so it should be minimized in device design. People usually reduce the energy loss during the propagation of light through the mode matching in appropriate materials and structures, or compensate the loss of devices through gain. According to quantum mechanics, a system is non-Hermitian if there is loss or gain. By using the similarity between the paraxial equation of electromagnetic field propagation under the slowly varying envelope approximation [25] and the Schrodinger equation of quantum mechanics, the concept of non-Hermite can be extended to the regime of optics [26; 27]. Relevant studies have contributed to the proposal of non-Hermitian photonics, which has become a hot topic in optics in recent years. In a non-Hermitian optical system, by simultaneously modulating the real part and the imaginary part of the refractive index of the material, single-mode laser [28], loss-induced transparency [29], and optical field localization [30] can be achieved. The parity-time (PT)-symmetric Hamiltonian can have real spectra, which extends quantum mechanics into a new research field [31; 32]. PT-symmetric Hermiticity is a new concept and PT symmetry in electronic systems has been widely concerned and aroused a lot of discussion [33]. PT symmetry is also of great theoretical and applied value in optics and other related regimes [34; 35; 36; 37; 38; 39; 40; 41]. The use of PT potentials in optics has been recently reviewed by [PERSON] [42], which provides a good background for non-Hermitian properties. The pioneering work to use gain and loss in optics to implement PT-symmetric Hamiltonian was [PERSON] et al. [43]. The work presented belongs to a set of explorations of non-Hermitian symmetries beyond PT, a vast and promising field that is just beginning to develop. It will surely provide many important results and applications [44]. The gain in the photon system can be obtained by nonlinear two-wave mixing [45], or Ge/Cr doped fiber [46], while the loss can be achieved by acoustic modulator [47]. Thus, optics has the natural advantage of being used to explore all sorts of non-Hermite predictions that used to be verified difficult in quantum mechanics. However, studies on non-Hermitian photonics have shown that loss and gain, satisfied with PT symmetry, provide a new degree of freedom for the system, which can effectively control photons and generate new optical effects, such as unidirectional reflectionless resonance, which can be used to prepare chip-level semiconductor lasers [48]. Anti-PT-symmetric systems, in which the refractive index of material obeys \(n(x)=-n^{*}(-x)\), have been proposed and attracted much attention [49; 50; 51].