## 2 Aperiodic PT Symmetry Lattices  
We explore the nonlinear optical properties in the aperiodic PT-symmetric lattices that were constructed by the Fibonacci sequence dielectrics. Aperiodic quasicrystals of Fibonacci multilayer dielectrics have many fascinating optical properties, such as mode beating, strong pulse stretching, and band-edge resonances [55]. The band-edge resonances can localize the electric field [56]. Moreover, the localization of field can be further enhanced by increasing the gain and loss in our PT-symmetric structure [19]. The strong resonance of cavity can greatly localized to the electric field. Therefore, the aperiodic PT-symmetric multilayers can be used to enhance the nonlinearity of graphene and achieve low-threshold optical bistability. The primitive aperiodic unit-cell dielectrics A and B are considered. The thicknesses of dielectrics A and B are \(d_{a,b}=\lambda_{0}/4\text{Re}(n_{a,b})\), where \(\lambda_{0}\) represents the midgap wavelength of aperiodic lattices with a defect and \(\text{Re}(n_{a,b})\) are the real parties of refractive indices. Fibonacci sequences are constructed in Figure 1a based on the production rule: \(F_{j}=\{F_{j-1}F_{j-2}\}\) for \(j\geq 3\), with \(F_{1}=\{\text{B}\}\), \(F_{2}=\{\text{BA}\}\)[55; 57]. Integrating two Fibonacci (\(F_{6}\)) sequence dielectrics together with graphene, in which the PT symmetry is satisfied for the whole structure, further incorporates the aperiodic PT symmetry lattices in Figure 1b. The thicknesses of dielectrics C and D are \(d_{c}=d_{b}\) and \(d_{d}=d_{a}\), respectively. The indices of the refraction of dielectrics are regulated with respect to PT symmetry \(n(z)=n^{\star}(-z)\). We define the parameter \(q\) as gain-loss factor. Dielectrics A and B are MgF\({}_{2}\) and ZnS, respectively. The gain in Fibonacci sequence dielectrics might be realized by doping Ge/Cr or nonlinear two-wave mixing [43; 47; 58; 59], and the loss of material could result from acoustic modulators [60].  
Graphene, as an ultrathin two-dimensional material, is embedded in the middle of the lattices \(z\) = 0; therefore, the addition of graphene has no effect on the PT symmetry. However, here, we view graphene as an equivalent dielectric with a thickness of \(d_{g}\) = 0.34 nm [61; 62]. The permittivity of the equivalent dielectric for graphene is expressed as \(\varepsilon_{g}=1+\mathrm{i}\sigma_{g}\eta_{0}/(kd_{g})\)[44], where \(\sigma_{g}\) is defined as the total surface conductivity of graphene. The parameter \(\sigma_{g}\) is composed by two parts, i.e., linear surface conductivity \(\sigma^{(1)}\) and nonlinear surface conductivity \(\sigma^{(3)}|E_{z}|^{2}\), where \(\sigma^{(3)}\) is the nonlinear coefficient of surface conductivity. The surface conductivity can be denoted by \(\sigma_{g}=\sigma^{(1)}+\sigma^{(3)}|E_{z}|^{2}\). The Kubo formula could estimate the linear surface conductivity [63]. In this study, the incident wavelength is defined in the range of terahertz waves, and we set the temperature as \(T\) = 300 K. The incident wave is a transverse magnetic (TM) polarization wave. We can derive the transmittance \(T\) and transmission coefficient \(t\) through the transfer matrix method (TMM), as a TM wave is normally incident upon the PT-symmetric lattices [18; 64].