## 2 Actively Tunable Topological Transition in Graphene/Dielectric Multilayers
According to Equation (5), the value and the sign of \(\varepsilon_{//}\) can be flexibly adjusted by changing the value of \(\mu_{c}\), as shown in Figure 2b. Considering a fixed frequency \(f=385\) THz, when the value of \(\mu_{c}\) increases from 0.1 eV to 0.5 eV, the sign of \(\varepsilon_{//}\) reverses at the critical point 0.439 eV, while the sign of \(\varepsilon_{\perp}\) remains positive, as displayed in Figure 3a. Figure 1: The scheme of the periodic structure composed of graphene/dielectric multilayers. The chemical potential of the graphene layer can be tuned by the external voltage. The dielectric layers (h-BN) are marked in yellow and the graphene layers are shown by the single layer of carbon atoms. Considering the light with transverse magnetic (TM) (\(H_{y}\), \(E_{x}\), \(E_{z}\)) polarization propagating in the \(x\)-\(z\) plane of the 2D uniaxial media, the IFC in such a strongly anisotropic metamaterial is given by [12]:  
\[\frac{k_{x}^{2}}{\varepsilon_{//}}+\frac{k_{z}^{2}}{\varepsilon_{\perp}}=\mu_{// }\cdot k_{0}^{2}, \tag{6}\]  
Figure 3: (**a**) Considering a fixed frequency \(f=385\) THz, the effective parameters \(\varepsilon_{//}\) and \(\varepsilon_{\perp}\) can be tuned by the different \(\mu_{c}\) values based on the multilayer structure. As the value of \(\mu_{c}\) increases from 0.1 eV to 0.5 eV, the sign of \(\varepsilon_{//}\) reverses, and the sign of \(\varepsilon_{\perp}\) remains positive. (**b**) When \(\mu_{c}=0.1\) eV, the iso-frequency contour (IFC) is a closed ellipsoid. The IFC will become an open hyperbola when \(\mu_{c}=0.5\) eV. The solid (dash)ed lines give the relationship of \(\mathrm{Re}(k_{z})\) (\(\mathrm{Im}(k_{z})\)) with \(k_{x}\). It should be noted that the imaginary part when \(\mu_{c}=0.5\) eV has been enlarged 20 times for clarification. (**c**) The emission patterns of the ellipsoid IFC (\(\mu_{c}=0.1\) eV) in the graphene/dielectric multilayers. (**d**) Same as in (**c**), but obtained for the hyperbola IFC (\(\mu_{c}=0.5\) eV). Figure 2: (**a**) Different effective complex permittivity values of graphene. The real and imaginary parts of \(\varepsilon_{g}\) are marked by solid and dashed lines, respectively; (**b**) The effective parallel permittivity tensor \(\varepsilon_{//}\) can be tuned by the different \(\mu_{c}\). where \(k_{x}\) and \(k_{z}\) are the \(x\) and \(z\) components of the wave vector, respectively. \(k_{0}=\omega/c\) is the vector in free space. \(c\) is speed of light and \(\mu_{//}\) is the effective magnetic permeability. The IFC in the wave-vector space (\(k_{x}-k_{z}\)) can be obtained according to Equation (6). In the low chemical potential case \(\mu_{c}=0.1\) eV, at \(f=385\) THz the IFC is a closed ellipsoid, and is shown by a red line in Figure 3b. The solid (dashed) lines give the relationship of \(\text{Re}(k_{z})\) (\(\text{Im}(k_{z})\)) with \(k_{x}\). By using COMSOL MULTI-PHYSICS, the emission pattern of the ellipsoid IFC is shown in Figure 3c (see Appendix A for details of the simulations). In the simulation, a current source is put at the center of the material. However, when the chemical potential is changed to \(\mu_{c}=0.5\) eV, the IFC of the graphene/dielectric multilayers will be in the form of a hyperbola, which is marked by the solid blue line in Figure 3b. The change of the IFC will strongly modify the propagation and emission properties of electromagnetic waves. The emission pattern is an open line because the density of states is at a maximum along the hyperbolic asymptote (Figure 3d) [40]. The chemical potential of graphene is modulated by an applied electric field and the topological transition of dispersion is realized with the changed chemical potential [25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35]. This actively controlled topological transition will greatly change the emission pattern of a source in the medium. From the simulated emission patterns, we can find that the emission patterns as shown in Figure 3c,d coincide with the IFCs based on the effective medium theory (Figure 3b).