## 2 Actively Tunable Topological Transition in Graphene/Dielectric Multilayers  
Voltage-controlled topological transition of dispersion can be realized by actively tuning the anisotropic parameters of permittivity or permeability. In this section, based on the multilayers composed of graphene and a dielectric, we theoretically study the voltage-controlled topological transition by varying the permittivity. Then, owing to the difficulty of related experiments, we experimentally demonstrate the voltage-controlled topological transition by varying the permeability in the microwave regime in Section 3. We consider the structure, shown in Figure 1, which is composed of graphene layers separated by dielectric slabs. For the graphene sheet, its electromagnetic properties are characterized by the surface conductivity \(\sigma(\omega,\mu_{c})\). The \(\sigma(\omega,\mu_{c})\) can be calculated using the Kubo formula [36; 37]:  
\[\sigma(\omega,\mu_{c})=\sigma_{\text{int }ra}(\omega,\mu_{c})+\sigma_{\text{int }er}(\omega,\mu_{c}), \tag{1}\]  
Here, \(\sigma_{\text{int}ra}(\omega,\mu_{c})\) and \(\sigma_{\text{inter}}(\omega,\mu_{c})\) denote the conductivity due to the intra-band and inter-band contribution, respectively. \[ \begin{array}{l}\sigma_{\text{intra}}(\omega,\mu_{c})=\frac{-i\epsilon^{2}} {\pi\hbar^{2}(\omega+i\tau^{-1})}[\int_{0}^{\infty}p(\frac{\partial f_{d}(p)} {\partial p}-\frac{\partial f_{d}(-p)}{\partial p})dp]\\ \sigma_{\text{inter}}(\omega,\mu_{c})=\frac{i\epsilon^{2}(\omega+i\tau)}{\pi \hbar^{2}}[\int_{0}^{\infty}\frac{f_{d}(-p)-f_{d}(p)}{(\omega+i\tau^{-1})^{2} -4(p/\hbar)^{2}}dp]\end{array} , \tag{2}\]  
where \(f_{d}(p)\) is the Fermi-Dirac distribution function: \(f_{c}(p)=\{1+\exp[(p-\mu_{c})/(k_{B}T)]\}^{-1}\), \(k_{B}\), \(\omega\), \(\hbar\), \(e\), \(T\), and \(\tau^{-1}\) are the Boltzmann's constant, angular frequency, Planck constant, charge of an electron, temperature, and collision rate, respectively. \(\tau^{-1}=10^{12}\) is assumed independent of the energy \(p\). \(\mu_{c}\) is the chemical potential that can be tuned by the external voltage \(v_{g}\)[25]. At high frequencies,the contribution from the inter-band can be neglected. For the Fermi-Dirac statistics, conductivity can be simplified as [36]:  
\[\sigma(\omega,\mu_{c})=\frac{ie^{2}\mu_{c}}{\pi\hbar^{2}(\omega+i\tau^{-1})}. \tag{3}\]  
When electronic band structure of a graphene sheet is unaffected by the neighboring layers, the effective permittivity of graphene can be calculated as follows [25]:  
\[\varepsilon_{g}=1+i\sigma(\omega,\mu_{c})/\varepsilon_{0}\omega t_{g}, \tag{4}\]  
where \(t_{g}=1\) nm and \(\varepsilon_{0}\) are the thickness of graphene and the permittivity of the vacuum, respectively. For the varied chemical potential \(\mu_{c}\), the different effective complex permittivity values of graphene based on Equation (4) are shown in Figure 2a. The real and imaginary parts of \(\varepsilon_{g}\) are marked by solid and dashed lines, respectively. We can clearly find that the complex value of \(\varepsilon_{g}\) is sensitive to the value of \(\mu_{c}\), and the real part of \(\varepsilon_{g}\) has a redshift with the \(\mu_{c}\) value increased. Owning to the flexibly regulated \(\varepsilon_{g}\), the active topological transition of dispersion can be easily realized. Considering the graphene/dielectric multilayer structure, the components of the effective dielectric permittivity tensor parallel (\(\varepsilon_{//}\)) and perpendicular (\(\varepsilon_{\perp}\)) to the anisotropic axis are given by the effective medium theory [38]:  
\[\varepsilon_{//}=\frac{t_{g}+t_{d}}{t_{g}/\varepsilon_{g}+t_{g}/\varepsilon_{ g}},\varepsilon_{\perp}=\frac{\varepsilon_{g}\cdot t_{g}+\varepsilon_{d} \cdot t_{d}}{t_{g}+t_{d}}, \tag{5}\]  
where \(t_{d}=10\) nm and \(\varepsilon_{d}=4\) are the thickness and permittivity, respectively, of the dielectric layer for hexagonal boron-nitride (h-BN) [30; 39].