## 3 Experimental Demonstration of the Actively Tunable Topological Transition Based on TLs  
In the visible range, the experimental realization of the actively tunable topological transition of dispersion is still a great challenge. For example, in Figure 1, besides the difficulty in fabricating the multilayers, it is not easy to precisely add the same external voltage to each graphene layer. In this section, we introduce a microwave platform based on 2D TLs to experimentally demonstrate the actively controlled topological transition. The experimental schematic of the TL-based metamaterial is shown in Figure 5a and the corresponding circuit model is shown in Figure 5b. The insets near Figure 5a  
Figure 4: (**a**) When the value of \(\mu_{c}\) increases from 15 eV to 21 eV, the sign of \(\varepsilon_{\perp}\) is reversed and the sign of \(\varepsilon_{//}\) remains positive. (**b**) When \(\mu_{c}=18\) eV, the IFC is a very flat elliptic curve, which can be regarded as a highly anisotropic ENZ medium. In contrast to this case, the IFC corresponding to the case of \(\mu_{c}=0.439\) eV is another flat elliptic curve along the vertical direction. (**c**) When a plane wave is incident on the structure (\(\mu_{c}=18\) eV) with a tiny perfect electric conductor (PEC) defect, the wave can perfectly pass through the structure almost without being affected by the defect. (**d**) The transmission behavior will change once the PEC defect is replaced by a perfect magnetic conductor (PMC) defect and the incident wave will be scattered by the defect. (**e**) For \(\mu_{c}=0.439\) eV, the incident wave is not affected by the PEC defect, as in the case of (**c**). However, the transmission will dramatically change for the PMC defect. (**f**) For \(\mu_{c}=0.439\) eV, the incident wave will be completely blocked by the tiny defect. The direction of the incident wave vector is marked by the direction of the arrow. show the enlarged lumped variable capacitance diodes and the protected elements, respectively. Figure 5c shows the schematic of the experimental platform to realize the actively control topological transition. For better visibility, the number of the unit cell (\(8\times 8\)) in Figure 5c is smaller than the experimental sample (\(20\times 20\)) in Figure 5a. In the experiment, a direct voltage source is connected to the sample from the top, which is marked by the red arrow. The grounding position is indicated by red dots. The signal is input at the center of the sample and the protection component is added to avoid the interaction between direct current (DC) source and the signal source, as is shown in Figure 5c. Here, our sample is composed of \(20\times 20\) unit cells with lumped elements that are fabricated on a commercial printed circuit board, F4B, with a relative permittivity of \(\varepsilon_{r}=2.2\) and thickness of \(h=1.6\) mm. The width of the micro-strip lines is \(w=2\) mm. The unit lengths of TL in the \(x\) and \(z\) direction are \(d=12\) mm. The structural factor of the TL is defined as \(g=Z_{0}/\eta_{eff}\), where \(Z_{0}\) and \(\eta_{eff}\) are the characteristic impedance and the effective wave impedance of the normal TL, respectively. When \(w>h\), the structural factor \(g=[1.393+w/h+0.667\ln(w/h+1.444)]^{-1}\approx 0.303\). Then, we load elements into the normal TL to realize the tunable metamaterials. Considering the transverse electric (TE) wave (\(E_{y}\), \(H_{x}\), \(H_{z}\)), According to the quasi-static TE polarized solution and Ampere's law, by loading the variable capacitance diode \(C\), the effective permittivity and permeability of 2D TLs in the long-wavelength limit can be written as [23; 45]:  
Figure 5: (**a**) The experimental schematic of the transmission line (TL)-based actively controlled metamaterial. The insets show the enlarged lumped variable capacitance diodes and protected elements, respectively. (**b**) The related anisotropic 2D-circuit model of the microwave media. (**c**) Schematic of part of the structure to realize an actively controlled topological transition.