## 3 Experimental Demonstration of the Actively Tunable Topological Transition Based on TLs
A direct voltage source is connected to the sample from the top, which is marked by the red arrow. \[\varepsilon=2C_{0}\cdot g/\varepsilon_{0},\mu_{x}=L_{0}/(p\cdot\mu_{0}),\] \[\mu_{z}=(\frac{L_{0}}{g}-\frac{1}{\omega^{2}\cdot C\cdot d\cdot g}) \big{/}\mu_{0},\quad, \tag{7}\]  
where \(\mu_{0}\) is permeability of vacuum. \(L_{0}\) and \(C_{0}\) are the per-unit-length inductance and capacitance of TL segment, respectively. The effective permittivity \(\varepsilon\approx 3.57\) is isotropic in our structure. For the varactors, one can change the external voltage to tune the capacitance value of \(C\). The reference relationship between bias voltage and the diode capacitance provided by the product manual is indicated by the blue dotted line in Figure 6. From Equation (7), by continuously changing the value of \(C\), the sign and value of \(\mu_{z}\) can be actively controlled at a fixed frequency. Through the modulation of capacitance, we can manipulate the sign of \(\mu_{z}\) from negative to positive while the signs of \(\varepsilon\) and \(\mu_{x}\) remain positive, as shown in Figure 6.  
The 2D TLs can be described by the effective medium theory and the dispersion relation is derived as [2; 17]:  
\[\frac{k_{x}^{2}}{\mu_{z}}+\frac{k_{z}^{2}}{\mu_{x}}=\varepsilon\cdot k_{0}^{2}. \tag{8}\]  
Based on Equation (8), we calculate different IFCs by varying the value of \(C\) at 0.8 GHz, as is shown in Figure 7a. One can see that the topological transition of IFC will happen once the value of \(C\) changes from 10 pF to 4 pF. Moreover, the IFC will become more and more flat with the decrease of \(C\). We further use the computer simulation technology (CST) microwave studio software to perform the simulation. A linearly polarized source is loaded near the center of the sample, as is marked by the blue point in Figure 5. In Figure 7b, we simulated \(E_{y}\) patterns of sample at 0.8 GHz by varying the value of \(C\). The emission patterns for different values of \(C\) coincide well with the IFCs in Figure 7a. For example, for \(C=10\) pF, the IFC is a closed ellipsoid and the source can propagate in all in-plane directions. However, for \(C=4.0\) pF, the IFC is an open hyperbolic curve and the source can only propagate in some directions. In particular, for \(C=1.0\) pF, the IFC is very flat. The source is collimated and can only propagate in the vertical direction. Now we perform an experiment to observe the emission patterns. In the experiment, a DC voltage source is connected to the sample from the top, which is marked by the red arrow in Figure 5. With the increase of the voltage, we measure the different emission patterns at 0.8 GHz to observe the changing process of IFCs (see Appendix B for the details of experimental operation). The measured \(E_{y}\) patterns in Figure 7c agree well with the numerically simulated one in Figure 7b. Hence, based on 2D TLs, we have experimentally demonstrated the actively controlled magnetic topological transition of dispersion by changing the external voltages. Figure 6: The variation of the effective parameters of TLs with the capacitances of varactors.