This thesis is mainly about the study of topological models and the corresponding proof-of-principle experimental demonstrations by transmission line networks. Due to the similarity between the network equation and the tight-binding model, the transmission line network is an ideal platform for mimicking the tight-binding model in both experiment and simulation.

In the first half of the thesis, we propose a topological model with angular-momentum-orbital coupling, similar to the spin-orbital coupling in the quantum spin Hall effect. It exhibits nontrivial topological property in the non-zero angular momentum subspace. A transmission line network is built based on this model, and the one-way topological edge states are observed experimentally. Furthermore, we introduce the angular-...[ Read more ]

This thesis is mainly about the study of topological models and the corresponding proof-of-principle experimental demonstrations by transmission line networks. Due to the similarity between the network equation and the tight-binding model, the transmission line network is an ideal platform for mimicking the tight-binding model in both experiment and simulation.

In the first half of the thesis, we propose a topological model with angular-momentum-orbital coupling, similar to the spin-orbital coupling in the quantum spin Hall effect. It exhibits nontrivial topological property in the non-zero angular momentum subspace. A transmission line network is built based on this model, and the one-way topological edge states are observed experimentally. Furthermore, we introduce the angular-momentum-preserved randomness to this tight-binding model to verify the topological states' robustness and study topological Anderson insulators theoretically.

In the other half of the thesis, we study the non-Abelian band topology using transmission lines. The non-Abelian band topology is a new theory proposed to study the topological property of multiple bands (>2) PT (parity and time-reversal) symmetric systems, in which the fundamental group of the order-parameter space is a non-Abelian group. Several quasi-1D transmission line networks with three sublattices (3-band) are built in the experiment. By observing the eigenstate rotations on the eigenstate sphere in momentum space, we directly detect different non-Abelian quaternion charges experimentally. In addition, we propose a non-Abelian edge-bulk correspondence theory to predict the energy positions of interface states. This new theory can be inferred from the traditional Abelian view. Furthermore, we prove this non-Abelian edge-bulk correspondence by using the Jackiw-Rebbi argument. Finally, we design a circuit experiment to mimic the nodal points braiding in a 2D PT-symmetric system. It is expected to observe that two nodal points carrying the same non-Abelian charge fail to annihilate after collision with each other. The simulation results verify the feasibility of this design.