THESIS
2020

1 online resource (xv, 123 pages) : illustrations (some color)

**Abstract**
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-...[

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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.

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