Nonlinear, non-Hermitian and high-dimensional classical topological systems

This thesis concerns classical topological systems with properties that are difficult to explore in solid state topological systems, namely nonlinearity, non-Hermiticity and high-dimensionality. In the first part, we study topological photonic structures with nonlinearity, with a particular focus on their application as optical isolators. We study three representative designs that respectively implement a nonlinear 1D Su-Schrieffer-Heeger (SSH) model, a nonlinear 2D Haldane model, and a 2D lattice of coupled-ring waveguides. We show that power thresholds and discontinuities due to nonlinearity induced topological phase transition can lead to isolator functionalities either with the emergence of self- induced topological solitons or with discontinuities in the transmittance curve. In the second part, we experimentally demonstrate an application of topological edge modes in a left-handed nonlinear circuit transmission line (NLTL), which is a nonlinear analogue of a 1D SSH lattice. We find that strong higher-harmonic signals that propagate into the lattice are produced by a topological electric edge mode at the first harmonic frequency and act as an effectively nonlocal cross-phasee nonlinearity. This process leads to strongly enhanced third-harmonic generation that outperforms a comparable left-handed NLTL of conventional design. This advances our fundamental understanding of the effects of nonlinearities on topological states and can be ultilized to design compact electronic frequency generators. In the third part, we turn to a non-Hermitian (rather than nonlinear) version of the SSH model. We analyze the sublattice/time-reversal symmetry breaking for large gain/loss magnitudes, and show that a pair of defect states appears, which corresponds to nontopologically protected defect states of the SSH model at Hermitian limit. This work advances our theoretical understanding of the interaction between topology and non-Hermicity. In the fourth part, I present an experimental realization of a new type of four- dimensional(4D) topological system in a circuit lattice, based on a recently theoreti- cally proposed 4D quantum Hall model. Surface states that correspond to the second Chern number can be directly observed, due to absence of nontrivial first Chern num- ber in this model. This work shows that circuit lattices can be a convenient platform to experimentally explore topological phenomena in high-dimensional lattices.Doctor of Philosoph