  Nontrivial topology in physical systems is the driving force behind many
phenomena. Notably, phases of matter must be classified in part by their
topological properties. Phases with topological order (TO), such as the
fractional quantum Hall effect (QHE), can support anyonic excitations obeying
fractional statistics with potential application in topological quantum
computing. States lacking intrinsic TO can still be in topological phases
provided certain symmetries are imposed. On their own, these symmetry-protected
phases do not support anyons, but they can still have other interesting
features, such as protected boundary states. In this thesis we present our
research into several nontrivial systems. First, we present the results on
light propagation in the valley modes of inversion-symmetry broken honeycomb
lattices. We find that a rotating spiral pattern, leading to Zitterbewegung,
arises in the intensity profile of the beam as a result of the nontrivial
topology of the valleys. Next, we present the numerical demonstration of
dynamical topological phase transitions driven by nonlinearity of the photonic
medium, which occur in soliton SSH lattices. The phase transitions, marked by
the appearance of topological edge states in the band gap, occur due to the
setup which enables continually changing values of the intracell and intercell
soliton couplings. Finally, we propose a scheme for creating and manipulating
synthetic anyons in a noninteracting system by perturbing it with specially
tailored localized probes. The external probes are needed because
noninteracting systems do not possess the kind of TO required to support
anyons. We start from a 2DEG in a uniform magnetic field (in an integer QHE
state) and introduce thin solenoids carrying a fractional magnetic flux. We
find a suitable ground state and demonstrate the fractional braiding statistics
in the coordinates of the solenoids.
