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Nontrivial topology in physical systems is the driving force behind manyphenomena. Notably, phases of matter must be classified in part by theirtopological properties. Phases with topological order (TO), such as thefractional quantum Hall effect (QHE), can support anyonic excitations obeyingfractional statistics with potential application in topological quantumcomputing. States lacking intrinsic TO can still be in topological phasesprovided certain symmetries are imposed. On their own, these symmetry-protectedphases do not support anyons, but they can still have other interestingfeatures, such as protected boundary states. In this thesis we present ourresearch into several nontrivial systems. First, we present the results onlight propagation in the valley modes of inversion-symmetry broken honeycomblattices. We find that a rotating spiral pattern, leading to Zitterbewegung,arises in the intensity profile of the beam as a result of the nontrivialtopology of the valleys. Next, we present the numerical demonstration ofdynamical topological phase transitions driven by nonlinearity of the photonicmedium, which occur in soliton SSH lattices. The phase transitions, marked bythe appearance of topological edge states in the band gap, occur due to thesetup which enables continually changing values of the intracell and intercellsoliton couplings. Finally, we propose a scheme for creating and manipulatingsynthetic anyons in a noninteracting system by perturbing it with speciallytailored localized probes. The external probes are needed becausenoninteracting systems do not possess the kind of TO required to supportanyons. We start from a 2DEG in a uniform magnetic field (in an integer QHEstate) and introduce thin solenoids carrying a fractional magnetic flux. Wefind a suitable ground state and demonstrate the fractional braiding statisticsin the coordinates of the solenoids.

Anyons, Zitterbewegung and dynamical phase transitions in topologically nontrivial systems