Abstract

The purpose of this Thesis is to theoretically investigate the power of topology on various problems in condensed-matter physics, as well as their possible applications for biological systems. We start the discussions with the so-called Laughlin’s gauge argument, i.e. quantized topological adiabatic transport in the integer quantum Hall cylinder. We
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describe singe-electron transport by using the mathematical equivalence between the two-dimensional integer quantum Hall effect and a one-dimensional charge-density wave, and reformulate the problem for the latter system. The obtained results yield adiabatic and nonadiabatic transport protocols for both setups. The theoretical predictions open prospects for experimental realizations in the context of ultracold atoms and photonic waveguide experiments. Then, we switch from topological insulators to the Josephson effect in a superconductor-insulator-superconductor setup. In experiment, an oscillatory current as a function of the phase difference between the two superconductors is observed, and the periodicity of this oscillation is set by the quantum of magnetic flux that has a soliton-like character. It can be described with the modified sine-Gordon equation. In real life, one cannot avoid the presence of inhomogeneities in a system. We consider the behavior of topological solitary waves (solitons and kinks) in the sine-Gordon and the phi4 classical field theories, in the presence of a step-like inhomogeneity. We observe that the wave evolution depends on the initial conditions, namely the initial velocity of a wave and the coupling constant of the potential. The soliton and kink movement can be either an elastic reflection or an inelastic propagation through the media with loss of kinetic energy and the excitation of small waves on top of the main wave, which can be interpreted as plasma waves in the Josephson effect. We observe that solitary wave behavior can be predicted by describing the latter as a relativistic particle. Most results are model-independent meaning that the integrable structure of the sine-Gordon model is irrelevant and robustness is rooted in the topology of these field configurations. Afterwards, we continue investigating single kink solutions of the phi4 classical field theory, but now we verify its interaction with a single impurity. We find that kink behavior depends on the exact shape of the defect and on whether it is attractive or repulsive. We numerically prove the existence of an internal impurity mode, which already had been predicted theoretically beforehand. Further, we focus on a single Gaussian impurity, which stays in between the kink and the antikink. For some parameters of impurity strength and initial velocities of the kink and the antikink, a transformation of a kink-impurity-antikink system into oscillon-impurity-oscillon occurs. The oscillon is another small-amplitude oscillating quasi-long-lived solution. We observe the bound state of two oscillons and the oscillons’ escape windows. The kinks of the phi4 model proved useful in the description of proteins. Despite the apparent simplicity of a problem of kink-impurity and kink-impurity-antikink interactions, these results might be useful in the future as a powerful tool for influence on a protein folding.
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