Interactions between topological defects in (1+1) dimensions

TL;DR

This thesis explores the complex interactions of topological defects called kinks in (1+1)D, including their resonance phenomena, long-range tails, collisions, and fermion interactions, revealing new dynamical behaviors and numerical methods.

Contribution

It introduces a toy model with quasinormal modes, develops numerical methods for long-range kinks, and studies fermion-kink and impurity interactions, advancing understanding of topological defect dynamics.

Findings

Resonance windows are suppressed when kink vibrational modes become quasinormal modes.

Kinks with long-range tails can annihilate without bion formation at high velocities.

Fermions escape from wobbling kinks as radiation at a constant rate.

Abstract

In this thesis, we study interactions between topological defects in two-dimensional spacetimes. These defects are called kinks. They are solutions of scalar field theories with localized energy which propagate without losing its shape. In order to understand the resonance phenomenon exhibited by those models, we built a toy model where the kink's vibrational mode turns into a quasinormal mode. This causes the suppression of resonance windows and, consequently, its fractal structure is lost. Considering a higher order polynomial as the scalar field potential, we find kinks with long-range tails, which decay as a power law. We developed a numerical method to correctly initialize this systems and applied it to a scalar field model containing kinks with long-range tails in both sides. After the collision, the kink-antikink pair is annihilated for velocities below an ultra-relativistic critical velocity without bion formation. We also investigated a collision between wobbling kinks of the double sine-Gordon model. When the kinks are already wobbling before colliding, there appears resonance windows with separation after a single bounce. On the second half of the thesis, we focused on fermion-kink interactions. We studied what happens when a fermion binds to a wobbling kink. The result is that the fermion escapes from the kink as radiation and at a constant rate. This occurs if the energy gap between the initial state and the continuum threshold is not too large. Lastly, we investigated the interaction of a fermion with a background scalar field with an impurity that preserves half of the Bogomol'nyi-Prasad-Sommerfield (BPS) property. We found an adiabatic evolution near the BPS regime, which means that the system is at a static BPS solution at every moment.