Experiments have pointed to the formation of the electron quadrupling condensate in Ba1-xKxFe2As2 at x similar to 0.8. The state spontaneously breaks time-reversal symmetry and is sandwiched between two critical points, separating it from the broken time-reversal symmetry (BTRS) superconducting state at and a normal-metal state at . We report a theory of the acoustic effects spectroscopy of systems with an electron quadrupling phase based on ultrasound-velocity measurements. We show that the experimental results are consistent with BTRS superconductivity at x similar to 0.8, fulfilling the necessary condition for the formation of electron quadrupling in Ba1-xKxFe2As2. We provide the theoretical basis and the experimental strategy to study the order parameter symmetry of emerging quadrupling condensates in superconductors.

We predict a topological defect in perfectly screened ferroelectric barium titanate which we call a skyrme line. These are linelike objects characterized by skyrmionic topological charge. As well as configurations with integer topological charge, the charge density can split into well-localized parts carrying a localized fraction of topological charge. We show that under certain conditions the fractional skyrme lines are stable. We discuss a mechanism to create fractional topological charge objects and investigate their stability.

We construct a modified non-Bogomol'nyi-Prasad-Sommerfield sine-Gordon theory which supports stable static kinks of arbitrary topological degree N. We use this toy model to study problems that are interesting for higher-dimensional soliton theories supporting multisolitons. We construct a 2-kink collective coordinate model and use it to generate scattering trajectories, which are compared to full-field dynamics. We find that the approximation works well, but starts to fail as radiation becomes more important, due to our model becoming less Bogomol'nyi-Prasad-Sommerfield or when the initial kink velocities are large. We also construct the quantum 2-kink using various approximations and consider how these quantum corrections affect the binding energy of the 2-kink.

We provide a step-by-step method to construct skyrmions from instanton ADHM data, including when the exact ADHM data is unknown. The configurations look like clusters of smaller skyrmions, and can be used to build manifolds of skyrmions with or without symmetries. Nuclei are described by quantum states on these manifolds. We describe the construction and quantization procedure generally, then apply the methods in detail to the 8-skyrmion which describes the Beryllium-8 nucleus.

A major problem in the Skyrme model is that the binding energy of skyrmions, which model nuclei, is too high by an order of magnitude. We show that the most popular solution to this problem, to construct models with zero classical binding energy, still produces large binding energies when spin energy is included. We argue that it is thus necessary to include quantum effects. We calculate the binding energy of skyrmions including the most simple quantum correction, that of vibrational modes in a harmonic approximation. We show that this can give physically reasonable results for nucleon numbers N=1-8 thanks to a remarkable cancellation between the strongly binding classical energy and the strongly unbinding zero-point energy from vibrational modes.

We study ferroelectric domain walls in barium titanate-type systems with perfect electric screening. We search for structurally nontrivial so-called non-Ising domain walls, where the polarization is nontrivial for all components. Our approach enables us to find solutions for domain walls in any orientation, and the existence and energy of these walls depend on their particular orientation. We find that, across all phases of the material, there are orientations where the non-Ising walls have lower energy than Ising walls. The most interesting property of these domain walls is their nonmonotonic interaction forces, allowing them to form stable domain-wall clusters rather than following standard behavior where domain walls annihilate or repel each other. We find the required external electric field to create the non-Ising configurations. Besides theoretical interest, this unconventional property of domain walls makes them a good candidate for memory applications.

We apply the collective coordinate model framework to describe collisions of a kink and an antikink with nonzero total momentum, i.e., when the solitons possess different velocities. The minimal moduli space with only two coordinates (the mutual distance and the position of the center of mass) is of a wormhole type, whose throat shrinks to a point for symmetric kinks. In this case, a singularity is formed. For non-zero momentum, it prohibits solutions where the solitons pass through each other. We show that this unphysical feature can be cured by enlarging the dimension of the moduli space, e.g., by the inclusion of internal modes.

We construct and study two kink theories. One contains a static 2-kink configu-ration with controllable binding energy. The other contains a locally stable non-top ological solution, which we call a lavi ' on. The new models are 1D analogs of non-integrable systems in higher dimensions such as the Skyrme model and realistic vortex systems. To help construct the theories, we derive a simple expression for the interaction energy between two kinks.

We derive the nucleon-nucleon interaction from the Skyrme model using the instanton and product approximations to skyrmion dynamics. In doing so, we also calculate the classical potential and metric for skyrmion dynamics in each of the approximations. This is the first time they have been compared in detail and the results show major disagreements between the approximations. We derive the eight low-energy nucleon-nucleon interaction potentials and compare them with the Paris model. For the instanton approximation we find strong negative isoscalar and isovector spin-orbit potentials, matching phenomenological models and our geometric intuition. Results for the other potentials are mixed, in part due to the zero pion mass limit used in this approximation.