Nonlinear Dynamics and Condensed States with Broken Symmetries
This thesis concerns two topics where symmetry and strong correlations in time and space are important. The first topic is the theory of dynamical systems, and this theory deals with generic phenomena which are often convenient to study in very simple models. Dynamical systems can be divided in two basic classes: dissipative and conservative systems.
Dissipation often effectively reduces the degrees of freedom in a dynamical system, and it is often sufficient to study low-dimensional models. An example is the one- dimensional circle map. The transition from ordered to chaotic motion in a circle map with extra symmetry is studied in Paper I using renormalization group techniques. Special emphasis is put into the possible breakdown of the extra symmetry.
An example of a conservative system is a particle moving in a potential field, and typical for the dynamics is the large number of periodic trajectories. A particularly simple class of conservative systems is the integrable systems. In such a system there are continuous manifolds of marginally stable periodic orbits, which are divided into discrete sets of stable and unstable periodic orbits if the potential of the system is nonlinearly perturbed. In Paper II we demonstrate how the stable and unstable orbits in a quartic potential exchange stability several times as the strength of the nonlinear perturbation is increased.
The second topic of this thesis is the ceramic high-temperature superconductors. Apart from being superconducting these materials show several other phases dependent on the doping. The 2-dimensional Hubbard model has been suggested as a highly simplified model for the electronic properties of these materials, and we study an extended version of this model at arbitrary temperature and doping using a theory capable of describing a rich variety of states simultaneously. Important is that we use no a priori assumptions of the state of the system. The high degree of freedom in the theory requires a group- theoretical treatment in order to classify the possible states, and this classification relies on the basic symmetries of the Hubbard model. Examples of states that we find are antiferromagnetic, ferromagnetic, superconducting (s- and d-wave), charge density wave and spiral spin-wave states.