This dissertation represents a study in selected topics in condensed matter physics. We study Fano resonance and tunneling of electrons on the surface of liquid helium and in photonic crystals, an exactly solvable su(n) Kondo model, and determination of the glass transition temperature of polymers from small-time simulations.
For electrons on the surface of helium in a perpendicular magnetic field we applied a small in-plane magnetic field to study Fano resonance. Certain states that were bound to the helium surface then dissolve into the continuum turning into long-lived resonances. As a result microwave absorption lines acquire an asymmetric Fano lineshape that is tunable
by varying the microwave polarization or the in-plane magnetic field. Electrons trapped in a formerly bound state will tunnel off the surface of helium; we show that under suitable circumstances this “radioactive decay” can show damped oscillations rather than a simple exponential decay. Since the formal derivation of asymmetric line shapes requires subtle analysis of some diagonalization procedure we study simple models that explain Fano physics without an elaborat analysis by using Padè approximation. As a specific example we study resonant transmission in a photonic crystal channel drop device and show its frequency dependence to have an asymmetric Fano lineshape.
In the second topic we study a single channel one dimensional Kondo Model where the impurity spin is replaced by an su(n) spin. Using Bosonization and canonical transformation we explicitly shown that this system has an exactly solvable point. The calculation also shows that for the model we consider the solvable point is the same for all n.
In a simulation study we devised techniques by which the glass transition temperature of a polymer can be predicted with minimal computational effort. Three polymers were studied in atomistic molecular dynamics simulations and the mean squared displacements of their molecules have been analyzed by two new techniques. These techniques, which utilize the convoluted-velocity autocorrelation and the curvature of the mean squared displacement, efficiently predict the glass transition temperature of the polymers from short-time simulations.