Topics in Condensed Matter Theory: I. Theory of Cyclotron Resonance Power Absorption in Electron-Photon Systems. I. Suppression of the Josephson Critical Current by Quantum Electrodynamic Nyquist Noise. III. Theory of Phase Transition in Quasicharge Space in Two-Dimensional Granular Superconductors

The following thesis will consist of three parts: (1) a theory of cyclotron resonance power absorption in electron-phonon systems, (2) a theory of the suppression of the Josephson critical current at ultra-low temperatures, (3) a theory of phase transition in quasicharge space in two-dimensional granular superconductors. In part (1), an expression for the cyclotron resonance power absorption by electrons interacting with phonons will be derived with the help of a projection operator that takes into account the effects of exchange. It will be shown that the correct evaluation of this expression requires the summation of an infinite number of terms in order to describe the effects of scattering to second order. These results contradict some recently proposed theories of this phenomenon, which will be discussed and criticized. In addition, they will constitute a generalization of Argyres and Sigel's theory of cyclotron resonance power absorption in electron-impurity systems. In part (2), a linear passive shunt model for superconducting weak links will be considered. It will be shown that for the case of a resistance-capacitance shunt model, the quantum electrodynamic Nyquist noise predicts a suppression of the Josephson critical current at ultra -low temperatures, and the existence of a critical resistance for the rate of suppression will be established. Both the suppression and the critical resistance are in agreement with recent experimental observations. In part (3), 1-dimensional and 2-dimensional granular superconductors will be modeled respectively by 1-dimensional and 2-dimensional arrays of superconducting weak links. These will be taken to be described by a hamiltonian with energy bands structure in quasicharge space. It will be shown that in quasicharge space, with the assumption of nearest neighbor interactions, and with the help of a transformation to the dual lattice, the arrays are quantum XY models. A Kosterlitz-Thouless phase transition will then be deduced, the critical temperature determined, and the phases discussed.