Phase Transitions and Phase Fluctuations in the Josephson-Junction Arrays

We study the effect of quantum and classical phase fluctuations on the phase transitions in the system of Josephson-junction arrays. We employed a variational method for calculating the Gaussian type fluctuation of the phase in the Josephson-junction array lattice systems without and with an external magnetic field. We investigate the spectrum of collective excitations and the effects of collective excitations on the transport properties of Josephson-junction arrays. We showed that the Hamiltonian for the lattice system of the Josephson junction is the same as the Hamiltonian for the classical or quantum two-dimensional interacting rotators. We also showed that the dynamics of fluctuations of the phase in the lattice system of Josephson junction is very similar to the lattice dynamics of the lattice in crystals. We also showed that in the lattice system of Josephson junctions there is the collective acoustic mode similar to the acoustic mode in the crystal lattice, and this mode may lead to the dissipation of the Josephson current in the superconducting array of Josephson junctions. The speed of sound of the collective acoustic mode of the phase fluctuation depends on the Josephson coupling energy and the Coulomb charging energy. The contribution of the collective acoustic mode to the low temperature specific heat is the same as the contribution of the acoustic phonons to the specific heat of crystals. We also discuss the future development of results and their application.