Quantum fluctuations in the two-dimensional XY model: Variational approach.

Abstract

Due to the small self-capacitance of individual junctions, two-dimensional arrays of ultrasmall Josephson junctions are sensitive to charging effects. Identifying the charging energy term with the kinetic part of a quantum Hamiltonian for a two-dimensional XY spin system, we study the effects of quantum fluctuations on the equilibrium thermodynamics of such a system. The study is based on a self-consistent evaluation of the free energy of the system, with both effective couplings K and angular thermal averages 〈\ensuremath{\theta}〉 as variational parameters. In the weakly fluctuating regime (high values of K), we use the self-consistent harmonic approximation for an array of coupled quantum harmonic oscillators. In the strong fluctuating regime (small values of K), the coupling term is treated as a perturbation of the free-planar-rotator system. The calculation is then performed to second order in the coupling constant K. When plotted versus K, this free energy shows a double-well structure that evolves with the temperature as well as the capacitance, leading to a first-order phase transition at low temperatures. Sensitivity to charging effects is found to be considerably enhanced in fully frustrated systems (i.e., when the array is placed in a transverse magnetic field with half a flux quantum per plaquette). Furthermore, in the fully frustrated case we found an unusual coherent phase at very low temperatures. Our results are in good qualitative agreement with those obtained by Jacobs et al. with quantum Monte Carlo simulations.