Josephson arrays in an incommensurate magnetic field

We study the irrationally frustrated two-dimensional XY model, which is a model for Josephson-junction arrays in an incommensurate magnetic field, and provide evidence that there is no phase transition at finite temperatures. We use the Hubbard-Stratanovich transform to construct the Landau-Ginzburg-Wilson Hamiltonian, characterized by an infinite number of order parameters, which may be expressed in the form of a continuous function of z (0$q_2$ component of the critical mode. It is suggested that the fluctuations may be expressed in terms of an infinite number of coupled XY models. The renormalization-group idea then allows us to obtain the effective Hamiltonian in the form of a bond-diluted XY model, which in turn leads to the conclusion that there is no phase transition at finite temperatures. Also indicated is the existence of an upper bound for the transition temperature of a system with a sufficiently small rational frustration.