The equilibrium behavior of vortices in a classical two-dimensional (2D) XY model with uncorrelated random phase shifts is investigated. The model describes Josephson-junction arrays with positional disorder and has ramifications in a number of other bond-disordered 2D systems. The vortex Hamiltonian is that of a Coulomb gas in a background of quenched random dipoles, which is capable of forming either a dielectric insulator or a plasma. We confirm a recent suggestion by Nattermann, Scheidl, Korshunov, and Li [J. Phys. (France) I 5, 565 (1995)] and by Cha and Fertig [Phys. Rev. Lett. 74, 4867 (1995)] that, when the variance σ of random phase shifts is sufficiently small, the system is in a phase with quasi-long-range order at low temperatures, without a reentrance transition. This conclusion is reached through a nearly exact calculation of the single-vortex free energy and a Kosterlitz-type renormalization group analysis of screening and random polarization effects from vortex-antivortex pairs. There is a critical disorder strength ${\mathrm{\sigma }}_{\mathit{c}}$, above which the system is in the paramagnetic phase at any nonzero temperature. The value of ${\mathrm{\sigma }}_{\mathit{c}}$ is found not to be universal, but generally lies in the range 0<${\mathrm{\sigma }}_{\mathit{c}}$<π/8. In the ordered phase, vortex pairs undergo a series of spatial and angular localization processes as the temperature is lowered. This behavior, which is common to many glass-forming systems, can be quantified through approximate mappings to the random energy model and to the directed polymer on the Cayley tree. Various critical properties at the order-disorder transition are calculated. © 1996 The American Physical Society.

• Received 12 April 1996

DOI:https://doi.org/10.1103/PhysRevB.54.3350