Three distinct types of quantum phase transitions in a (2+1)-dimensional   array of dissipative Josephson junctions
  We have performed large-scale Monte Carlo simulations on a model describing a (2+1)-dimensional array of dissipative Josephson junctions. We find three distinct stable quantum phases of the system. The most ordered state features long-range spatial ordering in the phase $\theta$ of the superconducting order parameter, but temporal ordering only in spatial gradients $\Delta \theta$, not in $\theta$. Significantly, the most ordered state therefore does not have 3D XY ordering. Rather, it features 2D spin waves coexisting with temporally disordered phases $\theta$. There is also an intermediate phase featuring quasi-long-range spatial order in $\theta$ coexisting with a gas of instantons in $\Delta \theta$. We briefly discuss possible experimental signatures of such a state, which may be viewed as a local metal and a global superconductor. The most disordered state has phase disorder in all spatio-temporal directions, and may be characterized as a gas of proliferated vortices coexisting with a gas of $\Delta \theta$-instantons. The phase transitions between these phases are discussed. The transition from the most ordered state to the intermediate state is driven by proliferation of instantons in $\Delta \theta$. The transition from the intermediate state to the most disordered state is driven by the proliferation of spatial point vortices in the background of a proliferated $\Delta \theta$-instanton gas, and constitutes a Berezinskii-Kosterlitz-Thouless phase transition. The model also features a direct phase transition from the most ordered state to the most disordered state, and this transition is neither in the 2D XY nor in the 3D XY universality class. It comes about via a simultaneous proliferation of point vortices in two spatial dimensions and $\Delta \theta$-instantons, with a complicated interplay between them. 
Einar B. Stiansen. Iver B. Sperstad, and Asle Sudb{\o}
cond-mat.stat-mech cond-mat.str-el