Superconductivity Versus Phase Separation, Stripes, and Checkerboard   Ordering: A Two-Dimensional Monte Carlo Study
  Using Monte Carlo techniques, we study a simple model which exhibits a competition between superconductivity and other types of order in two dimensions. The model is a site-diluted XY model, in which the XY spins are mobile, and also experience a repulsive interaction extending to one, two, or many shells of neighbors. Depending on the strength and range of the repulsion and spin concentration, the spins arrange themselves into a remarkable variety of patterns at low temperatures $T$, including phase separation, checkerboard order, and straight or labyrinthine patterns of stripes, which sometimes show hints of nematic or smectic order. This pattern formation profoundly affects the superfluid density, $\gamma$. Phase separation tends to enhance $\gamma$, checkerboard order suppresses it, and stripe formation increases the component of $\gamma$ parallel to the stripes and reduces the perpendicular one. We verify that $\gamma(T = 0)$ is proportional to the effective conductance of a random conductance network whose conductances equal the couplings of the XY system. Possible connections between the model and real materials, such as single high-T$_c$ cuprate layers, are briefly discussed. 
Daniel Valdez-Balderas and David Stroud