Theory of superconducting and nonsuperconducting stripe phases in the   hole-doped superconductor Ca{2-x}Na{x}CuO{2}Cl{2} (x=1/8,1/16)
  Based on the newly developed real-space spin-parallel pairing and superconducting theory, we explore a simple explanation for the observed checkerboard patterns in hole-doped Ca{2-x}Na{x}CuO{2}Cl{2} cuprate superconductor. At a hole concentration x=1/8=0.125, the analytical results show that there exists a phase competition between non-superconducting 4a*4b (a=b) checkerboard phase and 8a*2c superconducting vortex phase, where a, b and c are the lattice constants of the superconductor. At a lower hole concentration x=1/16=0.0625, it is revealed that the metastable 4\sqrt{2}a*4\sqrt{2}a checkerboard phase can reorganize itself into a more stable octahedron phase with the 4a*4a checkerboard symmetry. 
Xiuqing Huang