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Based on the newly developed real-space spin-parallel pairing andsuperconducting theory, we explore a simple explanation for the observedcheckerboard patterns in hole-doped Ca{2-x}Na{x}CuO{2}Cl{2} cupratesuperconductor. At a hole concentration x=1/8=0.125, the analytical resultsshow that there exists a phase competition between non-superconducting 4a*4b(a=b) checkerboard phase and 8a*2c superconducting vortex phase, where a, b andc are the lattice constants of the superconductor. At a lower holeconcentration x=1/16=0.0625, it is revealed that the metastable4\sqrt{2}a*4\sqrt{2}a checkerboard phase can reorganize itself into a morestable octahedron phase with the 4a*4a checkerboard symmetry.

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)