Theory of magic dopings in high Tc superconductors

Based on an effective bosonic model, we predict checkerboard- type ordering of the Cooper pairs at magic rational doping fractions $(2m+1)/2^n$, where m and n are integers. At the magic doping fraction $x = (2m+1)/2^n$, the charge unit-cell is $2^ {(n+1)/2}a \times 2^{(n+1)/2}a$, pointing along the original CuO bond direction when $n$ is odd, and along the diagonal direction when $n$ is even. It is generally expected that the charge ordering tendencies are stronger at higher levels of the hierarchy, with smaller n. Recently, a tendency towards charge ordering at particular rational hole doping fractions, 1/16, 3/32, 1/8, and 3/16, is reported in transport measurements of $La_{2-x}Sr_xCuO_4$ samples[1]. This observation is most consistent with our predicted hierarchy of charge ordering of hole-pairs.