Chiral d-Wave Superconductivity in coupled ladders

We study the Hubbard model on the trellis lattice, a two-dimensional frustrated lattice of coupled two-leg ladders, with hopping amplitude $t$ within ladders and $t'$ between ladders. For large $U/t$ this is a model for the cuprate Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$. We use the variational cluster approximation (VCA), with clusters of sizes 8 to 12. We investigate the phase diagram as a function of doping, $U/t$ and $t'/t$ and find a superconducting dome ending at roughly 20\% doping. Repulsion-induced spin singlet correlations within ladders block inter-ladder single electron tunneling, but allow pair tunneling and help establish 2D superconductivity. However, the nature of the order parameter depends on doping. At small doping ($\delta<3\%$ for $t'=0.15t$ and $U=10t$), the order parameter is real and its interladder component grows steeply with $\delta$. Beyond that value, the order parameter becomes complex for a finite range of doping and gives the bulk chiral, PT violating, two-dimensional superconductivity. In all cases, the ladder component of the order parameter has d-wave character.