Universality of transition temperatures in families of copper oxide superconductors

The transition temperature ($T_c$) of multi-layer cuprate superconductors has an unusual dependence on the number of layers ($n$) per unit cell: it forms a bell-shaped curve peaked at $n=3$. An explanation of this behavior is due to the combined effects of interlayer tunneling and a competing order, the latter effect being enhanced for $n \geq 3$ by a charge imbalance between the layers. We explore this proposal further by examining the mean-field theory of a superconducting order parameter and a competing $d$-density wave (DDW) order parameter. We focus on three effects: interlayer DDW coupling, increased charge imbalance in the five-layer system, and fluctuations of the superconducting order parameter. We find that (1) the DDW order parameters in neighboring layers prefer to couple ``anti-ferromagnetically''---and, surprisingly, the coupling vanishes identically for two layers with order parameters that are ``ferromagnetically'' aligned; (2) both the interlayer DDW coupling and the increased charge imbalance bring the calculation into better agreement with the experimental results; and (3) fluctuations can have a more pronounced effect when they occur in the presence of a competing order parameter.