Superconducting Critical Temperature of Homologous Series of High-T$\text{}_{c}$ Cuprates as a Function of Number of Layers

We have considered a model of n-layer high-temperature cuprates of homologous series like HgBa$\text{}_{2}$Ca$\text{}_{n-1}$Cu$\text{}_{n}$O$\text{}_{2+2n+δ}$ to determine the dependence of the critical temperature T$\text{}_{c}$(n) on the number n of Cu-O planes in an elementary cell. Focusing on the description of the high-temperature superconducting system in terms of the collective phase variables, we have studied a semi-microscopic anisotropic three-dimensional vector XY model of stacked copper-oxide layers with adjustable parameters representing microscopic in-plane and out-of-plane phase stiffnesses. The model captures the layered composition and block structure along c-axis of superconducting homologous series. Implementing the spherical closure relation we have solved the phase XY model exactly with the help of transfer matrix method for vector variables. The calculated dependence of the critical temperature T$\text{}_{c}$(n) on the block size n is monotonic with n.