Josephson lattice model for phase fluctuations of local pairs in copper oxide superconductors

We derive an expression for the effective Josephson coupling from the microscopic Hubbard model. It serves as a starting point for the description of phase fluctuations of local Cooper pairs in ${d}_{{x}²\ensuremath{-}{y}²}$-wave superconductors in the framework of an effective $XY$ model of plaquettes, the Josephson lattice. The expression for the effective interaction is derived by means of the local-force theorem, and it depends on local symmetry-broken correlation functions that we obtain using the cluster dynamical mean-field theory. Moreover, we apply the continuum limit to the Josephson lattice to obtain an expression for the gradient term in the Ginzburg-Landau theory and compare predicted London penetration depths and Kosterlitz-Thouless transition temperatures with experimental data for ${\mathrm{YBa}}₂{\mathrm{Cu}}₃{\mathrm{O}}_{7\ensuremath{-}x}\phantom{\rule{0.16em}{0ex}}$.