Ginzburg Landau Theory for Cuprate Superconductivity

We propose and develop the consequences of a theory in which the free energy $F$ of a cuprate is expressed as a functional of the complex nearest neighbour spin singlet bond pair order parameter $\Delta_{ij} \exp{(i\phi_{ij})}$. $F$ is a sum of two terms $\sum_m(a \Delta_m^2+b\Delta_m^4)$, and $F_1= c\sum_{mn}\Delta_{mn}\cos{(\phi_m-\phi_n)}$; here $m$ is the site corresponding to $ij$ on the dual lattice (also square) and $m, n$ are nearest neighbours. The doping $x$ and temperature $T$ dependences of $a, b$ and $c$ are rationalized ( eg, $c\propto x$ for small $x$). The pseudogap ( due to incoherent bond pairs) and the parabolic $x$ dependence of $T_c$ ( AF ordering of the 2d-XY spin $\Delta_m\exp{(i\phi_m)}$ leading to d wave superconductivity) are described. The observed $C_v(T)$ behaviour is shown to be due to order parameter fluctuations. Detailed calculations of the vortex structure show a crossover from a Josephson like to a BCS like form with increasing doping, mirroring a similar change in superconductivity.