Thermodynamics and Phase Diagram of High Temperature Superconductors
  Thermodynamic quantities are derived for superconducting and pseudogap regimes by taking into account both amplitude and phase fluctuations of the pairing field. In the normal (pseudogap) state of the underdoped cuprates, two domains have to be distinguished: near the superconducting region, phase correlations are important up to the temperature $T_\phi$. Above $T_\phi$, the pseudogap region is only determined by amplitudes, and phases are uncorrelated.   Our calculations show excellent quantitative agreement with specific heat and magnetic susceptibility experiments on cuprates. We find that the mean field temperature $T_0$ has a similar doping dependence as the pseudogap temperature $T^*$, whereas the pseudogap energy scale is given by the average amplitude above $T_c$. 
Philippe Curty and Hans Beck