This work is done with Philippe Curty and  Hans Beck.

The behaviour of classical superconductors is well described by the BCS theory which is a mean-field variational approach with electrons forming pairs and causing a gap in the energy spectrum of single electrons. 

High temperature superconductors as copper oxides have many features that are  not understood in the framework of BCS theory. One of the most important problems is the presence of a region in the normal regime above the critical temperature Tc and below a temperature T* where observable quantities deviate from the free electron gas behaviour. This region is called pseudogap region because it contains effects similar to superconductivity like a partial suppression of electronic density of states. 

Schematic experimental phase diagram of cuprates superconductors.  Anomalous effects are measured in the pseudogap region (below T*) of the copper oxides phase diagram. Superconductivity is present only below Tc and some signals persists until T'(Nernst and Hall effects). Eg is the energy scale of the pseudogap. This phase diagram is controversial since the Eg line crosses T' and T*. Usually people present either Eg or T' and T*.

The properties of high temperarture superconductors with low and high superfluid density can be described in term of phase fluctuations (i.e. variartions of the "velocity" of the superfluidity) and amplitude fluctuations (i.e. variation of the "pairing density"): based on a pairing mechanism between electrons, the pseudogap regime is described by  taking into account amplitude and phase fluctuations of the pairing field.

Our calculations show good quantitative agreement with specific heat and magnetic susceptibility experiments. We find that the mean-field temperature To has a similar doping dependence as the  pseudogap temperature T*, moreover the caracteristic pseudogap energy scale Eg is given by the average amplitude above T_\phi:
 

Comparison between theory (variational method) and experimental specific heat on underdoped YBCO_6.73.  The two peaks of the specific heat (thick blue) are the sum of the critical XY contribution (green) and the amplitude contribution (dashed blue).

 

Comparison between theory (average value method) and experimental specific heat on  YBCO_{6+x}. The specific heat (thick blue) is given by C= Co + Cxy, where Co is the BCS specific heat depending on the average amplitude Co = Co(<|\psi|>), and Cxy comes from the phase part of the Ginzburg-Landau (GL) action. (averages are performed by using Monte Carlo simulations of the GL action)
	This fits are obtained by using a Monte Carlo procedure in the parameters space, i.e. by varying parameters randomly until the best fit is obtained. With this method, you can find the absolute best fits. See the fitting  animation for doping x=0.73:  (433 Kb)
	Comparison between the measured spin susceptibility (points) of underdoped YBCO_{6.64} and theory. The d-wave (thick blue) fits experiments.  The average amplitude is shown in blue and its standard deviation as well. T'_\phi is the temperature where the amplitude start to be influence from phases. The green dashed line is the amplitude for uncorrelated phases.
In the underdoped regime especially, the averaged amplitude of the pairing field is of the order of the zero temperature gap  up to room temperature. Above some crossover temperature T_\phi larger than the critical temperature Tc, the pseudogap region is mainly determined by the amplitude whereas phase fluctuations are only important near Tc up to T_\phi.
	Extracted phase diagram of cuprates. To is the mean-field pairing temperature.  Vo is the phase stiffness and is proportionnal to the charge carrier density. T_phi is the temperature where correlations disappear. T_1 is the temperature where the phase specific heat contribution vanishes. The hatched region is where the pseudogap starts, i.e. where T* is located. Inset: Eg is the energy scale of the pseudogap, here defined as the average amplitude at 200K.

• Superconductivity and pseudogap must (can) have the same origin.

• Two regimes in the pseudogap: correlated (phase) and uncorrelated (amplitude). (is not the phase scenario.)

• The pseudogap energy scale is controled by the fluctuations. (There is no quantum critical point)

Publication:

Thermodynamics and Phase Diagram of High Temperature Superconductors (PDF)
Reference:
Ph. Curty, H. Beck 
Phys. Rev. Lett. 91, 257002 (2003)
cond-mat/0401124