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 temperature $T_{\varphi }$. Above $T_{\varphi }$, the pseudogap region is determined only 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 $T_c$.

Figure 1

Average amplitude $⟨|\tilde{\psi }|⟩$ (thick line) and the corresponding standard deviation $(⟨|\tilde{\psi }{|}^2⟩-⟨|\tilde{\psi }|{⟩}^2{)}^{1/2}$ (long-dashed line) from 2D GL simulations. Parameters are $V_0=95\mathrm{K}$, $T_0=260\mathrm{K}$. Dashed curves: amplitude distribution $P(|\tilde{\psi }|)$ (histogram) for $T=300\mathrm{K}$ (right) and $T=30\mathrm{K}$ (left).

Figure 2

The specific heat from the variational method (thick line), which is the sum of gradient (dashed line) and amplitude (dot-dashed line) contributions, reproduces measurements of ${\mathrm{Y}\mathrm{B}\mathrm{a}}_2{\mathrm{C}\mathrm{u}}_3{\mathrm{O}}_{6.73}$ (points). Inset: $T$ dependent amplitude $|\overline{\psi }|$ from Eq. (4).

Figure 3

Experimental specific heat (points) for ${\mathrm{Y}\mathrm{B}\mathrm{a}}_2{\mathrm{C}\mathrm{u}}_3{\mathrm{O}}_{6+x}$ compared to the average value method.

Figure 4

The measured spin susceptibility of ${\mathrm{Y}\mathrm{B}\mathrm{a}}_2{\mathrm{C}\mathrm{u}}_3{\mathrm{O}}_{6.63}$ (points) divided by ${\chi }_0$ is well fitted by the theoretical (thick line) susceptibility ${\chi }_p$. The dot-dashed line is the standard deviation of the average amplitude.

Figure 5

Phase diagram of ${\mathrm{Y}\mathrm{B}\mathrm{a}}_2{\mathrm{C}\mathrm{u}}_3{\mathrm{O}}_{6+x}$. Effects of amplitudes are large in the quasi-uncorrelated pseudogap region (below $T_0$), whereas phase correlations remain important only below $T_{\varphi }$. The temperature $T^{*}$, where $\gamma$ and ${\chi }_d$ cross over to normal behavior, is located in the hatched area. Inset: the pseudogap energy scale $E_g$.