Importance of local correlations for the order parameter of high-$T_c$ superconductors

It is shown that local temporal correlations in addition to nonlocal spatial correlations are important to understand the size and the doping dependence of the $d$-wave superconducting order parameter of high-temperature superconductors. To this end, the hole- and electron-doped two-dimensional Hubbard model at zero temperature is considered and treated by an extension of the variational cluster approximation. Within this approach, the effects of temporal correlations can be studied systematically and in a thermodynamically consistent way by comparing results obtained from different reference clusters. Contact can be made with previous cellular (plaquette) dynamical mean-field calculations. This shows that temporal correlations considerably decrease the order parameter and provide a substantial gain of binding energy. Besides, a few methodical insights regarding real-space quantum-cluster approaches are obtained in addition.

Figure 1

(Color online) Reference systems and variational parameters as used for our calculations. Additionally a corresponding reference system generating the C-DMFT is illustrated. Here, a continuous bath is optimized while the parameters of the correlated sites remain at their physical values.

Figure 2

(Color online) Superconducting ($d$-wave) order parameter ${\Delta }_{\text{SC}}$ [see Eq. (10)] as a function of the electron density $n$ within conventional VCA (blue line, variational parameters $\epsilon$ and $h_{\text{SC}}$) and VCA including bath sites (red line, variational parameters $\epsilon$, ${\epsilon }_b$, $V$, and $h_{b,\text{SC}}$). In the case of VCA with bath sites, a second solution with higher ground-state energy is found (dashed red line). Calculations have been performed for system sizes up to $L=6400$ sites. C-DMFT results from Kancharla et al. (green symbols, Ref. 20, Fig. 2) are shown for comparison.

Figure 3

(Color online) Ground-state energy as a function of the electron density $n$ within conventional VCA (blue line) and VCA including bath sites (red line). Dashed red line: second solution with higher ground-state energy.

Figure 4

(Color online) Difference $\left|n-n^{\prime }\right|$ of the electron density in the original and the reference system (correlated sites only) as a function of the electron density $n$. VCA calculations with and without bath sites corresponding to the calculations shown in Fig. 2. $n-n^{\prime }$ changes sign at $n=1$ for the conventional VCA calculation (blue line) and is positive for electron doping.

Figure 5

(Color online) Relative difference ${\delta }_{\Delta }$ according to Eq. (11) as a function of $n$ for VCA calculations with and without bath sites. We find the bath sites to partially compensate for the artificial breaking of translational symmetry.