Minimal-size real-space d -wave pairing operator in CuO2 planes

Abstract

A novel minimal-size pairing operator ${\mathrm{\ensuremath{\Delta}}}_{D0}^{\ifmmode\dagger\else\textdagger\fi{}}$ with $d$-wave symmetry in ${\mathrm{CuO}}_{2}$ planes is introduced. This pairing operator creates on-site Cooper pairs at the four oxygens that surround a copper atom. Via the time evolution of ${\mathrm{\ensuremath{\Delta}}}_{D0}^{\ifmmode\dagger\else\textdagger\fi{}}$, an additional interorbital pairing operator ${\mathrm{\ensuremath{\Delta}}}_{Dpd}^{\ifmmode\dagger\else\textdagger\fi{}}$ with $d$-wave symmetry is generated that pairs fermions located in a Cu and its four surrounding O atoms. The subsequent time evolution of ${\mathrm{\ensuremath{\Delta}}}_{Dpd}^{\ifmmode\dagger\else\textdagger\fi{}}$ generates an intraorbital $d$-wave pairing operator ${\mathrm{\ensuremath{\Delta}}}_{Dpp}^{\ifmmode\dagger\else\textdagger\fi{}}$ involving the four O atoms that surround a Cu atom, as well as the $d$-wave operator ${\mathrm{\ensuremath{\Delta}}}_{D}^{\ifmmode\dagger\else\textdagger\fi{}}$ traditionally used in single-band models for cuprates. Because we recover the larger size operators extensively used in the three-orbital Hubbard model, we suggest that long-range order using the canonical extended operators occurs together with long-range order in the new minimal operators. However, our minimal $d$-wave operators could be more practical to study $d$-wave superconductivity because in the finite-size relatively small systems accessible to computational techniques it is easier to observe long-range order using local operators. Moreover, an effective model with the usual tight-binding hopping of the ${\mathrm{CuO}}_{2}$ planes supplemented by an attractive potential $V$ in the $d$-wave channel is introduced. Using mean-field techniques, we show that a paired ground state is stabilized for any finite value of $V$. We observed that the values of $V$ that lead to gap sizes similar to those in the cuprates are smaller for $d$-wave pairing operators that include Cu $d$ orbitals than those that include only $p$ orbitals. In all cases the gap that opens in the spectrum has standard $d$-wave symmetry. Finally, a simpler effective model is introduced to study the phenomenology of multiorbital $d$-wave superconductors, similar to how the negative-$U$ Hubbard model is used for properties of $s$-wave superconductors.