Stability of the Superconducting d-Wave Pairing Toward the Intersite Coulomb Repulsion in $\hbox {CuO}_2$ Plane

Abstract

Taking into account the real crystalline structure of the $\hbox {CuO}_2$ plane and the strong spin-fermion coupling, we study the influence of the intersite Coulomb repulsion between holes on the Cooper instability of the spin-polaron quasiparticles in cuprate superconductors. The analysis shows that only the superconducting d-wave pairing is implemented in the whole region of doping, whereas the solutions of the self-consistent equations for the s-wave pairing are absent. It is shown that intersite Coulomb interaction $V_1$ between the holes located at the nearest oxygen ions does not affect the d-wave pairing, because its Fourier transform $V_q$ vanishes in the kernel of the corresponding integral equation. The intersite Coulomb interaction $V_2$ of quasiparticles located at the next-nearest oxygen ions does not vanish in the integral equations, however, but it is also shown that the d-wave pairing is robust toward this interaction for physically reasonable values of $V_2$.

Appendix

The functions $S_{ij}^{(l)}(k,{{\omega }})$ in anomalous Green’s functions $F_{ij}(k,{{\omega }})$ have the form

$$\begin{aligned} S^{(1)}_{11}(k,{{\omega }})= & {} Q_{3y}(k,-{{\omega }})Q_{3y}(k,{{\omega }}),\nonumber \\ S^{(2)}_{11}(k,{{\omega }})= & {} S^{(1)}_{21}(k,{{\omega }})= Q_{3}(k,-{{\omega }})Q_{3y}(k,{{\omega }}),\nonumber \\ S^{(3)}_{11}(k,{{\omega }})= & {} S^{(1)}_{12}(k,{{\omega }})= S^{(2)}_{11}(k,-{{\omega }}),\nonumber \\ S^{(4)}_{11}(k,{{\omega }})= & {} S^{(2)}_{12}(k,{{\omega }})=S^{(3)}_{21}(k,{{\omega }})= S^{(1)}_{22}(k,{{\omega }})=Q_{3}(k,-{{\omega }})Q_{3}(k,{{\omega }}),\nonumber \\ S^{(5)}_{11}(k,{{\omega }})= & {} -Q_{y}(k,-{{\omega }})Q_{y}(k,{{\omega }}),\nonumber \\ S^{(3)}_{12}(k,{{\omega }})= & {} Q_{3y}(k,-{{\omega }})Q_{3x}(k,{{\omega }}),\nonumber \\ S^{(2)}_{21}(k,{{\omega }})= & {} S^{(3)}_{12}(k,-{{\omega }}),\nonumber \\ S^{(4)}_{12}(k,{{\omega }})= & {} S^{(3)}_{22}(k,{{\omega }})= Q_{3}(k,-{{\omega }})Q_{3x}(k,{{\omega }}),\nonumber \\ S^{(4)}_{21}(k,{{\omega }})= & {} S^{(2)}_{22}(k,{{\omega }})=S^{(4)}_{12}(k,-{{\omega }}),\nonumber \\ S^{(5)}_{12}(k,{{\omega }})= & {} -Q_{y}(k,-{{\omega }})Q_{x}(k,{{\omega }}),\nonumber \\ S^{(5)}_{21}(k,{{\omega }})= & {} S^{(5)}_{12}(k,-{{\omega }}),\nonumber \\ S^{(4)}_{22}(k,{{\omega }})= & {} Q_{3x}(k,-{{\omega }})Q_{3x}(k,{{\omega }}),\nonumber \\ S^{(5)}_{22}(k,{{\omega }})= & {} -Q_{x}(k,-{{\omega }})Q_{x}(k,{{\omega }}),\nonumber \\ S^{(1)}_{31}(k,{{\omega }})= & {} -K_kQ_{y}(k,-{{\omega }})Q_{3y}(k,{{\omega }}),\nonumber \\ S^{(2)}_{31}(k,{{\omega }})= & {} -K_kQ_{x}(k,-{{\omega }})Q_{3y}(k,{{\omega }}),\nonumber \\ S^{(3)}_{31}(k,{{\omega }})= & {} S^{(1)}_{32}(k,{{\omega }})=-K_kQ_{y}(k,-{{\omega }})Q_{3}(k,{{\omega }}),\nonumber \\ S^{(4)}_{31}(k,{{\omega }})= & {} S^{(2)}_{32}(k,{{\omega }})=-K_kQ_{x}(k,-{{\omega }})Q_{3}(k,{{\omega }}),\nonumber \\ S^{(5)}_{31}(k,{{\omega }})= & {} Q_{xy}(k,-{{\omega }})Q_{y}(k,{{\omega }}),\nonumber \\ S^{(3)}_{32}(k,{{\omega }})= & {} -K_kQ_{y}(k,-{{\omega }})Q_{3x}(k,{{\omega }}),\nonumber \\ S^{(4)}_{32}(k,{{\omega }})= & {} -K_kQ_{x}(k,-{{\omega }})Q_{3x}(k,{{\omega }}),\nonumber \\ S^{(5)}_{32}(k,{{\omega }})= & {} Q_{xy}(k,-{{\omega }})Q_{x}(k,{{\omega }}),\nonumber \\ S^{(1)}_{33}(k,{{\omega }})= & {} -K_k^2S^{(5)}_{11}(k,{{\omega }}),\nonumber \\ S^{(2)}_{33}(k,{{\omega }})= & {} K_k^2S^{(5)}_{12}(k,-{{\omega }}),\nonumber \\ S^{(3)}_{33}(k,{{\omega }})= & {} S^{(2)}_{33}(k,-{{\omega }}),\nonumber \\ S^{(4)}_{33}(k,{{\omega }})= & {} K_k^2S^{(5)}_{22}(k,{{\omega }}),\nonumber \\ S^{(5)}_{33}(k,{{\omega }})= & {} Q_{xy}(k,-{{\omega }})Q_{xy}(k,{{\omega }}). \end{aligned}$$

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where

$$\begin{aligned} Q_{x(y)}(k,{{\omega }})= & {} ({{\omega }}-\xi _{x(y)})J_{y(x)}+t_kJ_{x(y)},\qquad ~~ Q_{xy}(k,{{\omega }})=({{\omega }}-\xi _x)({{\omega }}-\xi _y)-t_k^2,\\ Q_{3x(3y)}(k,{{\omega }})= & {} ({{\omega }}-\xi _L)({{\omega }}-\xi _{x(y)})-J_{x(y)}^2K_k,~~~ Q_3(k,{{\omega }})=({{\omega }}-\xi _L)t_k+J_xJ_yK_k. \end{aligned}$$