Strong local variational approach for superconductivity theory, and the principles of coherent interaction and action-counteraction

Abstract

For the two-mode electron pairing, we propose a local stacking force pairing mechanism driven by strong local fluctuations, with two straight pairing orbits where the tying Cooper pairing $C_{-k\downarrow }{C}_{k\uparrow }{e}^{ik\cdot r}$ replaces the itinerant pairing. Based on coherent interaction and action-counteraction principles, the strong local variational theory is constructed, with the energy extremum and gap equations forming self-consistent pairs, involving the local variational parameter $\hat{I}»$, energy gap $\hat{I}”$, and the energy cut-off $\hslash \ddot{I}{‰}_0$. As $\hslash \ddot{I}{‰}_0(j)$ approaches its cut-off, $\hat{I}»$ and $\hat{I}”$ converge to fixed values. The theory predicts that the coupling strength $Vg(0)$ reduces to $\tilde{V}g(0)=e^{-{\left(1-\hat{I}{\pm }_1\right)}^2{k}^2/4\hat{I}{»}^2}Vg(0)$, and the Cooper pair reduces similarly. For weak coupling, $\hat{I}{\pm }_1=1$, and when $Vg(0)=0.1$, $\hat{I}{”}_{\mathrm{A}\cdot \mathrm{C}}=108\hat{I}{”}_{\text{BCS}}$, but $\hat{I}{”}_{\mathrm{A}\cdot \mathrm{C}}$ decreases to $28\hat{I}{”}_{\text{BCS}}$ at $Vg(0)=0.2$. For strong coupling, $\hat{I}{\pm }_1=0$, if $Vg(0)=1.4$, $\tilde{V}g(0)$ reduces to 0.2, and the smaller Cooper pair $\widetilde{C_{k\uparrow }{C}_{-k\downarrow }}$ reduces to $0.14C_{k\uparrow }{C}_{-k\downarrow }$. Additionally, $\hat{I}{”}_{\mathrm{A}\cdot \mathrm{C}}=0.5676\text{eV}\gg \hslash \ddot{I}{‰}_{\text{D}}$, and the local stacking force is ${\tilde{V}}_{\text{st}}=0.264\text{eV}$. With $k^2/\hat{I}{»}^2=$ const, the local strength increases, causing the stacking force to grow significantly. Thus, $\hslash \ddot{I}{‰}_0$ and $\hat{I}”$ yield a unique solution.