Strong local variational approach for superconductivity theory, and the principles of coherent interaction and action-counteraction
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 $λ$, energy gap $Δ$, and the energy cut-off $\hbar ω_0$. As $\hbar ω_0(j)$ approaches its cut-off, $λ$ and $Δ$ converge to fixed values. The theory predicts that the coupling strength $Vg(0)$ reduces to $\tilde{V}g(0)=e^{-\left(1-α_{1}\right)^{2} k^{2} / 4 λ^{2}} Vg(0)$, and the Cooper pair reduces similarly. For weak coupling, $α_1=1$, and when $Vg(0)=0.1$, $Δ_{\mathrm{A \cdot C}}=108 Δ_{\text{BCS}}$, but $Δ_{\mathrm{A \cdot C}}$ decreases to $28 Δ_{\text{BCS}}$ at $Vg(0)=0.2$. For strong coupling, $α_1=0$, 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 $0.14 C_{k \uparrow} C_{-k \downarrow}$. Additionally, $Δ_{\mathrm{A \cdot C}} = 0.5676~\text{eV} \gg \hbar ω_{\text{D}}$, and the local stacking force is $\widetilde{V}_{\text{st}}=0.264 ~\text{eV}$. With $k^2/λ^2 =$ const, the local strength increases, causing the stacking force to grow significantly. Thus, $\hbar ω_0$ and $Δ$ yield a unique solution.