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

TL;DR

This paper introduces a local variational theory for two-mode electron pairing in superconductivity, emphasizing coherent interaction principles, and predicts how coupling strength and energy gaps behave under different conditions.

Contribution

It develops a novel local variational approach based on coherent interaction principles, providing self-consistent equations for energy gap and coupling parameters in superconductivity.

Findings

The theory predicts the reduction of coupling strength and Cooper pair size with increasing interaction.

It shows the energy gap can significantly exceed the Debye energy in strong coupling regimes.

The local stacking force and energy gap are interconnected and depend on the variational parameters.

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 $\lambda$, energy gap $\Delta$, and the energy cut-off $\hbar \omega_0$. As $\hbar \omega_0(j)$ approaches its cut-off, $\lambda$ and $\Delta$ converge to fixed values. The theory predicts that the coupling strength $Vg(0)$ reduces to $\tilde{V}g(0)=e^{-\left(1-\alpha_{1}\right)^{2} k^{2} / 4 \lambda^{2}} Vg(0)$, and the Cooper pair reduces similarly. For weak coupling, $\alpha_1=1$, and when $Vg(0)=0.1$, $\Delta_{\mathrm{A \cdot C}}=108 \Delta_{\text{BCS}}$, but $\Delta_{\mathrm{A \cdot C}}$ decreases to $28 \Delta_{\text{BCS}}$ at $Vg(0)=0.2$. For strong coupling, $\alpha_1=0$, 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, $\Delta_{\mathrm{A \cdot C}} = 0.5676~\text{eV} \gg \hbar \omega_{\text{D}}$, and the local stacking force is $\widetilde{V}_{\text{st}}=0.264 ~\text{eV}$. With $k^2/\lambda^2 =$ const, the local strength increases, causing the stacking force to grow significantly. Thus, $\hbar \omega_0$ and $\Delta$ yield a unique solution.