# The unbalanced phonon-induced superconducting state on a square lattice beyond the static boundary

## TL;DR

This study investigates the induction of superconductivity on a square lattice via electron-phonon interactions, revealing a lower unbalance parameter threshold than the static boundary and significant deviations from BCS theory predictions.

## Contribution

The paper demonstrates superconductivity induction at unbalance parameters below the static boundary using Eliashberg equations with momentum-dependent interactions, extending understanding beyond BCS theory.

## Key findings

- Superconductivity occurs for unbalance parameter $\, extless 0.42$
- Thermodynamic functions differ markedly from BCS predictions
- The static boundary $\, extgreater 0.93$ is not reached in the dynamic case

## Abstract

The paper presents our verification of induction of the superconducting state on a square lattice by the linear electron-phonon interaction for values of the unbalance parameter ($\gamma=\lambda_{D}/\lambda_{ND}$) less than $\gamma_{C}=0.42$. Symbols $\lambda_{D}$ and $\lambda_{ND}$ denote the values of the coupling constant in the diagonal and the non-diagonal channel of the self-energy. Calculations were carried out using the Eliashberg equations, in which the order parameter ($\Delta_{\bf k}\left(i\omega_{n}\right)$) and the wave function renormalising factor ($Z_{\bf k}\left(i\omega_{n}\right)$) depend explicitly on the Matsubara frequency ($\omega_{n}$) and the wave vector (${\bf k}$). The value of $\gamma_{C}$ in the static boundary ($\Delta_{\bf k}\left(i\omega_{n}\right)\rightarrow \Delta_{\bf k}\left(i\omega_{n=1}\right)$), equal to ($0.93$), is significantly greater than the obtained limit value of $0.42$. Values of the thermodynamic functions of the superconducting state determined for our assumptions are significantly different from the values calculated in accordance with the BCS theory. The results were obtained for the electron-phonon interaction function explicitly dependent on the momentum transfer between electron states.