# Superconducting states of the quasi-2D Holstein model: Effects of vertex and non-local corrections

## TL;DR

This study explores how spatial fluctuations and vertex corrections influence superconducting states in a quasi-2D Holstein model, revealing momentum-dependent order parameters and deviations from traditional theories.

## Contribution

It provides a detailed analysis of non-local and vertex correction effects on superconductivity beyond Migdal--Eliashberg theory in a quasi-2D Holstein model.

## Key findings

- Momentum-dependent superconducting order parameter
- Suppression of s-wave order at half-filling
- Effects beyond Migdal--Eliashberg theory

## Abstract

I investigate superconducting states in a quasi-2D Holstein model using the dynamical cluster approximation (DCA). The effects of spatial fluctuations (non-local corrections) are examined and approximations neglecting and incorporating lowest-order vertex corrections are computed. The approximation is expected to be valid for electron-phonon couplings of less than the bandwidth. The phase diagram and superconducting order parameter are calculated. Effects which can only be attributed to theories beyond Migdal--Eliashberg theory are present. In particular, the order parameter shows momentum dependence on the Fermi-surface with a modulated form and s-wave order is suppressed at half-filling. The results are discussed in relation to Hohenberg's theorem and the BCS approximation.