# TRILEX and $GW$+EDMFT approach to $d$-wave superconductivity in the Hubbard model

## TL;DR

This paper extends the TRILEX method to superconducting phases in the Hubbard model, successfully capturing $d$-wave superconductivity at strong coupling and exploring its dependence on doping, dispersion, and AF correlations.

## Contribution

The paper introduces a generalized TRILEX approach for superconductivity, capable of modeling $d$-wave pairing with a single-site impurity, and compares it with $GW$+EDMFT results.

## Key findings

- TRILEX produces a $d$-wave superconducting dome at strong coupling.
- The method captures $d$-wave pairing using only a single-site impurity model.
- Superconducting temperature depends on hopping amplitudes and AF correlations.

## Abstract

We generalize the recently introduced TRILEX approach (TRiply Irreducible Local EXpansion) to superconducting phases. The method treats simultaneously Mott and spin-fluctuation physics using an Eliashberg theory supplemented by local vertex corrections determined by a self-consistent quantum impurity model. We show that, in the two-dimensional Hubbard model, at strong coupling, TRILEX yields a $d$-wave superconducting dome as a function of doping. Contrary to the standard cluster dynamical mean field theory (DMFT) approaches, TRILEX can capture $d$-wave pairing using only a single-site effective impurity model. We also systematically explore the dependence of the superconducting temperature on the bare dispersion at weak coupling, which shows a clear link between strong antiferromagnetic (AF) correlations and the onset of superconductivity. We identify a combination of hopping amplitudes particularly favorable to superconductivity at intermediate doping. Finally, we study within $GW$+EDMFT the low-temperature $d$-wave superconducting phase at strong coupling in a region of parameter space with reduced AF fluctuations.