# SO(4) FLEX+DMFT formalism with SU(2)$\otimes$SU(2)-symmetric impurity solver for superconductivity in the repulsive Hubbard model

## TL;DR

This paper introduces a novel SO(4) symmetric FLEX+DMFT formalism with an SU(2)$_{
m spin} imes$SU(2)$_{
m eta spin}$ impurity solver, enabling simultaneous investigation of superconductivity and antiferromagnetism in the Hubbard model.

## Contribution

The paper develops a full-SU(2) slave-boson formalism integrated into FLEX+DMFT, allowing for a unified treatment of competing quantum orders.

## Key findings

- Incorporates SO(4) symmetry into FLEX+DMFT framework.
- Uses an SU(2)$_{
m spin} imes$SU(2)$_{
m eta spin}$ impurity solver.
- Enables study of coexistence and competition of quantum orders.

## Abstract

Here we have developed a FLEX+DMFT formalism, where the symmetry properties of the system are incorporated by constructing a SO(4) generalization of the conventional fluctuation-exchange approximation (FLEX) coupled self-consistently to the dynamical mean-field theory (DMFT). Along with this line, we emphasize that the SO(4) symmetry is the lowest group-symmetry that enables us to investigate superconductivity and antiferromagnetism on an equal footing. We have imposed this by decomposing the electron operator into auxiliary fermionic and slave-boson constituents that respect SU(2)$_{\rm spin}\otimes$SU(2)$_{\eta{\rm spin}}$. This is used not in a mean-field treatment as in the usual slave-boson formalisms, but instead in the DMFT impurity solver with an SU(2)$_{\rm spin}\otimes$SU(2)$_{\eta{\rm spin}}$ hybridization function to incorporate the FLEX-generated bath information into DMFT iterations. While there have been attempts such as the doublon-less SU(2) slave-boson formalism, the present "full-SU(2)" slave-boson formalism is expected to provide a new platform for addressing the underlying physics for various quantum orders, which compete with each other and can coexist.