# A mean-field approach to Kondo-attractive-Hubbard model

## TL;DR

This paper uses a mean-field approach to study a model combining Kondo interactions and attractive Hubbard coupling, revealing complex phase diagrams with coexistence and competition between magnetic order and superconductivity.

## Contribution

It introduces a semi-classical Hartree-Fock framework to analyze the coexistence of magnetism and superconductivity in a Kondo-attractive-Hubbard model on a cubic lattice.

## Key findings

- Coexistence of superconductivity and spiral antiferromagnetic order at intermediate U.
- Thermal fluctuations suppress magnetic order before superconductivity as temperature increases.
- Reentrant behavior of phases induced by temperature changes.

## Abstract

With the purpose of investigating coexistence between magnetic order and superconductivity, we consider a model in which conduction electrons interact with each other, via an attractive Hubbard on-site coupling $U$, and with local moments on every site, via a Kondo-like coupling, $J$. The model is solved on a simple cubic lattice through a Hartree-Fock approximation, within a `semi-classical' framework which allows spiral magnetic modes to be stabilized. For a fixed electronic density, $n_c$, the small $J$ region of the ground state ($T=0$) phase diagram displays spiral antiferromagnetic (SAFM) states for small $U$. Upon increasing $U$, a state with coexistence between superconductivity (SC) and SAFM sets in; further increase in $U$ turns the spiral mode into a N\'eel antiferromagnet. The large $J$ region is a (singlet) Kondo phase. At finite temperatures, and in the region of coexistence, thermal fluctuations suppress the different ordered phases in succession: the SAFM phase at lower temperatures and SC at higher temperatures; also, reentrant behaviour is found to be induced by temperature. Our results provide a qualitative description of the competition between local moment magnetism and superconductivity in the borocarbides family.