Competing Kondo Singlet and Magnetic Order Insulator in Graphene Lattices: A Variational Cluster Approximation Approach

Strongly correlated electron systems, where localized magnetic moments interact with conduction electrons, continue to challenge our understanding of quantum phases. In particular, the competition between the Kondo effect-which promotes the formation of singlet states via the screening of localized spins-and magnetic ordering driven by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, plays a crucial role in defining the electronic properties of materials such as graphene and other honeycomb lattice systems. In this work, we investigate the interplay between these competing mechanisms using a Kondo-Hubbard model on the hexagonal lattice. Our model incorporates key interactions including the Kondo coupling J⊥ between conduction electrons and localized spins, the Heisenberg exchange JH between localized moments, the onsite Coulomb repulsion U for conduction electrons, and a second nearest-neighbor hopping term t′. The study is conducted at half-filling, where each lattice site hosts one electron on average, and the system is analyzed via the variational cluster approximation (VCA) combined with an exact diagonalization solver at zero temperature. Our analysis focuses on mapping the phase diagrams in different parameter spaces, particularly the (JH, J⊥) and (J⊥, UJ⊥) planes. We find that the antiferromagnetic phase is favored at smaller J⊥ and larger JH, while an increase in J⊥ stabilizes the Kondo singlet phase. The transition between these phases occurs smoothly, indicating a second-order phase transition. Additionally, the inclusion of the hopping term t′ is shown to enhance the stability of the Kondo singlet phase. Overall, our results provide new insights into the delicate balance between magnetic order and Kondo singlet formation in low-dimensional correlated systems, potentially guiding future experimental and theoretical investigations in graphene-based materials and related compounds.