Non-perturbative indirect exchange in spin-valley coupled 2D crystals

We study indirect exchange interactions between localized spins of magnetic impurities in spin-valley coupled systems described with the Kane-Mele model. Our model captures the main ingredients of the energy bands of 1H transition metal dichalcogenides (TMDs) monolayers, such as 1H-MoS$_2$ and 1H-NbSe$_2$. To obtain the effective interactions, we use the exact diagonalization of the Hamiltonian, avoiding momentum cut-offs. We start by comparing the standard perturbation expansion in terms of the Kondo exchange with the exact calculation of the interaction, treating the local spins classically. We find that perturbation theory works well even beyond the regime where the relevant figure of merit, the ratio between the exchange $J$ and the hopping $t$, is small. We verify that the effective indirect exchange Hamiltonian derived from perturbation theory also works in the non-perturbative regime. Additionally, we analyse the interplay between the symmetry of the different terms of the interaction (Heisenberg, Ising, and Dzyaloshinskii$-$Moriya (DM)), the Fermi-surface topology, and the crystallographic direction in which the impurities are placed. We show that the indirect exchange along the armchair direction is actually Heisenberg-like, due to the reflection symmetry of the crystal structure around this direction. Finally, we explore the exploitation of indirect exchange, combined with atomic manipulation, to engineer the Majumdar-Ghosh Model. Our results show that TMDs provide an extremely versatile platform to engineer indirect exchange interactions.