Optimal array geometries for kinetic magnetism and Nagaoka polarons

Quantum dot (QD) platforms have enabled the direct observation of Nagaoka ferromagnetism (NFM) in small arrays and non-infinite interaction strength. However, optimizing the cluster connectivity characteristics that yield a ground state with maximal spin and their robustness against magnetic fields remains unexplored. Employing exact diagonalization of the Hubbard Hamiltonian, we find a connection between the existence of kinetic ferromagnetism and graph theory descriptions. Algebraic connectivity ($\lambda_2$) and Katz centrality (KC) are shown to be related to the spin-correlation over the system. In square arrays, the onset of NFM is found to be $t_c/U\simeq \lambda_2^2$. In optimal cluster geometries, large $\lambda_2$ and low KC fluctuation per site are found to enhance $t_c/U$, extending the NFM phase while diminishing the strength of spin correlation clouds. A perpendicular magnetic field introduces Aharonov-Bohm phases, and a critical flux for which NFM is destroyed. We further find that tuning the flux phase to $\pi$ results in a ground state that exhibits antiferromagnetic correlations (counter-Nagaoka state). Our results illustrate how NFM and polaron formation can be predicted from the array's connectivity ($\lambda_2$ and KC), and how the introduction of flux results in the counterintuitive destruction of kinetic ferromagnetism in the system.