Optimal array geometries for kinetic magnetism and Nagaoka polarons

TL;DR

This paper investigates how the geometry and connectivity of quantum dot arrays influence Nagaoka ferromagnetism, revealing that specific graph properties predict magnetic phases and that magnetic flux can suppress or induce different magnetic states.

Contribution

It establishes a connection between graph theory metrics and magnetic phases in quantum dot arrays, providing guidelines for optimizing array geometries for kinetic ferromagnetism.

Findings

Large algebraic connectivity enhances NFM stability.

Optimal geometries extend the NFM phase.

Magnetic flux can destroy or induce magnetic states.

Abstract

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.