Nagaoka Ferromagnetism in $3 \times 3$ Arrays and Beyond

Abstract

Nagaoka ferromagnetism (NF) is a long-predicted example of itinerant ferromagnetism (IF) in the Hubbard model that has been studied theoretically for many years. The condition for NF, an infinite on-site Coulomb repulsion and a single hole in a half-filled band, does not arise naturally in materials. NF was only realized recently for the first time in experiments on a $2\times 2$ array of gated quantum dots. Dopant arrays and gated quantum dots in Si allow for engineering controllable systems with complex geometries. This makes quantum dot arrays good candidates to study NF in different geometries through analog quantum simulation. Here we present theoretical simulations done for $3\times 3$ arrays and larger $N\times N$ arrays, and predict the emergence of different forms of ferromagnetism in different geometries. We find NF in perfect $3\times 3$ arrays, and in $N\times N$ arrays for one hole doping of a half-filled band. The ratio of the hopping $t$ to Hubbard on-site repulsion $U$ that defines the onset of NF scales as $1/N^{4}$ as $N$ increases, approaching the bulk limit of infinite $U$ for large $N$. Additional simulations are done for geometries made by removing sites from $N\times N$ arrays. Different forms of ferromagnetism are found for different geometries. Loops show ferromagnetism, but only for three electrons. For loops, the critical $t/U$ for the onset of ferromagnetism scales as $N$ as the loop length increases. We show that the different dependences on size for loops and $N\times N$ arrays can be understood by scaling arguments that highlight the different energy contributions to each form of ferromagnetism. Our results show how analog quantum simulation with small arrays can elucidate the role of effects including wavefunction connectivity; system geometry, size and symmetry; bulk and edge sites; and kinetic energy in determining quantum magnetism of small systems.