  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 onsite Coulomb repulsion and
a single hole in a half-filled band, does not arise naturally in materials, and
was only realized recently for the first time in experiments on a 2 \times 2
array of gated quantum dots. Atomically precise fabrication of dopant arrays in
Si allows for engineering highly controllable systems with complex geometries.
This makes dopant arrays a good candidate to study NF in different array
geometries through analog quantum simulation. Here we present theoretical
simulations done for $3 \times 3$ arrays and larger $N \times N $ arrays of
quantum dots, and predict the emergence of different forms of ferromagnetism in
different geometries. We find NF in perfect $3 \times 3$ arrays, as well as in
$N \times N $ arrays for one hole less than half-filling. The ratio of the
Hubbard on-site repulsion U to hopping t that defines the onset of NF increases
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 U/t for the onset of ferromagnetism decreases as the loop length
increases. We show that this different dependence on size for loops and $N
\times N $ arrays can be understood by a scaling argument. 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 the quantum magnetism of
small systems.
