Certain exact many-body results for Hubbard model ground states testable in small quantum dot arrays

We present several interesting phenomena related to flat-band ferromagnetism in the Hubbard model. The first is a mathematical theorem stating certain conditions under which a flat-band ferromagnetic must necessarily be degenerate with a nonferromagnetic state. This theorem is generally applicable and geometry independent but holds only for a small number of holes in an otherwise filled band. The second phenomenon is a peculiar example where the intuition fails that particles prefer to doubly occupy low-energy states before filling higher-energy states. Lastly, we show a pattern of ferromagnetism which appears in small pentagonal and hexagonal plaquettes at filling factors of roughly 3/10 and 1/4. These examples require only a small number of lattice sites and may be observable in quantum dot arrays currently available as laboratory spin qubit arrays.