Exact Results on Itinerant Ferromagnetism and the 15-puzzle Problem

TL;DR

This paper uses graph theory to extend Nagaoka's theorem, proving that certain lattice models with one hole exhibit fully spin-polarized ground states, linking ferromagnetism to the connectivity of tile configurations.

Contribution

It generalizes Nagaoka's ferromagnetism theorem to all non-separable graphs and $SU(N)$ systems using the 15-puzzle problem, providing exact results for complex lattice structures.

Findings

Ground state is fully spin-polarized on 2D honeycomb and 3D diamond lattices.

Extension of Nagaoka's theorem to $SU(N)$-symmetric fermion systems.

Connectivity conditions determine ferromagnetic ground states.

Abstract

We apply a result from graph theory to prove exact results about itinerant ferromagnetism. Nagaoka's theorem of ferromagnetism is extended to all non-separable graphs except single polygons with more than four vertices by applying the solution to the generalized 15-puzzle problem, which studies whether the hole's motion can connect all possible tile configurations. This proves that the ground state of a $U\to\infty$ Hubbard model with one hole away from the half filling on a 2D honeycomb lattice or a 3D diamond lattice is fully spin-polarized. Furthermore, the condition of connectivity for $N$-component fermions is presented, and Nagaoka's theorem is also generalized to $SU(N)$-symmetric fermion systems on non-separable graphs.