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arXiv:1702.04442 · Physical review. B./Physical review. B

Balance and frustration in strongly correlated itinerant electron systems: An extension of Nagaoka's theorem

Benjamin James Powell

Published Nov 27, 2017• 3 citations
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Abstract

We prove that Nagaoka's theorem, that the ground state of the large-$U$ Hubbard model with exactly one hole is ferromagnetic, holds for any balanced Hamiltonian. We argue that, in itinerant electron systems, a balanced Hamiltonian, rather than bipartite lattice, defines an unfrustrated system. The proof is valid for multiorbital models with arbitrary two-orbital interactions provided that no exchange interactions are antiferromagnetic: a class of models including the Kanomori Hamiltonian.

DOI
10.1103/physrevb.96.174435

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We prove that Nagaoka's theorem, that the ground state of the large-UUU Hubbard model with exactly one hole is ferromagnetic, holds for any balanced Hamiltonian. We argue that, in itinerant electron systems, a balanced Hamiltonian, rather than bipartite lattice, defines an unfrustrated system. The proof is valid for multiorbital models with arbitrary two-orbital interactions provided that no exchange interactions are antiferromagnetic: a class of models including the Kanomori Hamiltonian.
We apply a result from graph theory to prove exact results about itinerant ferromagnetism. Nagaoka's theorem is extended to all nonseparable 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 $U\ensuremath{\rightarrow}\ensuremath{\infty}$ Hubbard model with one hole away from the half filling on a two-dimensional honeycomb lattice or a three-dimensional diamond lattice is fully spin polarized. Furthermore, the condition of connectivity for NNN-component fermions is presented, and Nagaoka's theorem is also generalized to SU(N)\text{SU}(N)SU(N)-symmetric fermion systems on nonseparable graphs.
In the present paper, we investigate the existence of ferromagnetism in a two-band Hubbard model, by applying a recently-introduced method by us to study ferromagnetism in metallic clusters. We prove rigorously that the ground state of this model is ferromagnetic if the intra-orbital Coulomb repulsion between electrons is infinitely strong and only one hole exists in the system. Our theorem improves a previous result. Furthermore, our method can also be applied to deal with the case of multiple holes.
We consider the Hubbard model on a finite set of sites with nonpositive hopping matrix elements and infinitely strong on-site repulsion. Nagaoka's theorem states that in this model the relative ground state in the sector with one unoccupied site is maximally ferromagnetic. We show that this phenomenon is a consequence of a combinatorial coincidence valid in the one-hole regime only. In the case of more than one hole there is no reason to expect maximally ferromagnetic ground states. We prove this claim for the case of two holes for models defined on a class of graphs which contains all tori that are not too small.
We consider the Hubbard model on a finite set of sites with nonpositive hopping matrix elements and infinitely strong on-site repulsion. Nagaoka's theorem states that in this model the relative ground state in the sector with one unoccupied site is maximally ferromagnetic. We show that this phenomenon is a consequence of a combinatorial coincidence valid in the one-hole regime only. In the case of more than one hole there is no reason to expect maximally ferromagnetic ground states. We prove this claim for the case of two holes for models defined on a class of graphs which contains all tori that are not too small.
We study itinerant ferromagnetism in multiorbital Hubbard models in certain two-dimensional square and three-dimensional cubic lattices. In the strong coupling limit where doubly occupied orbitals are not allowed, we prove that the fully spin-polarized states are the unique ground states, apart from the trivial spin degeneracies, for a large region of filling factors. Possible applications to ppp-orbital bands with ultracold fermions in optical lattices, and electronic 3d3d3d-orbital bands in transition-metal oxides, are discussed.