We provide numerical evidence of the Nagaoka's theorem in the
SU(N) Fermi-Hubbard model on various cluster geometries, such as the square, the honeycomb, and the triangular lattices. In particular, by diagonalizing several finite-size clusters, we show that for one hole away from filling
1/N, the itinerant ferromagnetism arises for
U (the positive on-site interaction) larger than
Uc (the value at the transition), which depends strongly on the coordination number
z and on
N, the number of degenerate orbitals, that we vary from
N=2 to 6 in our simulations. We prove that
Uc is a nondecreasing function of
N. In addition, we find that the lattice dependency is rooted in the kinetic energy of the hole. We find that large coordination numbers
z lower the value of
Uc. Complementary, we explore the effect of long-range hopping on the appearance of itinerant ferromagnetism, and we demonstrate that it acts as an increased coordination number, protecting the ferromagnetic phase at small
U. Finally, the effects of both the presence of some additional holes and the finite size of the clusters are briefly discussed.
We study the magnetic properties of itinerant quantum magnetic particles, described by a generalized Hubbard model with large spin
(S>1/2), which may be realized in optical lattices of laser-cooled atom systems. In fermion systems (half-integer spins), an extended form of Nagaoka ferromagnetism may be realized. However, as novel aspects of the large-spin cases, we found that the condition on the lattice connectivity is more stringent than in the case of
S=1/2 particles and that the system shows a peculiar degenerate structure of the ground state in which the ferromagnetic state is included. In contrast, it turns out that the ground state of itinerant bosonic systems (integer spins) has a degenerate structure similar to that of fermion system with
S>1/2 regardless of the shape, connectivity, or filling of the lattice, and that the state with the maximum total spin is always one of the ground states. Because the system consists of
2S+1 types of particles and we study a
SU(2S+1) invariant model, the degeneracy of the ground state is given by the multiplets of the fully symmetric Young tableau of
SU(2S+1) if the state with maximum total spin belongs to the ground state.
We present an extension of Nagaoka's theorem in the SU(
n) generalization of the infinite-
U Hubbard model. It is shown that, when there is exactly one hole, the fully polarized states analogous to the ferromagnetic states in the SU(2) Hubbard model are ground states. For a restricted class of models satisfying the connectivity condition, these fully polarized states are the unique ground states up to the trivial degeneracy due to the SU(
n) symmetry. We also give examples of lattices in which the connectivity condition can be verified explicitly. The examples include the triangular, Kagom\'e, and hypercubic lattices in $d\phantom{\rule{0.28em}{0ex}}(\ensuremath{\ge}\phantom{\rule{-0.16em}{0ex}}2)$ dimensions, among which the cases of
d=2 and 3 are experimentally realizable in ultracold atomic gases loaded into optical lattices.
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.
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
p-orbital bands with ultracold fermions in optical lattices, and electronic
3d-orbital bands in transition-metal oxides, are discussed.
We provide numerical evidence of the Nagaoka's theorem in the
SU(N) Fermi-Hubbard model on various cluster geometries, such as the square, the honeycomb and the triangular lattices. In particular, by diagonalizing several finite-size clusters, we show that for one hole away from filling
1/N, the itinerant ferromagnetism arises for
U (the positive on-site interaction) larger than
Uc (the value at the transition), which strongly depends on the coordination number
z and on
N, the number of degenerate orbitals, that we vary from
N=2 to
N=6 in our simulations. We prove that
Uc is a non decreasing function of
N. In addition, we find that the lattice dependency is rooted in the kinetic energy of the hole. We find that large coordination numbers
z lower the value of
Uc. Complementary, we explore the effect of long-range hopping on the appearance of itinerant ferromagnetism and demonstrate that it acts as an increased coordination number, protecting the ferromagnetic phase at small
U. Finally, both the effects of the presence of some additional holes and of the finite size of the clusters are briefly discussed.