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Journal of Physics A Mathematical and General

Ferromagnetic ground states for the Hubbard model on line graphs

Andreas Mielke

Published Jan 21, 1991• 458 citations
Publisher

Abstract

The author discusses some of the properties of the Hubbard model on a line graph with n vertices. It is shown that the model has ferromagnetic ground states if the interaction is repulsive (U)0) and if the number of electrons N satisfies 2n>or=N>or=M. M is a natural number that depends on the line graph. For example, the Kagome lattice is a line graph, it has M=5n/3-1.

DOI
10.1088/0305-4470/24/2/005

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The author discusses some of the properties of the Hubbard model on a line graph with n vertices. It is shown that the model has ferromagnetic ground states if the interaction is repulsive (U)0) and if the number of electrons N satisfies 2n>or=N>or=M. M is a natural number that depends on the line graph. For example, the Kagome lattice is a line graph, it has M=5n/3-1.
Let L(G) be the line graph of a graph G=(V,E). The Hubbard model on L(G) has ferromagnetic ground states with a saturated spin if the interaction is repulsive (U>0) and if the number of electrons N satisfies N>or=M. M= mod E mod + mod V mod -1 if G is bipartite and M= mod E mod + mod V mod otherwise. The author shows that the ferromagnetic ground state is unique if N=M. Further he gives a sufficient condition for the existence of other ground states if N>M. The results are valid also for a multi-band Hubbard model on a bipartite graph. In the case of a periodic lattice, the results are related to the existence of a flat energy band.
It is well known that the Hubbard model on a line graph has a flat band and ferromagnetic ground states in a certain density range. We show that for a Hubbard model on a line graph of a planar bipartite graph the ferromagnetic ground state is stable if one adds a special contribution to the kinetic energy which lifts the degeneracy of the lowest single particle state. Stability holds for sufficiently strong repulsion U. The model has extended single particle eigenstates, no degeneracy, and no band gap. It is therefore a good candidate for metallic ferromagnetism.
The Hubbard model on line graphs has a flat band and highly degenerate ferromagnetic ground states. Here, we study the Hubbard model on a line graph of a planar bipartite graph by adding a special contribution to the kinetic energy which lifts the degeneracy of the lowest single particle state to a general Hamiltonian. We prove that, at half-filling of the lowest band and for sufficiently strong repulsion U , the ground states of this Hamiltonian remain saturated ferromagnetic for a class of line graphs of planar bipartite graphs.
We consider the one-orbital N -site repulsive Hubbard model on two kagome like chains, both of which yield a completely dispersionless (flat) one-electron band.Using exact many-electron ground states in the subspaces with n ≤ nmax (nmax ∝ N ) electrons, we calculate the square of the total spin in the ground state to discuss magnetic properties of the models.We have found that although for n < nmax the ground states contain fully polarized states, there is no finite region of electron densities n/N < 1 (N = N/3 or N = N/5) where ground-state ferromagnetism survives for N → ∞.
In this work, the ground states of the Hubbard model on complete graph are studied, for a finite lattice size LLL and arbitrary on-site energy UUU. We construct explicitly the ground states of the system when the number of the electrons Ne≥L+1N_e \ge L+1Ne​≥L+1. In particular, for Ne=L+1N_e=L+1Ne​=L+1, the ground state is ferromagnetic with total spin sg=(Ne−2)/2s_g=(N_e-2)/2sg​=(Ne​−2)/2.
The author investigates the U= infinity Hubbard model on a large class of lattices which are line graphs. The most interesting lattices in this class are line graphs of regular bipartite lattices with N s sites and coordination number k>or=4. The ground state energy and some ground states are given. If the number of electrons N satisfies N s >or=N>or=2N s /k-2, the ground state energy is -4 mod t mod (N s -N). The ground states have no magnetic ordering, they are projections of the ground states at U=0 onto the subspace of states without doubly occupied sites.