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arXiv:cond-mat/9910385 · Journal of Physics A Mathematical and General

Stability of ferromagnetism in Hubbard models with degenerate single-particle ground states

Andreas Mielke

Published Nov 17, 1999• 42 citations
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Abstract

A Hubbard model with a N d -fold degenerate single-particle ground state has ferromagnetic ground states if the number of electrons is less than or equal to N d . It is shown rigorously that the local stability of ferromagnetism in such a model implies global stability: the model has only ferromagnetic ground states, if there are no single spin-flip ground states. If the number of electrons is equal to N d , it is well known that the ferromagnetic ground state is unique if and only if the single-particle density matrix is irreducible. We present a simplified proof for this result.

DOI
10.1088/0305-4470/32/48/304

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A Hubbard model with a N d -fold degenerate single-particle ground state has ferromagnetic ground states if the number of electrons is less than or equal to N d . It is shown rigorously that the local stability of ferromagnetism in such a model implies global stability: the model has only ferromagnetic ground states, if there are no single spin-flip ground states. If the number of electrons is equal to N d , it is well known that the ferromagnetic ground state is unique if and only if the single-particle density matrix is irreducible. We present a simplified proof for this result.
Accepted for publication in J. Phys. A A Hubbard model with a Nd-fold degenerate single-particle ground state has ferromagnetic ground states if the number of electrons is less or equal to Nd. It is shown rigorously that the local stability of ferromagnetism in such a model implies global stability: The model has only ferromagnetic ground states, if there are no single spin-flip ground states. If the number of electrons is equal to Nd, it is well known that the ferromagnetic ground state is unique if and only if the single-particle density matrix is irreducible. We present a simplified proof for this result. PACS-numbers: 75.10.Lp, 71.10.Fd 1
Whether spin-independent Coulomb interaction can be the origin of a realistic ferromagnetism in an itinerant electron system has been an open problem for a long time. Here we study a class of Hubbard models on decorated lattices, which have a special property that the corresponding single-electron Schrödinger equation has NdN_{\rm d}Nd​-fold degenerate ground states. The degeneracy NdN_{\rm d}Nd​ is proportional to the total number of sites $\absΛ$. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to $ρ_0=N_{\rm d}/(2\absΛ)$, and paramagnetism when the filling factor is sufficiently small. An important feature of the present work is that it provides examples of three dimensional itinerant electron systems which are proved to exhibit ferromagnetism in a finite range of the electron filling factor.
We present a Hubbard model with a special hopping matrix, which has highly degenerate single-electron energy eigenstates but has different properties than the flat-band Hubbard model. Our model has a paramagnetic ground state when the electron filling factor is less than or equal to quarter-filling. In the case of quarter-filling with one extra electron, the present model exhibits saturated ferromagnetism in the ground states for a sufficiently large value of on-site Coulomb interaction but does not for a small one.
We study a class of Hubbard models in which the corresponding single-electron ground states have bulk degeneracy. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to ${\mathrm{\ensuremath{\rho}}}_{0}$=\ensuremath{\Vert}V\ensuremath{\Vert}/2\ensuremath{\Vert}\ensuremath{\Lambda}\ensuremath{\Vert} (where \ensuremath{\Vert}V\ensuremath{\Vert} is the dimension of degeneracy and \ensuremath{\Vert}\ensuremath{\Lambda}\ensuremath{\Vert} is the number of sites), and exhibit paramagnetism when the filling factor is sufficiently small. This is the first time that a three-dimensional itinerant-electron system is proved to exhibit ferromagnetism in a finite range of the electron filling factor.
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.