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

Andreas Mielke

Published Jan 1, 2008• 14 citations
OpenAlex

Abstract

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

Related papers

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

Andreas Mielke

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.

arXiv:cond-mat/9910385•Journal of Physics A Mathematical and General•Nov 17, 1999•42 citations

Ferromagnetism in Single-Band Hubbard Models with a Partially Flat Band

Andreas Mielke

A Hubbard model with a single, partially flat band has ferromagnetic ground states. It is shown that local stability of ferromagnetism implies its global stability in such a model: The model has only ferromagnetic ground states if there are no single spin-flip ground states. Since a single-band Hubbard model away from half filling describes a metal, this result may open a route to metallic ferromagnetism in single-band Hubbard models.

arXiv:cond-mat/9904258•Physical Review Letters•May 24, 1999•75 citations

Ferromagnetism in the Hubbard Model-- Examples from Models with Degenerate Single-Electron Ground States

Andreas Mielke, Hal Tasaki

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 $N_{\rm d}$-fold degenerate ground states. The degeneracy $N_{\rm d}$ 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.

arXiv:cond-mat/9305026•Communications in Mathematical Physics•May 24, 1993•191 citations

Stability of Ferromagnetism in the Hubbard Model on the Kagome Lattice

Akinori Tanaka, Hiromitsu Ueda

The Hubbard model on the kagome lattice has highly degenerate ground states (the flat lowest band) in the corresponding single-electron problem and exhibits the so-called flat-band ferromagnetism in the many-electron ground states as was found by Mielke [J. Phys. A 24, L73 (1991)]]. Here we study the model obtained by adding extra hopping terms to the above model. The lowest single-electron band becomes dispersive, and there is no band gap between the lowest band and the other band. We prove that, at half filling of the lowest band, the ground states of this perturbed model remain saturated ferromagnetic if the lowest band is nearly flat.

arXiv:cond-mat/0209423•Physical Review Letters•Feb 14, 2003•69 citations

Low-Density Ferromagnetism in the Hubbard Model

Jin An, Chang-De Gong

We study analytically the stability of the saturated ferromagnetic ground state (the Nagaoka state) of the Hubbard model in the low electron density limit with the mean-field approach. We find that the critical U c is just 4π t , where t is the hopping integral between the nearest neighbours, i.e., when U is larger than 4π t , the paramagnetic state is unstable with respect to the ferromagnetic state, and vice versa. PACS: 75. 30. Kz, 71. 10. Fd

Chinese Physics Letters•Dec 13, 2000•2 citations

Ferromagnetism in the Hubbard models with degenerate single-electron ground states

Hal Tasaki

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.

Physical Review Letters•Sep 7, 1992•584 citations
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
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.
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.
The Hubbard model on the kagome lattice has highly degenerate ground states (the flat lowest band) in the corresponding single-electron problem and exhibits the so-called flat-band ferromagnetism in the many-electron ground states as was found by Mielke [J. Phys. A 24, L73 (1991)]]. Here we study the model obtained by adding extra hopping terms to the above model. The lowest single-electron band becomes dispersive, and there is no band gap between the lowest band and the other band. We prove that, at half filling of the lowest band, the ground states of this perturbed model remain saturated ferromagnetic if the lowest band is nearly flat.
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.