© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Magnetic Properties of Some Itineraat-Electron Systems at T > 0

Michael Aizenman, Élliott H. Lieb

Published Jan 1, 2004• 0 citations

Abstract

To many physicists the study of magnetism begins with a Heisenberg model, or something similar, in which localized spins interact witb eacb other. Unfortunately, nature docs not present us witb sucb a starting point. In stead, at a reasonably (thougb not absolutely) funda mental level, we are presented witb the Scbriidinger equation for electrons wbose interaction is, to a very good approximation, spin independent. It is left to the Pauli principle to induce a spin dependence, and for tbis it is essential tbat the electrons are itinerant, i.e., rwt lo calized a priori. Given the subtlety of the phenomenon, it is crucial to elucidate the conditions which encourage or discourage the emergence of ferromagnetism. Some years ago Lieb and Mattis 1 showed that one never gets ferromagnetism in the ground states of itinerant electrons on a line. The results presented here include an extension of that earlier theorem (by a different method) to positive tempera tures, and an opposite statement for systems in any di mension with a certain parity t:onstraint (requiring hard -core repulsions)- for which we show that parallel spin alignments are favored at all temperatures. Exam ples of the second kind are one-dimensional systems of odd numbers of electrons on a ring (j.e., periodit: bound ary conditionS>, for which we extend the T-O result of Herring, 2 and a particular case of the Hubbard model whose ground state was shown to be ferromagnetic by Nagaoka J and Thouless. 45 We deal with the following general (spin-independent) Hamiltonian for N identical, itinerant particles of mass

Related papers

Magnetic properties of some itinerant-electron systems atT>0

Michael Aizenman, Élliott H. Lieb

The Lieb-Mattis theorem on the absence of one-dimensional ferromagnetism is extended here from ground states to T>0 by proving, inter alia, that M(\ensuremath{\beta},h), the magnetization of a quantum system in a field h>0, is always less than the pure paramagnetic value ${\mathit{M}}_{0}$(\ensuremath{\beta},h)=tanh(\ensuremath{\beta}h), with \ensuremath{\beta}==1/kT. Our proof rests on a new formulation in terms of path integrals that holds in any dimension; another of its applications is that the Nagaoka-Thouless theorem on the Hubbard model also extends to T>0 in the sense that M(\ensuremath{\beta},h) exceeds ${\mathit{M}}_{0}$(\ensuremath{\beta},h).

Physical Review Letters•Sep 17, 1990•44 citations

Two theorems on the Hubbard model

Élliott H. Lieb

In the attractive Hubbard Model (and some extended versions of it), the ground state is proved to have spin angular momentum S=0 for every (even) electron filling. In the repulsive case, and with a bipartite lattice and a half-filled band, the ground state has S=(1/2\ensuremath{\parallel}B\ensuremath{\Vert}-\ensuremath{\Vert}A\ensuremath{\Vert}\ensuremath{\Vert}, where \ensuremath{\Vert}B\ensuremath{\Vert} (\ensuremath{\Vert}A\ensuremath{\Vert}) is the number of sites in the B (A) sublattice. In both cases the ground state is unique. The second theorem confirms an old, unproved conjecture in the \ensuremath{\Vert}B\ensuremath{\Vert}=\ensuremath{\Vert}A\ensuremath{\Vert} case and yields, with \ensuremath{\Vert}B\ensuremath{\Vert}\ensuremath{\ne}\ensuremath{\Vert}A\ensuremath{\Vert}, the first provable example of itinerant-electron ferromagnetism. The theorems hold in all dimensions without even the necessity of a periodic lattice structure.

Physical Review Letters•Mar 6, 1989•2,222 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

Magnetic Properties of Some Itinerant-Electron Systems at T > 0

Michael Aizenman, Élliott H. Lieb

Springer eBooks•Jan 1, 2004•0 citations

Ground-state spin structure of strongly interacting disordered 1D Hubbard model

Eli Eisenberg, Richard Berkovits

Abstract. We study the influence of on-site disorder on the magnetic properties of the ground state of the infinite-U one-dimensional (1D) Hubbard model. We find that the ground state is not ferromagnetic. This is analysed in terms of the algebraic structure of the spin dependence of the Hamiltonian. A simple explanation is derived for the 1=N periodicity in the persistent current for this model. The Hubbard model is the simplest model one can study to examine the effects of correlations between electrons in narrow energy bands. The Hamiltonian consists of a nearest neighbour hopping term and an electron–electron repulsion, U, which acts only when two electrons are on the same site. The Hubbard model is also the canonical model for the study of itinerant ferromagnetism. The strong coupling regime is of special importance for the study of ferromagnetism, since a theorem by Nagaoka [1] states that in theU D 1 limit, the ground state (GS) is ferromagnetic given some connectivity property of the lattice (which holds in most cases for d> 1). The model is solvable in one dimension and it was shown [2] that for open boundary conditions (BCs) the GS for finiteU is a singlet, i.e. there can be no ferromagnetism unless one postulates explicitly spin- or velocity-dependent forces. For infinite U, the GS of

•Jan 1, 1999•4 citations

Magnetic Properties of Some Itinerant-Electron Systems at T > 0

Élliott H. Lieb, Michael Aizenman

Springer eBooks•Jan 1, 2002•0 citations

To many physicists the study of magnetism begins with a Heisenberg model, or something similar, in which localized spins interact witb eacb other. Unfortunately, nature docs not present us witb sucb a starting point. In stead, at a reasonably (thougb not absolutely) funda mental level, we are presented witb the Scbriidinger equation for electrons wbose interaction is, to a very good approximation, spin independent. It is left to the Pauli principle to induce a spin dependence, and for tbis it is essential tbat the electrons are itinerant, i.e., rwt lo calized a priori. Given the subtlety of the phenomenon, it is crucial to elucidate the conditions which encourage or discourage the emergence of ferromagnetism. Some years ago Lieb and Mattis 1 showed that one never gets ferromagnetism in the ground states of itinerant electrons on a line. The results presented here include an extension of that earlier theorem (by a different method) to positive tempera tures, and an opposite statement for systems in any di mension with a certain parity t:onstraint (requiring hard -core repulsions)- for which we show that parallel spin alignments are favored at all temperatures. Exam ples of the second kind are one-dimensional systems of odd numbers of electrons on a ring (j.e., periodit: bound ary conditionS>, for which we extend the T-O result of Herring, 2 and a particular case of the Hubbard model whose ground state was shown to be ferromagnetic by Nagaoka J and Thouless. 45 We deal with the following general (spin-independent) Hamiltonian for N identical, itinerant particles of mass

The Lieb-Mattis theorem on the absence of one-dimensional ferromagnetism is extended here from ground states to T>0 by proving, inter alia, that M(\ensuremath{\beta},h), the magnetization of a quantum system in a field h>0, is always less than the pure paramagnetic value

{\mathit{M}}_{0}

(\ensuremath{\beta},h)=tanh(\ensuremath{\beta}h), with \ensuremath{\beta}==1/kT. Our proof rests on a new formulation in terms of path integrals that holds in any dimension; another of its applications is that the Nagaoka-Thouless theorem on the Hubbard model also extends to T>0 in the sense that M(\ensuremath{\beta},h) exceeds

{\mathit{M}}_{0}

(\ensuremath{\beta},h).

In the attractive Hubbard Model (and some extended versions of it), the ground state is proved to have spin angular momentum S=0 for every (even) electron filling. In the repulsive case, and with a bipartite lattice and a half-filled band, the ground state has S=(1/2\ensuremath{\parallel}B\ensuremath{\Vert}-\ensuremath{\Vert}A\ensuremath{\Vert}\ensuremath{\Vert}, where \ensuremath{\Vert}B\ensuremath{\Vert} (\ensuremath{\Vert}A\ensuremath{\Vert}) is the number of sites in the B (A) sublattice. In both cases the ground state is unique. The second theorem confirms an old, unproved conjecture in the \ensuremath{\Vert}B\ensuremath{\Vert}=\ensuremath{\Vert}A\ensuremath{\Vert} case and yields, with \ensuremath{\Vert}B\ensuremath{\Vert}\ensuremath{\ne}\ensuremath{\Vert}A\ensuremath{\Vert}, the first provable example of itinerant-electron ferromagnetism. The theorems hold in all dimensions without even the necessity of a periodic lattice structure.

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.

Abstract. We study the influence of on-site disorder on the magnetic properties of the ground state of the infinite-U one-dimensional (1D) Hubbard model. We find that the ground state is not ferromagnetic. This is analysed in terms of the algebraic structure of the spin dependence of the Hamiltonian. A simple explanation is derived for the 1=N periodicity in the persistent current for this model. The Hubbard model is the simplest model one can study to examine the effects of correlations between electrons in narrow energy bands. The Hamiltonian consists of a nearest neighbour hopping term and an electron–electron repulsion, U, which acts only when two electrons are on the same site. The Hubbard model is also the canonical model for the study of itinerant ferromagnetism. The strong coupling regime is of special importance for the study of ferromagnetism, since a theorem by Nagaoka [1] states that in theU D 1 limit, the ground state (GS) is ferromagnetic given some connectivity property of the lattice (which holds in most cases for d> 1). The model is solvable in one dimension and it was shown [2] that for open boundary conditions (BCs) the GS for finiteU is a singlet, i.e. there can be no ferromagnetism unless one postulates explicitly spin- or velocity-dependent forces. For infinite U, the GS of