Physical review. B, Condensed matter

Bounds on ferromagnetism atT>0 in the hard-core electron lattice gas

András Sütö

Published Apr 1, 1991• 17 citations

Abstract

In a recent paper by Aizenman and Lieb [Phys. Rev. Lett. 65, 1470 (1990)], it was reported that the Nagaoka ferromagnet preserves its ferromagnetic character at temperatures T>0 in any finite volume. It is shown here that at any T>0 the Nagaoka ferromagnet becomes a perfect paramagnet in the thermodynamic limit. By using an inequality for the free energy, this result is extended to the general case when the number of electrons per site goes to 1. At any T>0 the spontaneous magnetization m(T,\ensuremath{\rho}) tends to zero as the electron density \ensuremath{\rho} tends to 1. Thus, a rigorous upper bound on m(T,\ensuremath{\rho}) is obtained. The results are independent of the statistics.

DOI: 10.1103/physrevb.43.8779

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Physical review. B, Condensed matter

Bounds on ferromagnetism atT>0 in the hard-core electron lattice gas

András Sütö

Published Apr 1, 1991• 17 citations

Publisher

Abstract

DOI: 10.1103/physrevb.43.8779

Related papers

Upper bound for spontaneous magnetization in the Hubbard model

Gang Su

A rigorous upper bound for the spontaneous magnetization per site of a general Hubbard model is obtained. The bound is independent of the on-site Coulomb potential ${\mathit{U}}_{\mathbf{r}}$ of the model, and is valid for arbitrary band fillings. From it one may find that the magnetic (ferromagnetic or antiferromagnetic) long-range order disappears in the atomic limit at finite temperatures.A rigorous upper bound for the spontaneous magnetization per site of a general Hubbard model is obtained. The bound is independent of the on-site Coulomb potential ${\mathit{U}}_{\mathbf{r}}$ of the model, and is valid for arbitrary band fillings. From it one may find that the magnetic (ferromagnetic or antiferromagnetic) long-range order disappears in the atomic limit at finite temperatures.

Physical review. B, Condensed matterMay 1, 19943 citations

Divergence of magnetic susceptibility in the SU( N ) Nagaoka-Thouless ferromagnet

Rajiv R. P. Singh, J. Oitmaa

Using finite-temperature strong-coupling expansions for the SU( $N$ ) Hubbard model, we calculate the thermodynamic properties of the model in the infinite- $U$ limit for arbitrary density $0\ensuremath{\le}\ensuremath{\rho}\ensuremath{\le}1$ and all $N$ . We express the ferromagnetic susceptibility of the model as a Curie term plus $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$, an excess susceptibility above the Curie behavior. We show that, on a bipartite lattice, graph by graph the contributions to $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$ are non-negative in the limit that the hole density $\ensuremath{\delta}=1\ensuremath{-}\ensuremath{\rho}$ goes to zero. By summing the contributions from all graphs consisting of closed loops we find that the low-hole-density ferromagnetic susceptibility diverges exponentially as $exp\mathrm{\ensuremath{\Delta}}/T$ as $T\ensuremath{\rightarrow}0$ in two and higher dimensions. This demonstrates that the Nagaoka-Thouless ferromagnetic state exists as a thermodynamic state of matter at low enough density of holes and sufficiently low temperatures. The constant $\mathrm{\ensuremath{\Delta}}$ scales with the SU( $N$ ) parameter $N$ as $1/N$ implying that ferromagnetism is gradually weakened with increasing $N$ as the characteristic temperature scale for ferromagnetic order goes down.Using finite-temperature strong-coupling expansions for the SU($N$) Hubbard model, we calculate the thermodynamic properties of the model in the infinite-$U$ limit for arbitrary density $0\ensuremath{\le}\ensuremath{\rho}\ensuremath{\le}1$ and all $N$. We express the ferromagnetic susceptibility of the model as a Curie term plus $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$, an excess susceptibility above the Curie behavior. We show that, on a bipartite lattice, graph by graph the contributions to $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$ are non-negative in the limit that the hole density $\ensuremath{\delta}=1\ensuremath{-}\ensuremath{\rho}$ goes to zero. By summing the contributions from all graphs consisting of closed loops we find that the low-hole-density ferromagnetic susceptibility diverges exponentially as $exp\mathrm{\ensuremath{\Delta}}/T$ as $T\ensuremath{\rightarrow}0$ in two and higher dimensions. This demonstrates that the Nagaoka-Thouless ferromagnetic state exists as a thermodynamic state of matter at low enough density of holes and sufficiently low temperatures. The constant $\mathrm{\ensuremath{\Delta}}$ scales with the SU($N$) parameter $N$ as $1/N$ implying that ferromagnetism is gradually weakened with increasing $N$ as the characteristic temperature scale for ferromagnetic order goes down.

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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 massTo 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

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A finite temperature version of the Nagaoka--Thouless theorem in the $\mathrm{SU}(n)$ Hubbard modelA finite temperature version of the Nagaoka--Thouless theorem in the $\mathrm{SU}(n)$ Hubbard model

Tadahiro Miyao

The Aizenman--Lieb theorem for the $\mathrm{SU}(2)$ Hubbard model expands upon the Nagaoka--Thouless theorem for the ground state to encompass finite temperatures. It can be succinctly stated that the magnetization $m(β, b)$ of the system in the presence of a field $b$ surpasses the pure paramagnetic value $m_0(β, b)=\tanh(βb)$ . In this manuscript, we present an extension of the Aizenman--Lieb theorem to the $\mathrm{SU}(n)$ Hubbard model. Our proof relies on a random-loop representation of the partition function, which becomes accessible when expressing the partition function in terms of path integrals.The Aizenman--Lieb theorem for the $\mathrm{SU}(2)$ Hubbard model expands upon the Nagaoka--Thouless theorem for the ground state to encompass finite temperatures. It can be succinctly stated that the magnetization $m(β, b)$ of the system in the presence of a field $b$ surpasses the pure paramagnetic value $m_0(β, b)=\tanh(βb)$. In this manuscript, we present an extension of the Aizenman--Lieb theorem to the $\mathrm{SU}(n)$ Hubbard model. Our proof relies on a random-loop representation of the partition function, which becomes accessible when expressing the partition function in terms of path integrals.

arXiv:2112.14475Dec 29, 20210 citations

A rigorous upper bound for the spontaneous magnetization per site of a general Hubbard model is obtained. The bound is independent of the on-site Coulomb potential

{\mathit{U}}_{\mathbf{r}}

of the model, and is valid for arbitrary band fillings. From it one may find that the magnetic (ferromagnetic or antiferromagnetic) long-range order disappears in the atomic limit at finite temperatures.

Using finite-temperature strong-coupling expansions for the SU(

N

) Hubbard model, we calculate the thermodynamic properties of the model in the infinite-

U

limit for arbitrary density $0\ensuremath{\le}\ensuremath{\rho}\ensuremath{\le}1$ and all

N

. We express the ferromagnetic susceptibility of the model as a Curie term plus $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$, an excess susceptibility above the Curie behavior. We show that, on a bipartite lattice, graph by graph the contributions to $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$ are non-negative in the limit that the hole density $\ensuremath{\delta}=1\ensuremath{-}\ensuremath{\rho}$ goes to zero. By summing the contributions from all graphs consisting of closed loops we find that the low-hole-density ferromagnetic susceptibility diverges exponentially as $exp\mathrm{\ensuremath{\Delta}}/T$ as $T\ensuremath{\rightarrow}0$ in two and higher dimensions. This demonstrates that the Nagaoka-Thouless ferromagnetic state exists as a thermodynamic state of matter at low enough density of holes and sufficiently low temperatures. The constant $\mathrm{\ensuremath{\Delta}}$ scales with the SU(

N

) parameter

N

1/N

implying that ferromagnetism is gradually weakened with increasing

N

as the characteristic temperature scale for ferromagnetic order goes down.

{\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).

A finite temperature version of the Nagaoka--Thouless theorem in the

\mathrm{SU}(n)

Hubbard model

The Aizenman--Lieb theorem for the

\mathrm{SU}(2)

Hubbard model expands upon the Nagaoka--Thouless theorem for the ground state to encompass finite temperatures. It can be succinctly stated that the magnetization

m(β, b)

of the system in the presence of a field

b

surpasses the pure paramagnetic value

m_0(β, b)=\tanh(βb)

. In this manuscript, we present an extension of the Aizenman--Lieb theorem to the

\mathrm{SU}(n)

Hubbard model. Our proof relies on a random-loop representation of the partition function, which becomes accessible when expressing the partition function in terms of path integrals.