Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9304015 · Journal of Mathematical Physics

Uniform density theorem for the Hubbard model

Élliott H. Lieb, Michael Loss, Robert J. McCann

Published Mar 1, 1993• 35 citations
arXivHTMLPDF

Abstract

A general class of hopping models on a finite bipartite lattice is considered, including the Hubbard model and the Falicov–Kimball model. For the half-filled band, the single-particle density matrix ρ(x,y) in the ground state and in the canonical and grand canonical ensembles is shown to be constant on the diagonal x=y, and to vanish if x≠y and if x and y are on the same sublattice. For free electron hopping models, it is shown in addition that there are no correlations between sites of the same sublattice in any higher order density matrix. Physical implications are discussed.

DOI
10.1063/1.530199

Related papers

The Falicov-Kimball Model with Long--Range Hopping Matrices

Oscar Bolina, Domingos H. U. Marchetti

The ground state nature of the Falicov-Kimball model with unconstrained hopping of electrons is investigated. We solve the eigenvalue problem in a pedagogical manner and give a complete account of the ground state energy both as a function of the number of electrons and nuclei and as a function of the total number of particles for any value of interaction U. We also study the energy gap and show the existence of a phase transition characterized by the absence of gap at the half--filled band for U<0. The model in consideration was proposed and solved by Farkasovsky for finite lattices and repulsive on-site interaction U>0. Contrary to his proposal we conveniently scale the hopping matrix to guarantee the existence of the thermodynamic limit. We also solve this model with bipartite unconstrained hopping matrices in order to compare with the Kennedy--Lieb variational analysis.

arXiv:cond-mat/9807232•Jul 15, 1998•1 citation

Thermodynamics of the one-dimensional half-filled-band Falicov-Kimball model

Cláudio Andrade Macedo, L. G. Azevedo, André M. C. Souza

In order to contribute to understand well the finite-temperature properties of the Falicov-Kimball model, we study the temperature dependence of the specific heat, internal energy, entropy, and mean square fluctuation of the difference between the ion and electron numbers, and of the correlation functions of the one-dimensional half-filled-band Falicov-Kimball model. For this purpose we used the method of small-cluster exact-diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain. Our results show that the examined Falicov-Kimball model exhibits behavior of alternate electron-ion short-range ordering at low temperature, and gradual metal-insulator transition at high temperature, but, does not exhibit the crossing point of the specific heat curves, characteristic of the Hubbard model.

Physical review. B, Condensed matter•Oct 25, 2001•20 citations

Density-matrix functional study of the Hubbard model on one- and two-dimensional bipartite lattices

Matthieu Saubanère, G. M. Pastor

The inhomogeneous Hubbard model is investigated in one and two dimensions by using lattice density-functional theory (LDFT). The ground-state energy $E=T+W$ is regarded as a functional of the single-particle density matrix $\ensuremath{\gamma}$, where $T[\ensuremath{\gamma}]$ represents the kinetic and crystal-field energy, and $W[\ensuremath{\gamma}]$ the interaction energy. Besides the known functional $T[\ensuremath{\gamma}]$ we propose a simple scaling approximation to the interaction energy $W[\ensuremath{\gamma}]$, which is based on exact results for the Hubbard dimer and on a scaling hypothesis within the domain of representability of $\ensuremath{\gamma}$. As applications we consider the Hubbard model on one- and two-dimensional bipartite lattices. Several ground-state properties are determined including the kinetic and Coulomb energy, density distribution, nearest-neighbor bond order, and charge gap. Comparison with exact Lanczos diagonalizations shows that LDFT with the scaled dimer approximation to $W[\ensuremath{\gamma}]$ yields a quite accurate description of the interplay between correlations and charge redistributions, from weak to strong coupling regimes, and for all band fillings. Goals and limitations of the present approach are discussed.

Physical Review B•Jul 20, 2011•29 citations

Two Theorems on the Hubbard-Hirsch Model

Wang Zhijun, Qiu Xiaoming

In this paper, the two theorems on the Hubbard–Hirsch model are shown rigorously. For positive on-site interaction , the model on some bipartite lattices has a weak ferromagnetic ground state at half filling. For negative , the ground state of the model on some bipartite lattices possesses off-diagonal long-range order within a finite range of the electron filling factor.

Communications in Theoretical Physics•Jul 30, 1997•9 citations

Long-Range Correlation of Electron Pairs in the Hubbard Model at Finite Temperatures in Three Dimensions

Gang Su, Bao‐Heng Zhao

We show that in the translation invariant case and in the antiferromagnetic phase, the reduced density matrix ρ2 has no off-diagonal long-range order of on-site electron pairs for the single-band Hubbard model on a cubic lattice away from half filling at finite temperatures both for the positive coupling and for the negative coupling. In these cases the model can not give a mechanism for the superconductivity caused by the condensation of on-site electron pairs and the nearestneighbor electron pairs. PACS numbers: 71.10.+x, 74.20.-z, 05.30.-d. 1 It was proposed the possibility that the Hubbard model can give a mechanism to characterize high temperature superconductivity[1]. It has been shown that the η pairing states for the single band Hubbard model with constant hopping matrix element for nearest neighbor sites have ODLRO[2]. It has also been shown that in the ground state the model with negative coupling has off-diagonal long-range order (ODLRO) on some bipartite lattice[3]. However, as the temperature T> 0 whether the reduced density matrix ρ2 has ODLRO is inconclusive. The purpose of this work is to study the ODLRO of ρ2 in the single-band Hubbard model with constant hopping matrix element for nearest neighbor pairs on a cubic lattice at finite temperatures. In the translation invariant case we obtain an equation ( (10) in the following) which is satisfied by the long-range correlation functions of on-site electron pairs. From which we find that there is no ODLRO of on-site electron pairs. Then we consider the case with an antiferromagnetic background. In this case we will show that there is still no ODLRO of on-site electron pairs. The Hamiltonian of the model is H = −t ∑ (a † rar ′ + b † rb ∑

•Jan 1, 1994•2 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
A general class of hopping models on a finite bipartite lattice is considered, including the Hubbard model and the Falicov–Kimball model. For the half-filled band, the single-particle density matrix ρ(x,y) in the ground state and in the canonical and grand canonical ensembles is shown to be constant on the diagonal x=y, and to vanish if x≠y and if x and y are on the same sublattice. For free electron hopping models, it is shown in addition that there are no correlations between sites of the same sublattice in any higher order density matrix. Physical implications are discussed.
The ground state nature of the Falicov-Kimball model with unconstrained hopping of electrons is investigated. We solve the eigenvalue problem in a pedagogical manner and give a complete account of the ground state energy both as a function of the number of electrons and nuclei and as a function of the total number of particles for any value of interaction U. We also study the energy gap and show the existence of a phase transition characterized by the absence of gap at the half--filled band for U<0. The model in consideration was proposed and solved by Farkasovsky for finite lattices and repulsive on-site interaction U>0. Contrary to his proposal we conveniently scale the hopping matrix to guarantee the existence of the thermodynamic limit. We also solve this model with bipartite unconstrained hopping matrices in order to compare with the Kennedy--Lieb variational analysis.
In order to contribute to understand well the finite-temperature properties of the Falicov-Kimball model, we study the temperature dependence of the specific heat, internal energy, entropy, and mean square fluctuation of the difference between the ion and electron numbers, and of the correlation functions of the one-dimensional half-filled-band Falicov-Kimball model. For this purpose we used the method of small-cluster exact-diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain. Our results show that the examined Falicov-Kimball model exhibits behavior of alternate electron-ion short-range ordering at low temperature, and gradual metal-insulator transition at high temperature, but, does not exhibit the crossing point of the specific heat curves, characteristic of the Hubbard model.
The inhomogeneous Hubbard model is investigated in one and two dimensions by using lattice density-functional theory (LDFT). The ground-state energy E=T+WE=T+WE=T+W is regarded as a functional of the single-particle density matrix $\ensuremath{\gamma}$, where $T[\ensuremath{\gamma}]$ represents the kinetic and crystal-field energy, and $W[\ensuremath{\gamma}]$ the interaction energy. Besides the known functional $T[\ensuremath{\gamma}]$ we propose a simple scaling approximation to the interaction energy $W[\ensuremath{\gamma}]$, which is based on exact results for the Hubbard dimer and on a scaling hypothesis within the domain of representability of $\ensuremath{\gamma}$. As applications we consider the Hubbard model on one- and two-dimensional bipartite lattices. Several ground-state properties are determined including the kinetic and Coulomb energy, density distribution, nearest-neighbor bond order, and charge gap. Comparison with exact Lanczos diagonalizations shows that LDFT with the scaled dimer approximation to $W[\ensuremath{\gamma}]$ yields a quite accurate description of the interplay between correlations and charge redistributions, from weak to strong coupling regimes, and for all band fillings. Goals and limitations of the present approach are discussed.
We show that in the translation invariant case and in the antiferromagnetic phase, the reduced density matrix ρ2 has no off-diagonal long-range order of on-site electron pairs for the single-band Hubbard model on a cubic lattice away from half filling at finite temperatures both for the positive coupling and for the negative coupling. In these cases the model can not give a mechanism for the superconductivity caused by the condensation of on-site electron pairs and the nearestneighbor electron pairs. PACS numbers: 71.10.+x, 74.20.-z, 05.30.-d. 1 It was proposed the possibility that the Hubbard model can give a mechanism to characterize high temperature superconductivity[1]. It has been shown that the η pairing states for the single band Hubbard model with constant hopping matrix element for nearest neighbor sites have ODLRO[2]. It has also been shown that in the ground state the model with negative coupling has off-diagonal long-range order (ODLRO) on some bipartite lattice[3]. However, as the temperature T> 0 whether the reduced density matrix ρ2 has ODLRO is inconclusive. The purpose of this work is to study the ODLRO of ρ2 in the single-band Hubbard model with constant hopping matrix element for nearest neighbor pairs on a cubic lattice at finite temperatures. In the translation invariant case we obtain an equation ( (10) in the following) which is satisfied by the long-range correlation functions of on-site electron pairs. From which we find that there is no ODLRO of on-site electron pairs. Then we consider the case with an antiferromagnetic background. In this case we will show that there is still no ODLRO of on-site electron pairs. The Hamiltonian of the model is H = −t ∑ (a † rar ′ + b † rb ∑
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.