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/0205554 · Europhysics Letters (EPL)

Density-functional study of the Mott gap in the Hubbard model

Neemias Alves de Lima, L. Nunes de Oliveira, Klaus Capelle

Published Nov 1, 2002• 79 citations
arXivHTMLPDF

Abstract

We study the Mott insulating phase of the one-dimensional Hubbard model using a local-density approximation (LDA) that is based on the Bethe Ansatz (BA).Unlike conventional functionals, the BA-LDA has an explicit derivative discontinuity.We demonstrate that, as a consequence of this discontinuity, the BA-LDA yields the correct Mott gap, independently of the strength of the correlations.A convenient analytical formula for the Mott gap in the thermodynamic limit is also derived.We find that in one-dimensional quantum systems the contribution of the discontinuity to the full gap is more important than that of the band-structure gap, and discuss some consequences this finding has for electronic-structure calculations.

DOI
10.1209/epl/i2002-00261-y

Related papers

Solving lattice density functionals close to the Mott regime

Zu‐Jian Ying, Valentina Brosco, J. Lorenzana

We study a lattice version of the local density approximation (LDA) based on Bethe ansatz (BALDA). Contrary to what happens in density functional theory in the continuum and despite its name, BALDA displays some very nonlocal features and it has a discontinuous functional derivative. The same features prevent the convergence of the self-consistent Kohn-Sham cycle thus hindering the study of BALDA solutions close to a Mott phase or in the Coulomb blockade regime. Here, we propose a numerical approach which, differently from previous works, does not introduce ad hoc parameters to smear out the singularity. Our results are relevant for all lattice models where BALDA is applied ranging from Kondo systems to harmonically trapped Hubbard fermions. As an example, we apply the method to the study of a one-dimensional lattice model with Hubbard interaction and a staggered potential which can be driven from an ionic to a Mott-insulating state. In the Mott regime, the presence of a ``vacuum'' allows us to calculate the different contribution to the gap and to highlight an ultranonlocality of BALDA.

arXiv:1403.5080•Physical Review B•May 27, 2014•14 citations

Tracing the Mott-Hubbard transition in one-dimensional Hubbard models without Umklapp scattering

Florian Gebhard, Örs Legeza

We apply the density-matrix renormalization group (DMRG) method to a one-dimensional Hubbard model that lacks Umklapp scattering and thus provides an ideal case to study the Mott-Hubbard transition analytically and numerically. The model has a linear dispersion and displays a metal-to-insulator transition when the Hubbard interaction $U$ equals the bandwidth ${U}_{\mathrm{c}}=W$, where the single-particle gap opens linearly, $\mathrm{\ensuremath{\Delta}}(U\ensuremath{\ge}W)=U\ensuremath{-}W$. The simple nature of the elementary excitations permits us to determine numerically with high accuracy the critical interaction strength and the gap function in the thermodynamic limit. The jump discontinuity of the momentum distribution ${n}_{k}$ at the Fermi wave number ${k}_{\mathrm{F}}=0$ cannot be used to locate accurately ${U}_{\mathrm{c}}$ from finite-size systems. However, the slope of ${n}_{k}$ at the band edges ${k}_{\mathrm{B}}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$, reveals the formation of a single-particle bound state which can be used to determine ${U}_{\mathrm{c}}$ reliably from ${n}_{k}$ using accurate finite-size data.

arXiv:2109.01423•Physical review. B./Physical review. B•Dec 13, 2021•6 citations

Thermodynamics of the one-dimensional half-filled Hubbard model in the spin-disordered regime

Satoshi Ejima, Fabian H. L. Eßler, Florian Gebhard

We analyse the thermodynamic Bethe Ansatz equations of the one-dimensional half-filled Hubbard model in the ‘spin-disordered regime’, which is characterized by the temperature being much larger than the magnetic energy scale, but small compared to the Mott–Hubbard gap. In this regime the thermodynamics of the Hubbard model can be thought of in terms of gapped charged excitations with an effective dispersion and spin degrees of freedom that only contribute entropically. In particular, the internal energy and the effective dispersion become essentially independent of temperature. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal–insulator transition at a finite interaction strength above which the gap opens linearly. We relate these observations to studies of the Mott–Hubbard transition in the limit of infinite dimensions.

arXiv:cond-mat/0602310•Journal of Physics A Mathematical and General•Apr 19, 2006•7 citations

Mott-Hubbard Insulator in Infinite Dimensions

Eva Kalinowski, Florian Gebhard

We calculate the one-particle density of states for the Mott-Hubbard insulating phase of the Hubbard model on a Bethe lattice in the limit of infinite coordination number. We employ the Kato-Takahashi perturbation theory around the strong-coupling limit to derive the Green function. We show that the Green function for the lower Hubbard band can be expressed in terms of polynomials in the bare hole-hopping operator. We check our technique against the exact solution of the Falicov-Kimball model and give explicit results up to and including second order in the inverse Hubbard interaction. Our results provide a stringent test for analytical and numerical investigations of the Mott-Hubbard insulator and the Mott-Hubbard transition within the dynamical mean-field theory. We find that the Hubbard-III approximation is not satisfactory beyond lowest order, but the local-moment approach provides a very good description of the Mott-Hubbard insulator at strong coupling.

arXiv:cond-mat/0109535•Sep 28, 2001•0 citations

Density Functionals Not Based on the Electron Gas: Local-Density Approximation for a Luttinger Liquid

Neemias Alves de Lima, M. F. Silva, L. Nunes de Oliveira +1 more

By shifting the reference system for the local-density approximation (LDA) from the electron gas to other model systems, one obtains a new class of density functionals, which by design account for the correlations present in the chosen reference system. This strategy is illustrated by constructing an explicit LDA for the one-dimensional Hubbard model. While the traditional ab initio LDA is based on a Fermi liquid (the three-dimensional interacting electron gas), this one is based on a Luttinger liquid. First applications to inhomogeneous Hubbard models, including one containing a localized impurity, are reported.

arXiv:cond-mat/0112428•Physical Review Letters•Apr 10, 2003•168 citations

Anomalous scaling and breakdown of conventional density functional theory methods for the description of Mott phenomena and stretched bonds

Zu‐Jian Ying, Valentina Brosco, Giorgia Maria Lopez +3 more

Density functional theory provides the most widespread framework for the realistic description of the electronic structure of solids, but the description of strongly correlated systems has remained so far elusive. We consider a particular limit of electrons and ions in which a one-band description becomes exact all the way from the weakly correlated metallic regime to the strongly correlated Mott-Hubbard regime. We provide a necessary condition a density functional should fulfill to describe Mott-Hubbard behavior in this one-band limit and show that it is not satisfied by standard and widely used local, semilocal, and hybrid functionals. We illustrate the condition in the case of few-atom systems and provide an analytic approximation to the exact exchange-correlation potential based on a variational wave function which shows explicitly the correct behavior, combining in a neat way lattice and continuum methods.

arXiv:1510.03425•Physical review. B./Physical review. B•Aug 29, 2016•25 citations
We study the Mott insulating phase of the one-dimensional Hubbard model using a local-density approximation (LDA) that is based on the Bethe Ansatz (BA).Unlike conventional functionals, the BA-LDA has an explicit derivative discontinuity.We demonstrate that, as a consequence of this discontinuity, the BA-LDA yields the correct Mott gap, independently of the strength of the correlations.A convenient analytical formula for the Mott gap in the thermodynamic limit is also derived.We find that in one-dimensional quantum systems the contribution of the discontinuity to the full gap is more important than that of the band-structure gap, and discuss some consequences this finding has for electronic-structure calculations.
We study a lattice version of the local density approximation (LDA) based on Bethe ansatz (BALDA). Contrary to what happens in density functional theory in the continuum and despite its name, BALDA displays some very nonlocal features and it has a discontinuous functional derivative. The same features prevent the convergence of the self-consistent Kohn-Sham cycle thus hindering the study of BALDA solutions close to a Mott phase or in the Coulomb blockade regime. Here, we propose a numerical approach which, differently from previous works, does not introduce ad hoc parameters to smear out the singularity. Our results are relevant for all lattice models where BALDA is applied ranging from Kondo systems to harmonically trapped Hubbard fermions. As an example, we apply the method to the study of a one-dimensional lattice model with Hubbard interaction and a staggered potential which can be driven from an ionic to a Mott-insulating state. In the Mott regime, the presence of a ``vacuum'' allows us to calculate the different contribution to the gap and to highlight an ultranonlocality of BALDA.
We apply the density-matrix renormalization group (DMRG) method to a one-dimensional Hubbard model that lacks Umklapp scattering and thus provides an ideal case to study the Mott-Hubbard transition analytically and numerically. The model has a linear dispersion and displays a metal-to-insulator transition when the Hubbard interaction UUU equals the bandwidth Uc=W{U}_{\mathrm{c}}=WUc​=W, where the single-particle gap opens linearly, $\mathrm{\ensuremath{\Delta}}(U\ensuremath{\ge}W)=U\ensuremath{-}W$. The simple nature of the elementary excitations permits us to determine numerically with high accuracy the critical interaction strength and the gap function in the thermodynamic limit. The jump discontinuity of the momentum distribution nk{n}_{k}nk​ at the Fermi wave number kF=0{k}_{\mathrm{F}}=0kF​=0 cannot be used to locate accurately Uc{U}_{\mathrm{c}}Uc​ from finite-size systems. However, the slope of nk{n}_{k}nk​ at the band edges ${k}_{\mathrm{B}}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$, reveals the formation of a single-particle bound state which can be used to determine Uc{U}_{\mathrm{c}}Uc​ reliably from nk{n}_{k}nk​ using accurate finite-size data.
We analyse the thermodynamic Bethe Ansatz equations of the one-dimensional half-filled Hubbard model in the ‘spin-disordered regime’, which is characterized by the temperature being much larger than the magnetic energy scale, but small compared to the Mott–Hubbard gap. In this regime the thermodynamics of the Hubbard model can be thought of in terms of gapped charged excitations with an effective dispersion and spin degrees of freedom that only contribute entropically. In particular, the internal energy and the effective dispersion become essentially independent of temperature. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal–insulator transition at a finite interaction strength above which the gap opens linearly. We relate these observations to studies of the Mott–Hubbard transition in the limit of infinite dimensions.
We calculate the one-particle density of states for the Mott-Hubbard insulating phase of the Hubbard model on a Bethe lattice in the limit of infinite coordination number. We employ the Kato-Takahashi perturbation theory around the strong-coupling limit to derive the Green function. We show that the Green function for the lower Hubbard band can be expressed in terms of polynomials in the bare hole-hopping operator. We check our technique against the exact solution of the Falicov-Kimball model and give explicit results up to and including second order in the inverse Hubbard interaction. Our results provide a stringent test for analytical and numerical investigations of the Mott-Hubbard insulator and the Mott-Hubbard transition within the dynamical mean-field theory. We find that the Hubbard-III approximation is not satisfactory beyond lowest order, but the local-moment approach provides a very good description of the Mott-Hubbard insulator at strong coupling.
By shifting the reference system for the local-density approximation (LDA) from the electron gas to other model systems, one obtains a new class of density functionals, which by design account for the correlations present in the chosen reference system. This strategy is illustrated by constructing an explicit LDA for the one-dimensional Hubbard model. While the traditional ab initio LDA is based on a Fermi liquid (the three-dimensional interacting electron gas), this one is based on a Luttinger liquid. First applications to inhomogeneous Hubbard models, including one containing a localized impurity, are reported.
Density functional theory provides the most widespread framework for the realistic description of the electronic structure of solids, but the description of strongly correlated systems has remained so far elusive. We consider a particular limit of electrons and ions in which a one-band description becomes exact all the way from the weakly correlated metallic regime to the strongly correlated Mott-Hubbard regime. We provide a necessary condition a density functional should fulfill to describe Mott-Hubbard behavior in this one-band limit and show that it is not satisfied by standard and widely used local, semilocal, and hybrid functionals. We illustrate the condition in the case of few-atom systems and provide an analytic approximation to the exact exchange-correlation potential based on a variational wave function which shows explicitly the correct behavior, combining in a neat way lattice and continuum methods.