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/0211451 · Physical review. B, Condensed matter

Equivalence of the Falicov-Kimball and Brandt-Mielsch forms for the free energy of the infinite-dimensional Falicov-Kimball model

Andrij M. Shvaika, James K. Freericks

Published Apr 2, 2003• 11 citations
arXivHTMLPDF

Abstract

Falicov and Kimball proposed a real-axis form for the free energy of the Falicov-Kimball model that was modified for the coherent potential approximation by Plischke. Brandt and Mielsch proposed an imaginary-axis form for the free energy of the dynamical mean field theory solution of the Falicov-Kimball model. It has long been known that these two formulas are numerically equal to each other; an explicit derivation showing this equivalence is presented here.

DOI
10.1103/physrevb.67.153103

Related papers

Falicov-Kimball model and the breaking of Fermi-liquid theory in infinite dimensions

Qimiao Si A, Gabriel Kotliar, Antoine Georges

We reexamine the Brandt-Mielsch solution of the Falicov-Kimball model in infinite dimensions, to elucidate the mechanism for the breakdown of Fermi-liquid theory in this model. We find that this breakdown is due to the pinning of the chemical potential at the localized particle level. When this pinning does not occur, Fermi-liquid theory is restored. Using a Lorentzian density of states, we obtain analytic expressions for spectral densities and local response functions. These expressions give new insights into the formation of upper and lower Hubbard bands, the metal-insulator transition, the softening of the collective modes, and the infrared singularities of this model.

Physical review. B, Condensed matter•Jul 1, 1992•70 citations

Ground-state configurations of the one-dimensional Falicov-Kimball model

Christian Gruber, Joel L. Lebowitz, Nicolas Macris

We consider the one-dimensional Falicov-Kimball lattice model as well as a continuum model, where the attractive potential between the electrons and the nuclei is a \ensuremath{\delta} function, with an equal number N of classical ``nuclei'' and fermion ``electrons.'' We find the exact leading behavior of the ground-state energy as a function of U for U>${\mathit{U}}_{\mathit{c}}$. The error term is rigorously estimated. For fixed N the system forms ``atoms'' which have an effective repulsion between them. From this we argue that when N becomes infinite the ground-state configurations of the nuclei should be periodic. The low-temperature structure of the model is also discussed.

Physical review. B, Condensed matter•Aug 15, 1993•22 citations

Correlated hopping in the Falicov-Kimball model: A large-dimensions study

Avraham Schiller

The Falicov-Kimball model with a correlated-hopping interaction is solved using an extended dynamical mean-field theory that becomes exact in the limit of large dimensions. The effect of correlated hopping is to introduce nonlocal self-energy components that retain full dynamics as $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\infty},$ thus introducing an explicit k dependence to the single-particle self-energy. An explicit solution for the homogeneous phase at $D=2$ reveals significant nonlocal dynamical contributions in the physically relevant regime of a moderately large correlated-hopping amplitude, indicating that important nonlocal correlations are omitted in Hubbard-like models upon neglecting the correlated-hopping interaction.

arXiv:cond-mat/9911118•Physical review. B, Condensed matter•Dec 15, 1999•27 citations

Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach

Andrij M. Shvaika, Oleg Farenyuk

The Falicov-Kimball model with correlated hopping is studied in the limit of infinite spatial dimensions.Dynamical susceptibilities are calculated using the generalization of the Dynamical Mean-Field Theory (DMFT), which is based on expansion over electron hopping around the single-site limit.A special case of semi-elliptic density of states and diagonal correlated hopping is considered numerically and the absence of phase transition for all temperatures except T = 0 is demonstrated, which corresponds to the known results.

Condensed Matter Physics•Jan 1, 2009•0 citations

Inhomogeneous phases in the Falicov-Kimball model: Dynamical mean-field approximation

Minh-Tien Tran

An inhomogeneous dynamical mean-field approximation is proposed to investigate inhomogeneous phases of strongly correlated electron systems. The approximation is based on the dynamical mean-field theory and the direct lattice space representation. Within the approximation the self-energy although is a local function of frequency, but it still depends on the lattice site. Applied the inhomogeneous dynamical mean-field approximation to the Falicov-Kimball model on a square lattice, various inhomogeneous phases are obtained. For intermediate couplings these phases are characterized by charge ordered clusters embedding in an almost homogeneous surrounding. The topology of the charge configuration of the clusters depends on the electron density and the number of ions. The clustered phases are a realization of the coexistence of the charge checkerboard ordering and segregation.

Physical Review B•May 1, 2006•17 citations

Lower bound for the segregation energy in the Falicov Kimball model

Pedro S Goldbaum

In this work, a lower bound for the ground-state energy of the Falicov–Kimball model for intermediate densities is derived. The explicit derivation is important in the proof of the conjecture of segregation of the two kinds of fermions in the Falicov–Kimball model, for sufficiently large interactions. This bound is given by a bulk term, plus a boundary term of the form α 1 ( n )|∂Λ|, where Λ is the region devoid of classical particles and n is the density of electrons. A detailed proof is presented for n = 1/2, where the coefficient α 1 (1/2) = 10 −13 is obtained, for the two-dimensional case. Although clearly not optimal in terms of order of magnitude, this is the largest explicitly calculated coefficient in the range of intermediate densities. With suitable modifications the method can also be used to obtain a coefficient for all densities. That is the topic of the last section, where a sketch of the proof for n <1/2 is shown.

arXiv:cond-mat/0211183•Journal of Physics A Mathematical and General•Feb 19, 2003•7 citations
Falicov and Kimball proposed a real-axis form for the free energy of the Falicov-Kimball model that was modified for the coherent potential approximation by Plischke. Brandt and Mielsch proposed an imaginary-axis form for the free energy of the dynamical mean field theory solution of the Falicov-Kimball model. It has long been known that these two formulas are numerically equal to each other; an explicit derivation showing this equivalence is presented here.
We reexamine the Brandt-Mielsch solution of the Falicov-Kimball model in infinite dimensions, to elucidate the mechanism for the breakdown of Fermi-liquid theory in this model. We find that this breakdown is due to the pinning of the chemical potential at the localized particle level. When this pinning does not occur, Fermi-liquid theory is restored. Using a Lorentzian density of states, we obtain analytic expressions for spectral densities and local response functions. These expressions give new insights into the formation of upper and lower Hubbard bands, the metal-insulator transition, the softening of the collective modes, and the infrared singularities of this model.
We consider the one-dimensional Falicov-Kimball lattice model as well as a continuum model, where the attractive potential between the electrons and the nuclei is a \ensuremath{\delta} function, with an equal number N of classical ``nuclei'' and fermion ``electrons.'' We find the exact leading behavior of the ground-state energy as a function of U for U>Uc{\mathit{U}}_{\mathit{c}}Uc​. The error term is rigorously estimated. For fixed N the system forms ``atoms'' which have an effective repulsion between them. From this we argue that when N becomes infinite the ground-state configurations of the nuclei should be periodic. The low-temperature structure of the model is also discussed.
The Falicov-Kimball model with a correlated-hopping interaction is solved using an extended dynamical mean-field theory that becomes exact in the limit of large dimensions. The effect of correlated hopping is to introduce nonlocal self-energy components that retain full dynamics as $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\infty},$ thus introducing an explicit k dependence to the single-particle self-energy. An explicit solution for the homogeneous phase at D=2D=2D=2 reveals significant nonlocal dynamical contributions in the physically relevant regime of a moderately large correlated-hopping amplitude, indicating that important nonlocal correlations are omitted in Hubbard-like models upon neglecting the correlated-hopping interaction.
The Falicov-Kimball model with correlated hopping is studied in the limit of infinite spatial dimensions.Dynamical susceptibilities are calculated using the generalization of the Dynamical Mean-Field Theory (DMFT), which is based on expansion over electron hopping around the single-site limit.A special case of semi-elliptic density of states and diagonal correlated hopping is considered numerically and the absence of phase transition for all temperatures except T = 0 is demonstrated, which corresponds to the known results.
An inhomogeneous dynamical mean-field approximation is proposed to investigate inhomogeneous phases of strongly correlated electron systems. The approximation is based on the dynamical mean-field theory and the direct lattice space representation. Within the approximation the self-energy although is a local function of frequency, but it still depends on the lattice site. Applied the inhomogeneous dynamical mean-field approximation to the Falicov-Kimball model on a square lattice, various inhomogeneous phases are obtained. For intermediate couplings these phases are characterized by charge ordered clusters embedding in an almost homogeneous surrounding. The topology of the charge configuration of the clusters depends on the electron density and the number of ions. The clustered phases are a realization of the coexistence of the charge checkerboard ordering and segregation.
In this work, a lower bound for the ground-state energy of the Falicov–Kimball model for intermediate densities is derived. The explicit derivation is important in the proof of the conjecture of segregation of the two kinds of fermions in the Falicov–Kimball model, for sufficiently large interactions. This bound is given by a bulk term, plus a boundary term of the form α 1 ( n )|∂Λ|, where Λ is the region devoid of classical particles and n is the density of electrons. A detailed proof is presented for n = 1/2, where the coefficient α 1 (1/2) = 10 −13 is obtained, for the two-dimensional case. Although clearly not optimal in terms of order of magnitude, this is the largest explicitly calculated coefficient in the range of intermediate densities. With suitable modifications the method can also be used to obtain a coefficient for all densities. That is the topic of the last section, where a sketch of the proof for n <1/2 is shown.