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Lower bound for the segregation energy in the Falicov-Kimball model

Pedro S. Goldbaum

Published Jan 1, 2003• 8 citations
OpenAlex

Abstract

Abstract. 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 α1(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 α1(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. PACS numbers: 71.10.Fd, 71.10.Hf, 71.30.+h. 1.

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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.

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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
Abstract. 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 α1(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 α1(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. PACS numbers: 71.10.Fd, 71.10.Hf, 71.30.+h. 1.
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.
The ground state of the one-dimensional Falicov-Kimball model (1969) is considered for large values of the interaction strength U. If the ion density rho i =p/q (p and q relatively prime) is rational and equals the electron density rho e the author proves that for U)U hom c (q) the ion configurations is the most homogeneous one. If 0( rho e ( rho i and U)U seg c ( rho e / rho i ) w prove that all the ions condense into one large cluster. This last result proves a recent conjecture known as the segregation principle.
In this work, we study two models of strongly correlated fermions: Hubbard and FalicovKimball. We present a proof of the existence of real and ordered solutions to the nested Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any U > 0 to U = ∞ is also shown, where U is the strength of the interaction between fermions at the same site. We use this continuity property, combined with the proof that the norm of the wavefunction obtained with the generalized Bethe Ansatz is not zero, to prove that the solution gives us the ground state of the finite system, as assumed by Lieb and Wu. For the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the Lieb-Wu solution of the 1D Hubbard model. We also obtain a lower bound for the ground state energy of the Falicov-Kimball model. This bound is given by a bulk term, plus a term proportional to the boundary. A numerical value for the coefficient of the boundary term is determined. 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.
The authors have studied energetic and structural ground-state properties of the one-dimensional Falicov-Kimball model in the strong-coupling limit. Using standard perturbation theory they recover some results already obtained (the phase segregation at large interactions) and present some new results concerning the ground state, e.,g., the analytic expression for the boundary, below which the segregated configuration cannot be the ground state, and the phase diagram of the model calculated for special periodic (aperiodic) configurations of ions and for the case when the ion concentration goes to zero.
We consider the Falicov-Kimball model in the case of an equal number of classical "nuclei" and quantum "electrons" with Fermi statistics. In one dimension we find an exact formula for the leading behavior of the ground-state energy as a function of the attractive potential U between electrons and nuclei. For U > U c the system forms "atoms" which have an effective repulsion between them. Similar results hold for a continuum model provided there is a sufficiently large hard core between the nuclei. Some low-temperature properties are also discussed.
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.