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Journal of Physics A Mathematical and General

Segregation in the Falicov-Kimball model

Pirmin Lemberger

Published Feb 21, 1992• 78 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/25/4/012

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•Jan 1, 1998•1 citation
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.
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.
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 states of the one-dimensional Falicov-Kimball model are investigated in the small-coupling limit, using nearly degenerate perturbation theory. For rational electron and ion densities, respectively, equal to p/q, pi{\mathit{p}}_{\mathit{i}}pi​/q, with p relatively prime to q and pi{\mathit{p}}_{\mathit{i}}pi​/q close enough to 1/2, we find that in the ground state the ion configuration has a period q. The situation is analogous to the Peierls instability, where the usual arguments predict a period-q state that produces a gap at the Fermi level and is insulating. However for pi{\mathit{p}}_{\mathit{i}}pi​/q far enough from 1/2, this phase becomes unstable against phase separation. The ground state is a mixture of a period-q ionic configuration and an empty (or full) configuration, where both configurations have the same electron density to leading order. Combining these results with those previously obtained for strong coupling, it follows that a phase transition occurs in the ground state, as a function of the coupling, for ion densities far enough from 1/2. \textcopyright{} 1996 The American Physical Society.
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.
We focus on the two-dimensional Falicov-Kimball model in the neutral case with U >> 0. We determine the ground state ion configurations for all rational densities in (1/3, 2/5), as well as densities 1/6 and 2/11. We also determine the ground states for a sequence of densities starting at 1/5 and converging to 0. On the interval (1/3, 2/5) we show that the ground states are periodic, having the same structure as the one-dimensional ground states. For densities between 1/6 and 2/11 we show that the ground states exhibit a phase separation.