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arXiv:cond-mat/9612137 · Journal of Physics A Mathematical and General

A complete devil's staircase in the Falicov - Kimball model

C Micheletti, A B Harris, J M Yeomans

Published Nov 7, 1997• 5 citations
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Abstract

We consider the neutral, one-dimensional Falicov - Kimball model at zero temperature in the limit of a large electron - ion attractive potential, U. By calculating the general n-ion interaction terms to leading order in 1/U we argue that the ground state of the model exhibits the behaviour of a complete devil's staircase.
DOI
10.1088/0305-4470/30/21/002

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The spinless Falicov-Kimball model on a two-dimensional square lattice is studied using the method of restricted phase diagrams constructed in the grand canonical ensemble. The results are compared with the one-dimensional model. Although the two-dimensional phase diagrams are more complex, with several distinct families of ion configurations occurring as ground states, there are surprising similarities with the one-dimensional case. Within each family of configurations, the ground states form a devil`s staircase structure and the configurations are constructed according to a composition rule identical to that in one dimension. It is also found that, as in one dimension, segregation occurs in the non-neutral model for large ion-electron interaction strength. Some features of the phase diagrams are understood by examining the effective two-body ion interaction.
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.
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.
The two-state spinless Falicov-Kimball model on a one-dimensional lattice is studied by means of well-controlled numerical procedures. Restricted phase diagrams in the grand-canonical ensemble and at zero temperature are constructed. The evolution of these phase diagrams, as the interaction parameter U is varied, along with the band structures corresponding to configurations of localized particles (ions) and densities of mobile particles (electrons), is monitored. The changes observed enable us to draw conclusions for the complete phase diagram and to make a qualitative plot of such a diagram.
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.