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Physical review. B, Condensed matter

Spinless Falicov-Kimball model (annealed binary alloy) from large to small dimensions

James K. Freericks

Published Apr 15, 1993• 52 citations
Publisher

Abstract

The phase diagram for the spinless Falicov-Kimball model on a hypercubic lattice is reexamined in the limits of large and small dimensions (for the half-filled ion case). This model is identical to the thermodynamical problem of an annealed binary alloy with diagonal disorder. The phase diagram for the infinite-dimensional case is remarkably similar to the conjectured phase diagram for the one-dimensional case. The system orders in short-period phases, orders in long-period (possibly incommensurate) phases, or segregates, depending on the interaction strength and the electron concentration. The analysis for this simple model provides hope that recently proposed solutions for other interacting fermion models in infinite dimensions will accurately represent the solutions in two and three dimensions.

DOI
10.1103/physrevb.47.9263

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The phase diagram for the spinless Falicov-Kimball model on a hypercubic lattice is reexamined in the limits of large and small dimensions (for the half-filled ion case). This model is identical to the thermodynamical problem of an annealed binary alloy with diagonal disorder. The phase diagram for the infinite-dimensional case is remarkably similar to the conjectured phase diagram for the one-dimensional case. The system orders in short-period phases, orders in long-period (possibly incommensurate) phases, or segregates, depending on the interaction strength and the electron concentration. The analysis for this simple model provides hope that recently proposed solutions for other interacting fermion models in infinite dimensions will accurately represent the solutions in two and three dimensions.
The spinless Falicov-Kimball model is solved exactly in the limit of infinite dimensions on both the hypercubic and Bethe lattices. The competition between segregation, which is present for large U,U,U, and charge-density-wave order, which is prevalent at moderate U,U,U, is examined in detail. We find a rich phase diagram that displays both of these phases. The model also shows nonanalytic behavior in the charge-density-wave transition temperature when UUU is large enough to generate a correlation-induced gap in the single-particle density of states.
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.
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.
The Falicov-Kimball model of spinless itinerant fermions and localized classical ions, on a one-dimensional lattice and at zero temperature, has been studied by means of restricted phase diagrams in the grand-canonical ensemble. Compared with earlier investigations a more efficient numerical procedure has been applied and consequently more precise phase diagrams have been obtained. For small and large values of the unique coupling parameter our study has confirmed the results obtained in other works; however for intermediate values we have found new kinds of periodic phases and structural phase transitions.
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.
The simplest statistical-mechanical model of crystalline formation (or alloy formation) that includes electronic degrees of freedom is solved exactly in the limit of large spatial dimensions and infinite interaction strength. The solutions contain both second-order phase transitions and first-order phase transitions (that involve phase separation or segregation) which are likely to illustrate the precursor physics behind the static charge-stripe ordering in cuprate systems. In addition, we find that the spinodal-decomposition temperature satisfies an approximate scaling law.