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Physical review. B, Condensed matter

Two-state one-dimensional spinless Fermi gas

James K. Freericks, L. M. Falicov

Published Feb 1, 1990• 80 citations
Publisher

Abstract

The two-state, one-dimensional, spinless Fermi gas (Falicov-Kimball model) is studied exactly by numerical calculation and perturbation theory. Rigorous results are presented for small interaction strength and (restricted) coherent and incoherent phase diagrams are calculated for two specific examples. The numerical calculations are extrapolated to provide a qualitative picture of the complete solution. The result includes a fractal structure in which the ground-state changes discontinuously as a function of the parameters.

DOI
10.1103/physrevb.41.2163

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The two-state, one-dimensional, spinless Fermi gas (Falicov-Kimball model) is studied exactly by numerical calculation and perturbation theory. Rigorous results are presented for small interaction strength and (restricted) coherent and incoherent phase diagrams are calculated for two specific examples. The numerical calculations are extrapolated to provide a qualitative picture of the complete solution. The result includes a fractal structure in which the ground-state changes discontinuously as a function of the parameters.
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 two-state spinless Falicov-Kimball model on a one-dimensional lattice is studied by means of well-controlled numericał 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, including 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 global structure of the phase diagram and to make a qualitative plot of such a diagram.Particular attention is paid to the possibility of conductor-insulator transitions.The results obtained suggest that the system is an insulator for all U only if the electron and ion densities pc, p; satisfy the conditions: pe+ pi =1, 'pi -1/2|< δ < 4. For all other pairs (Pc , pi) there is a critical U at which the system undergoes a conductor-insulator transition.
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 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.
Two-dimensional spinless Falicov-Kimball model (FKM) with correlated hopping is studied perturbatively in the limit of large on-site Coulomb interaction U. In the half filled case (i.e., ${\ensuremath{\rho}}_{i}+{\ensuremath{\rho}}_{e}=1,$ where ${\ensuremath{\rho}}_{i},{\ensuremath{\rho}}_{e}$ are densities of ions and electrons, respectively) the effective Hamiltonian in spin variables is derived up to terms proportional to ${U}^{\ensuremath{-}3}.$ Unlike the simplest FKM case, it contains odd parity terms (resulting from the correlated hopping) in addition to even parity ones. The ground-state phase diagram of the effective Hamiltonian is examined in the (a/t,h)(a/t,h)(a/t,h) plane, where a/ta/ta/t is a parameter characterizing strength of the correlated hopping and h is a difference of chemical potentials of two sorts of particles present in the system. It appears to be asymmetric with respect to the change $\stackrel{\ensuremath{\rightarrow}}{h}\ensuremath{-}h$ and an additional ordered phase is found for a certain interval of a/t.a/t.a/t.
The extended Falicov-Kimball model is analyzed exactly in the ground state at half filling in the limit of large dimensions. In the model the on-site and the intersite density-density interactions between all particles are included. We determined the model's phase diagram and found a discontinuous transition between two different charge-ordered phases. Our analytical calculations show that the ground state of the system is insulating for any nonzero values of the interaction couplings. We also show that the dynamical mean-field theory and the static broken-symmetry Hartree-Fock mean-field approximation give the same results for the model at zero temperature. In addition, we prove using analytical expressions that at infinitesimally small, but finite, temperatures the system can be metallic.