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Helvetica Physica Acta (Switzerland)

Spingless fermi gas on one-dimensional lattice : rigorous results

C. Gruber

Published Jan 1, 1991• 8 citations
OpenAlex

Abstract

The ground state configurations of the one-dimensional Fermi gas (Falicov-Kimball model) are investigated using expansion in powers of U −1 , where U is the interaction strength. For a given ion density ρ i =P/q we consider only those periodic configurations defined by νp ions followed by ν(q-p) holes. We then show that there exists critical values ρ' e and ρ″ e such that for electron densities ρ e ρ″ e

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The ground state configurations of the one-dimensional Fermi gas (Falicov-Kimball model) are investigated using expansion in powers of U −1 , where U is the interaction strength. For a given ion density ρ i =P/q we consider only those periodic configurations defined by νp ions followed by ν(q-p) holes. We then show that there exists critical values ρ' e and ρ″ e such that for electron densities ρ e ρ″ e
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.
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 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 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 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.