Extended Falicov–Kimball model: Hartree–Fock vs DMFT approach

Abstract

In this work, we study the extended Falicov-Kimball model at half-filling within the Hartree-Fock approach (HFA) (for various crystal lattices) and compare the results obtained with the rigorous ones derived within the dynamical mean field theory (DMFT). The model describes a system, where electrons with spin-↓ are itinerant (with hopping amplitude t), whereas those with spin-↑ are localized. The particles interact via on-site U and intersite V density-density Coulomb interactions. We show that the HFA description of the ground state properties of the model is equivalent to the exact DMFT solution and provides a qualitatively correct picture also for a range of small temperatures. It does capture the discontinuous transition between ordered phases at U = 2V for small temperatures as well as correct features of the continuous order-disorder transition. However, the HFA predicts that the discontinuous boundary ends at the isolated-critical point (of the liquidgas type) and it does not merge with the continuous boundary. This approach cannot also describe properly a change of order of the continuous transition for large V as well as various metal-insulator transitions found within the DMFT.

Extended Falicov-Kimball model at weak onsite and intersite Coulomb interactions

“…Here, we focus on the dependence of d(U, V; Θ) ≡ d(Θ) (where Θ = T/T C is a reduced temperature) for the EFKM with small U and V interactions. In this limit of weak interactions, the model exhibits the second order (continuous) order-disorder transition occurring at the transition temperature T C (d vanishes continously to 0 at T C ) [7,[12][13][14]37]. Obviously, the HFA overestimates T C and it is always larger than that determined within the DMFT, but for weak U interaction (U → 0), the difference between them is really small [14] (for U, V → 0 also T C → 0, but for any |U| + V 0, one gets T C > 0, whereas for U = 0 and V = 0, it is obvious that T C = 0).…”

“…It is worth to note that for large |U| or V, the dependence d(Θ) follows the m(T C ; Θ) curve (presisely, for U/V → ±∞, |U| t, and any V; or for U = 0 and V t), but the DMFT predicts [12,13,37,55,56], whereas the HFA gives T C → +∞ for U → ±∞ (any fixed V) [14,31]. For U = 0 and V t, one gets T C = V/k B in both approaches [12-14, 52, 55, 56].…”

“…In the ground state, a transition between two ordered phases has been found for U = 2V, cf. phase diagrams presented in references [11][12][13][14]. For U > 2V, the antiferromagnetism dominates over the charge order (m Q > ∆ Q or d 1 < 0), whereas U < 2V the charge order is dominant (m Q < ∆ Q or d 1 < 0).…”

“…Thus, one defines two mean-field order parameters (averages) d and d 1 as differences between average occupation of sublattices by localized and itinerant particles, respectively (cf. references [7,[11][12][13][14]37]…”

“…However, there are some limits in which it gives rigorous results for the EFKM (e.g., in the ground state and for U = 0), cf. references [11,14]. The equations derived within the HFA for the EFKM are presentend in [11,14,52].…”