Minimal length implications on the Hartree–Fock theory

Abstract

Hartree–Fock approximation suffers from two shortcomings including i) the divergence of the electron Fermi velocity, and ii) the existence of bandwidth which is not confirmed experimentally. Here, we study the effects of the minimal length on the ground state energy of the electron gas in the Hartree–Fock approximation. Our results indicate that, mathematically, the correction of minimal length to the phase space, which plays a vital, and predominant role below the Fermi surface, eliminates the weaknesses of the Hartree–Fock approximation. On the other hand, the effect of the Hamiltonian correction, which has the same form as the relativistic correction of electrons in solids, becomes dominant at energy levels above the Fermi surface. Physically, it is concluded that electrons in metals may be employed to test the quantum gravity scenario, if the value of its parameter (β) lies within the range of 2 to 10, depending on the used metal. Indeed, the latter addresses an upper bound on β parameter which is comparable with previous works meaning that these types of systems may be employed as a benchmark to examine quantum gravity scenarios. To overcome the Fermi velocity divergence in the Hartree–Fock method, the screening potential is used based on the Lindhard theory. In the context of this theory, we also find that considering the generalized Heisenberg uncertainly leads to some additional oscillating terms in the Friedel oscillations.