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 minimal length on the ground state energy of the electron gas in the Hartree-Fock approximation. Our results indicate that, mathematically, the corrections of minimal length to the phase space and Hamiltonian eliminate the weaknesses of the HartreeFock approximation. The Hamiltonian corrections, on the other hand, can be considered as the relativistic corrections of electron in solids. It is also obtained that corrections to the phase space have vital and crucial roles in eliminating the weaknesses of the Hartree-Fock approximation compared to the role of Hamiltonian correction. Physically, it is obtained that electrons in metals can 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 divergency of Fermi velocity in 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.
