Hartree-Fock approximation suffers from two inabilities including i) the divergence of electron Fermi velocity , and ii) existence of bandwidth not con?rfimed 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 considering some mathematical terms, similar to those of used for the minimal length correction to the Hamiltonian of system, can eliminate the weaknesses of Hartree-Fock approximation. These corrections, on the other hand, can be considered as relativistic corrections of electron in solids. Physically, it is obtained that electrons in metals can be employed to test the quantum gravity scenario, if the value of its parameter (?$beta$) lies within the range of 2 to 10, depending on the used metal. Indeed, the latter addresses an upper bound on ?$beta$? which is comparable with previous works meaning that these types of systems may be employed in testing quantum gravity scenarios. To overcome the in?nite Fermi velocity in Hartree-Fock method, the screening potential is used based on the Lindhard theory. We also ?nd that considering the generalized Heisenberg uncertainly leads to some additional oscillating terms in the Friedel oscillations.
It is argued that Planck mass may be considered as a candidate for the mass content of degrees of freedom of holographic screen. In addition, employing the Verlinde hypothesis on emergent gravity and considering holographic screen degrees of freedom
as a $q$-deformed fermionic system, it is obtained that the heat capacity per degree of freedom inspires the MOND interpolating function. Moreover, the MOND acceleration is achieved as a function of Planck acceleration. Both ultra-relativistic and non-relativistic statistics are studied.
