Fractional spectrum due to disorder and its effect on the ground state in the ensembles of interacting fermions

We consider a disordered system of interacting fermions, for which we demonstrate, that depending on the relationship between the degree of disorder and real physical dimensionality d, this system undergoes a delocalization (e.g. a metal-insulator) phase transition. The effect of the disorder is accounted for by replacing the ordinary single-particle kinetic energy operator by that with a fractional spectrum. These fractional spectra represent the microscopic equivalent of the well-known non-trivial power-law decay of the low-energy density of states of non-interacting disordered systems. We develop a powerful approach to analyze the density of states in interacting disordered system, combining both the saddle-point and renormalization group methods. These two methods complement each other effectively. Through this approach, we identify a fixed point in the renormalization group flow of the model’s parameters, which dictates the behavior of the system in the conducting phase. Additionally, we extend our analysis to include static conductivity. Our examination of the system’s conductivity allows us to establish several strict conditions regarding the stability of the metallic state across different spatial dimensions d. Notably, we emphasize the unique significance of the two-dimensional case d=2. We also compare our findings with salient experimental features observed in paradigmatic physical systems.