States-conserving density of states for Altshuler-Aronov effect: Heuristic derivation

Altshuler and Aronov (AA) have shown that the electron-electron interaction in a weakly-disordered metal suppresses the single-particle density of states (DOS) in the vicinity of the Fermi level ($E_F$). According to the AA theory the suppressed DOS exhibits the energy dependence $\propto \sqrt {|E-E_F|}$ valid for $|E-E_F|$ smaller than a certain correlation energy $U_{co}$. Recent experiments have shown that at energies larger than $U_{co}$ the DOS exhibits a states-conserving dependence on energy, namely, the states removed from near the Fermi level are found at energies above $U_{co}$ in the energy range of about $3 U_{co}$. In this work the AA effect is studied beyond the low energy limit theoretically. We consider the AA model in which the electrons interact via the statically screened Coulomb interaction and the modification of the DOS is due to the exchange part of the electron self-energy. We derive the states-conserving DOS heuristically. Namely, we show that the self-energy consists of a diverging part (which we skip on physical grounds) and of the small part of the order of the pair Coulomb energy. This small part gives the states-conserving DOS which is in qualitative accord with experimental observations at energies above $U_{co}$ and which reproduces the AA result at energies below $U_{co}$.