Conditions where random phase approximation becomes exact in the high-density limit

It is shown that, in $d$-dimensional systems, the vertex corrections beyond the random phase approximation (RPA) or $GW$ approximation scales with the power $d\ensuremath{-}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}$ of the Fermi momentum if the relation between Fermi energy and Fermi momentum is ${\ensuremath{\epsilon}}_{\mathrm{f}}\ensuremath{\sim}{p}_{\mathrm{f}}^{\ensuremath{\beta}}$ and the interacting potential possesses a momentum power law of $\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}$. The condition $d\ensuremath{-}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}<0$ specifies systems where RPA is exact in the high-density limit. The one-dimensional structure factor is found to be the interaction-free one in the high-density limit for contact interaction. A cancellation of RPA and vertex corrections render this result valid up to second order in contact interaction. For finite-range potentials of cylindrical wires a large-scale cancellation appears and is found to be independent of the width parameter of the wire. The proposed high-density expansion agrees with the quantum Monte Carlo simulations.