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 $d-\beta -\alpha$ of the Fermi momentum if the relation between Fermi energy and Fermi momentum is ${\epsilon }_{\mathrm{f}}\sim p_{\mathrm{f}}^{\beta }$ and the interacting potential possesses a momentum power law of $\sim p^{-\alpha }$. The condition $d-\beta -\alpha <0$ 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.

Figure 1

Five Hedin equations where the thin line is $G_0$. The numbers give the space-time variable in general and the letters denote the frequency-momentum ones after Fourier transform in equilibrium. Please note that the $\mp$ sign for Fermi/bosons will be absorbed into closed propagator lines when expanded to get the standard Feynman diagrams.

Figure 2

Expansion of the vertex function (5) up to first-order interaction.

Figure 3

First-order expansion of the polarization function (4) in terms of interaction when using the vertex of Fig. 2.

Figure 4

First-order corrections in $r_s$ to the structure factor according to the denominator (21), (A2) (black, lower lines) and the vertex corrections (24), (25) (red, upper lines) for a cylindrical potential. The sum of both corrections is nearly independent of $b$ (blue, middle line) and visually not distinguishable.

Figure 5

Interaction-free pair-correlation function, $g_0$, which is the high-density limit (28) together with the ones from RPA (10), and the first-order expansion (27) in cylindrical potential with thickness parameter $b=0.1$.

Figure 6

Pair-correlation function $g\left(r\right)$ with first order in $r_s$ including the vertex correction is plotted as a function of $r$ at several densities and compared with recent quantum Monte Carlo (QMC) simulation data [19].