# Conditions where RPA becomes exact in the high-density limit

## TL;DR

This paper identifies conditions under which the random phase approximation (RPA) becomes exact in high-density limits across different dimensions, supported by theoretical analysis and Quantum Monte Carlo simulations.

## Contribution

It derives a general criterion for RPA's exactness in high-density regimes based on system dimensionality and interaction potential properties.

## Key findings

- Vertex corrections scale with system parameters as derived.
- RPA is exact when the condition d - β - α < 0 is satisfied.
- Quantum Monte Carlo simulations confirm the theoretical predictions.

## Abstract

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 $\epsilon_{\rm f}\sim p_{\rm f}^\beta$ 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 found to be independent of the width parameter of the wire. The proposed high-density expansion agrees with the Quantum Monte Carlo simulations.