  The functional-renormalization-group aided density-functional theory
(FRG-DFT) is applied to the two-dimensional homogeneous electron gas (2DHEG).
The correlation energy of the 2DHEG is derived as a function of the
Wigner-Seitz radius $ r_{\rm s} $ directly. We find that our correlation energy
completely reproduces the exact behavior at the high-density limit. For finite
density, the result of FRG-DFT shows good agreement with the Monte Carlo (MC)
results in the high-density region, although the discrepancy between FRG-DFT
and MC results becomes larger as the system becomes more dilute. Our study is
the first example in which the FRG-DFT is applied to more-than-one-dimensional
models, and shows that the FRG-DFT is a feasible and promising method even for
the analysis of realistic models for quantum many-body systems.
