Constraints of reduced density-matrix functional theory for the two-dimensional homogeneous electron gas

Reduced density-matrix functional theory (RDMFT) has become an appealing alternative to density-functional theory to describe electronic properties of highly-correlated systems. Here we derive exact conditions for the suitability of RDMFT to describe the two-dimensional homogeneous electron gas, which is the base system for, e.g., semiconductor quantum dots and quantum Hall devices. Following the method of Cioslowski and Pernal [J. Chem. Phys. {\bf 111}, 3396 (1999)] we focus on the properties of power functionals of the form $f(n,n')=(n n')^\alpha$ for the scaling function in the exchange-correlation energy. We show that in order to have stable and analytic solutions, and for $f$ to satisfy the homogeneous scaling constraint, the power is restricted to $1/4 \leq \alpha \leq 3/4$. Applying a reasonable ansatz for the momentum distribution and the lower bound for the exchange-correlation energy tightens the physical regime further to $0.64 \lesssim \alpha \leq 0.75$.