Sum rules for electron-hole bilayer and two-dimensional point dipole systems

We formulate and analyze the third-frequency-moment sum rules for the two-dimensional (point) dipole system (2DDS) and the mass-symmetric electron-hole bilayer (EHB) in their strongly coupled liquid phases. The former, characterized by the repulsive interaction potential ${\phi }_D\left(r\right)={\mu }^2/r^3$ ($\mu$ is the electric dipole moment), reasonably well approximates the latter in the $d\to 0$ limit ($d$ is the interlayer spacing), a conjecture that is further supported by the findings of the present work. We explore the extent to which the in-phase sum rule for the closely spaced EHB may or may not reconcile with its 2DDS sum-rule counterpart. This is the main emphasis of the present work.