  We consider theoretically, using the random phase approximation (RPA),
low-energy intrinsic plasmons for two-dimensional (2D) systems obeying
Dirac-like linear chiral dispersion with the chemical potential set precisely
at the charge neutral Dirac point. The "intrinsic Dirac plasmon" energy has the
characteristic $q^{1/2}$ dispersion in the 2D wave-vector $q$, but vanishes as
$T^{1/2}$ in temperature for both monolayer and bilayer graphene. The intrinsic
plasmon becomes overdamped for a fixed $q$ as $T -> 0$ since the level
broadening (i.e. the decay of the plasmon into electron-hole pairs due to
Landau damping) increases as $T^{-1/2}$ as temperature decreases, however, the
plasmon mode remains well-defined at any fixed $T$ (no matter how small) as $q
-> 0$. We find the intrinsic plasmon to be well-defined as long as $q < k_B
T/e^2$. We give analytical results for low and high temperatures, and numerical
RPA results for arbitrary temperatures, and consider both single-layer and
double-layer intrinsic Dirac plasmons. We provide extensive comparison and
contrast between intrinsic and extrinsic graphene plasmons, and critically
discuss the prospects for experimentally observing intrinsic Dirac point
graphene plasmons.
