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Temperature-dependent many-body effects in Dirac-Weyl materials: Interacting compressibility and quasiparticle velocity

We calculate, within the single-loop or equivalently the Hartree-Fock Approximation (HFA), the finite-temperature interacting compressibility for three-dimensional (3D) Dirac materials and renormalized quasiparticle velocities for 3D and two-dimensional (2D) Dirac materials. We find that in the extrinsic (i.e., doped) system, the inverse compressibility (incompressibility) and renormalized quasiparticle velocity at $k=0$ show nonmonotonic dependences on temperature. At low temperatures the incompressibility initially decreases to a shallow minimum with a $T^2 \ln T$ dependence. As the temperature increases further, the incompressibility rises to a maximum and beyond that it decreases with increasing temperature. On the other hand, the renormalized quasiparticle velocity at $k=0$ for both 2D and 3D Dirac materials first increases with $T^2$, rises to a maximum, and after reaching the maximum it decreases with increasing temperature. We also find that within the HFA, the leading-order temperature correction to the low-temperature renormalized extrinsic Fermi velocity for both 2D and 3D doped Dirac materials is $\ln (1/T)$.