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We calculate, within the single-loop or equivalently the Hartree-FockApproximation (HFA), the finite-temperature interacting compressibility forthree-dimensional (3D) Dirac materials and renormalized quasiparticlevelocities for 3D and two-dimensional (2D) Dirac materials. We find that in theextrinsic (i.e., doped) system, the inverse compressibility (incompressibility)and renormalized quasiparticle velocity at $k=0$ show nonmonotonic dependenceson temperature. At low temperatures the incompressibility initially decreasesto a shallow minimum with a $T^2 \ln T$ dependence. As the temperatureincreases further, the incompressibility rises to a maximum and beyond that itdecreases with increasing temperature. On the other hand, the renormalizedquasiparticle velocity at $k=0$ for both 2D and 3D Dirac materials firstincreases with $T^2$, rises to a maximum, and after reaching the maximum itdecreases with increasing temperature. We also find that within the HFA, theleading-order temperature correction to the low-temperature renormalizedextrinsic Fermi velocity for both 2D and 3D doped Dirac materials is $\ln(1/T)$.

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