Effects of the two-dimensional Coulomb interaction in both Fermi velocity and energy gap for Dirac-like electrons at finite temperature

We describe both the Fermi velocity and the mass renormalization due to the two-dimensional Coulomb interaction in the presence of a thermal bath. To achieve this, we consider an anisotropic version of pseudoquantum electrodynamics, within a perturbative approach in the fine-structure constant $\ensuremath{\alpha}$. Thereafter, we use the so-called imaginary-time formalism for including the thermal bath. In the limit $T\ensuremath{\rightarrow}0$, we calculate the renormalized mass ${m}^{R}(\mathrm{p})$ and compare this result with the experimental findings for the energy band gap in monolayers of transition metal dichalcogenides, namely, ${\mathrm{WSe}}_{2}$ and ${\mathrm{MoS}}_{2}$. In these materials, the quasiparticle excitations behave as massive Dirac-like particles in the low-energy limit, hence, its mass is related to the energy band gap of the material. In the low-temperature limit $T\ensuremath{\ll}{v}_{F}\mathrm{p}$, where ${v}_{F}\mathrm{p}$ is taken as the Fermi energy, we show that ${m}^{R}(\mathrm{p})$ decreases linearly on the temperature, i.e., ${m}^{R}(\mathrm{p},T)\ensuremath{-}{m}^{R}(\mathrm{p},T\ensuremath{\rightarrow}0)\ensuremath{\approx}\ensuremath{-}{A}_{\ensuremath{\alpha}}T+O({T}^{3})$, where ${A}_{\ensuremath{\alpha}}$ is a positive constant. On the other hand, for the renormalized Fermi velocity, we find that ${v}_{F}^{R}(\mathrm{p},T)\ensuremath{-}{v}_{F}^{R}(\mathrm{p},T\ensuremath{\rightarrow}0)\ensuremath{\approx}\ensuremath{-}{B}_{\ensuremath{\alpha}}{T}^{3}+O({T}^{5})$, where ${B}_{\ensuremath{\alpha}}$ is a positive constant. We also perform numerical tests which confirm our analytical results.