Renormalization of the optical band gap through an effective Thirring interaction for massive Dirac-like electrons

We analyze mass renormalization in massive Dirac-like systems in (2+1) dimensions arising from electron-phonon interactions at finite temperatures, employing the large-$N$ expansion. Our model combines the low-energy description of charge carriers in a buckled honeycomb lattice with the low-energy approximation for phonons and electron-phonon interactions in two-dimensional materials. Consequently, the system is modeled as a massive Dirac-like field coupled to a two-component vector field $\mathcal{A}_i$, representing the phonon modes. This framework allows us to compute the one-loop electron self-energy at finite temperature, from which we derive the renormalized band gap, $m^R$. The effective model is subsequently applied to describe the renormalized optical band gap in monolayers of transition metal dichalcogenides (TMDs), including MoS$_2$, MoSe$_2$, WS$_2$, and WSe$_2$. A good agreement is observed with experimental data for reasonable values of the ultraviolet cutoff, $\Lambda \approx 1$ eV. Our main findings indicate that $m^R$ remains nearly constant at low temperatures, whereas at higher temperatures it decreases linearly with the temperature $T$. Specifically, we find that $m^R$ reduces by approximately $\approx [0.1,0.2]$ eV as the temperature increases from $\approx 4$ K to $500$ K, consistent with recent experimental observations. Furthermore, we estimate the temperature range at which the transition to the linear regime occurs, obtaining typical values within $\approx [110,150]$ K for the four materials under consideration.