One-loop renormalization of the two-dimensional linear σ model and Gross-Neveu model at finite temperature

We study the one-loop renormalization of the two-dimensional linear σ model and Gross-Neveu model at finite temperature. To do this, we use thermo field dynamics and regard the Gross-Neveu model as a Z3=Z4=0 limit of the linear σ model. We also derive the renormalization group equations with respect to a dimensional parameter M.