The formalism of finite-temperature quantum field theory, as developed by Matsubara, is applied to a Hamiltonian of N scalar fields with a quartic self-interaction at large N. A renormalized expression in the lowest quantum approximation is obtained for the squared mass m2 of the field, as a function of the temperature T, from which we study the process of spontaneous symmetry breaking. We find that in a range of values around the critical temperature Tc, the squared mass can be approximated by a linear relation m2 [Formula: see text] (T − Tc). We thus demonstrate the compatibility of the finite-temperature formalism for scalar fields, in the vicinity of criticality, with respect to the Ginzburg–Landau model. We also discuss the effects caused by the presence of a chemical potential and of an external magnetic field applied to the finite-temperature system, which however do not affect the linearity of the relation between the squared mass and the temperature.
Symmetry breaking patterns for two coupled complex scalar fields at finite temperature and in an external magnetic field