Thermal effects in the Ginzburg–Landau model in a magnetic field

We consider the Ginzburg–Landau N-component model in the large N limit, concurrently with the Matsubara thermal field formalism, the system being embedded in an external constant magnetic field. In this double framework, we investigate the behaviour of the coupling constant and the mass as functions of temperature. Closed exact formulas for the thermal renormalized coupling constant can be obtained in even spacial dimensions, in particular, d=4. We are also able, after performing a mass renormalization, to obtain in spacial dimension d=3, a critical equation relating temperature to the external magnetic field. This equation is compatible with a second-order transition.