Thermal and quantum phase transitions of the ϕ4 model

In this paper we discuss and revisit the finite temperature extension of the renormalization group (RG) treatment of T = 0 field theories, focusing as a case study on the ϕ4 model.We first discuss the extension of RG equations of the very same model from T = 0 to finite T in the usual way by resorting to sums on the Matsubara frequencies and fixing the physical temperature parameter T. We show that this approach, although useful for a variety of applications, may lead to the disappearance of the critical points as extracted from the RG flow. Since the identification of fixed points is key in the study of classical and quantum phase transitions, we propose a modification of the usual finite-temperature RG approach by relating the temperature parameter to the running RG scale, T ≡ kT = τk, where kT is the running cutoff for thermal and k is for the quantum fluctuations. Once this dimensionless temperature τ is introduced, we investigate the consequences on the thermal RG approach for the ϕ4 model and construct its phase diagram. Finally, we formulate requirements for the phase diagram of the ϕ4 theory based on known properties of the quantum and classical phase diagrams of the Ising model.