Thermal and Quantum Phase Transitions of the $\phi^4$ 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 $\phi^4$ 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, wepropose a modification of the usual finite-temperature RG approach by relating the temperature parameter to the running RG scale, $T \equiv k_T = \tau k$ where $k_T$ is the running cutoff for thermal, and $k$ is for the quantum fluctuations. Once introduced this dimensionless temperature $\tau$, we investigate the consequences on the thermal RG approach for the $\phi^4$ model and construct its phase diagram. Finally, we formulate requirements for the phase diagram of the $\phi^4$ theory based on known properties of the quantum and classical phase diagrams of the Ising model.