Thermal and Quantum Phase Transitions of the φ⁴ Model

The critical behaviour of the classical (thermal) phase transition (CPT) of the Ising model is identical to the quantum phase transition (QPT) of the φ⁴ Quantum Field Theory (QFT). Our goal in this work is to map out the QPT-CPT diagram of the φ⁴ QFT, and study its relation to the well-known QPT-CPT phase diagram of the Ising model. To do this, we propose a modification of the usual finite-temperature Renormalization Group (RG) approach by relating the temperature parameter to the running RG scale, T ≡ kT = τ k where kT is the running cutoff for thermal, and k is for the quantum fluctuations. In this case, the temperature is related to the dimensionless quantity τ. We apply this new thermal RG approach for the φ⁴ model in lower dimensions and construct its QPT-CPT phase diagram. We formulate requirements for the QPT-CPT phase diagram of the φ⁴ theory based on known properties of the QPT-CPT phase diagram of the Ising model. Finally, we use these requirements to check the viability of the thermal RG method proposed in this work.