The Φ⁴ Critical Point

Abstract

To fix the notation, we consider the interaction

$$V\left( \phi \right) = \lambda \phi ^4 + \sigma \phi ^2 - \mu \phi $$

(17.1.1)

with λ, σ, and μ real and 0 < λ. By the Lee-Yang theorem, there is no phase transition for μ ≠ 0. The high temperature series expansions (Chapter 18) show that there is no phase transition for μ = 0 and σ sufficiently large. By Section 16.2, there is a phase transition for μ = 0 and σ sufficiently negative. In the latter region one expects exactly two phases, and a unique value of σ, σ = σ_c, which separates the one and two phase regions. Throughout this chapter we define σ_c as the infinum of the values of σ for which (17.1.1) has a unique phase and exponential decay of correlations. (Thus H has a gap in its spectrum, which separates 0, the spectrum of the vacuum Ω, from the rest of the spectrum. We call this gap a mass m > 0.)