Scaling Limits and Critical Behaviour of the 4-Dimensional n-Component |φ|4 Spin Model

• Published in: Journal of statistical physics 2014, Vol.157 (4-5), p.692-742
• Main Authors: Bauerschmidt, Roland, Brydges, David C., Slade, Gordon
• Format: Article
• Language: English
• Subjects: Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-014-1060-5

We consider the n -component | φ | 4 spin model on Z 4 , for all n ≥ 1 , with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent n + 2 n + 8 for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for n = 1 , 2 , 3 ; double logarithmic scaling for n = 4 ; and is bounded when n > 4 . In addition, for the model defined on the 4 -dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the | φ | 4 model.