Three-dimensional tricritical spins and polymers

Abstract

We consider two intimately related statistical mechanical problems on ${\mathbb{Z}}^3$: (i) the tricritical behaviour of a model of classical unbounded $n$-component continuous spins with a triple-well single-spin potential (the $|\phi {|}^6$ model), and (ii) a random walk model of linear polymers with a three-body repulsion and two-body attraction at the tricritical theta point (critical point for the collapse transition) where repulsion and attraction effectively cancel. The polymer model is exactly equivalent to a supersymmetric spin model which corresponds to the $n=0$ version of the $|\phi {|}^6$ model. For the spin and polymer models, we identify the tricritical point, and prove that the tricritical two-point function has Gaussian long-distance decay, namely $|x{|}^{-1}$. The proof is based on an extension of a rigorous renormalisation group method that has been applied previously to analyse the $|\phi {|}^4$ and weakly self-avoiding walk models on ${\mathbb{Z}}^4$.

Accepted version

References in corpus (19)

• Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis
• Scaling limits and critical behaviour of the 4-dimensional n-component $|\phi {|}^4$ spin model
• Finite range Decomposition of Gaussian Processes
• Critical two-point function of the 4-dimensional weakly self-avoiding walk
• Critical exponents for long-range O(n) models below the upper critical dimension
• A renormalisation group method. II. Approximation by local polynomials
• A renormalisation group method. III. Perturbative analysis
• A simple method for finite range decomposition of quadratic forms and Gaussian fields
• A renormalisation group method. IV. Stability analysis
• Critical two-point function for long-range $O(n)$ models below the upper critical dimension
• A renormalisation group method. I. Gaussian integration and normed algebras
• A renormalisation group method. V. A single renormalisation group step
• Structural stability of a dynamical system near a non-hyperbolic fixed point
• Critical correlation functions for the 4-dimensional weakly self-avoiding walk and n-component $|\phi {|}^4$ model
• Renormalization group approach to interacting polymerised manifolds
• Mean-field tricritical polymers
• Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents
• Four-dimensional weakly self-avoiding walk with contact self-attraction
• Critical exponent for the magnetization of the weakly coupled ${\varphi }_4^4$ model

Cited by in corpus (3)

• Mean-field tricritical polymers
• Statistical Mechanics of Confined Polymer Networks
• Existence of a tricritical point for the Blume-Capel model on ${\mathbb{Z}}^d$