Mean-field tricritical polymers

TL;DR

This paper analyzes a mean-field random walk model of polymers to map out a tricritical phase diagram, identifying the nature of density transitions and the critical point separating different phase transition types.

Contribution

It introduces a detailed phase diagram for a mean-field polymer model, clarifying the tricritical behavior and transition types using a supersymmetric and renormalization group approach.

Findings

Identified the phase boundary curve separating dilute and dense phases.

Located the tricritical point dividing first-order and second-order transitions.

Applied a supersymmetric representation and renormalization group to analyze the model.

Abstract

We provide an introductory account of a tricritical phase diagram, in the setting of a mean-field random walk model of a polymer density transition, and clarify the nature of the density transition in this context. We consider a continuous-time random walk model on the complete graph, in the limit as the number of vertices $N$ in the graph grows to infinity. The walk has a repulsive self-interaction, as well as a competing attractive self-interaction whose strength is controlled by a parameter $g$. A chemical potential $\nu$ controls the walk length. We determine the phase diagram in the $(g,\nu)$ plane, as a model of a density transition for a single linear polymer chain. A dilute phase (walk of bounded length) is separated from a dense phase (walk of length of order $N$) by a phase boundary curve. The phase boundary is divided into two parts, corresponding to first-order and second-order phase transitions, with the division occurring at a tricritical point. The proof uses a supersymmetric representation for the random walk model, followed by a single block-spin renormalisation group step to reduce the problem to a 1-dimensional integral, followed by application of the Laplace method for an integral with a large parameter.