An Exactly Soluble Model of Directed Polymers with Multiple Phase Transitions

TL;DR

This paper introduces exactly solvable models of directed polymers with multiple phase transitions, revealing identical phase diagrams for intersecting and non-intersecting walks, and analyzing their critical properties and correlation functions.

Contribution

It provides the first exact solutions for models of directed polymers with hard-core interactions, comparing intersecting and non-intersecting walk scenarios.

Findings

Identical phase diagrams for both models.

Three distinct phases with specific transition behaviors.

Power-law decay of correlation functions in the liquid phase.

Abstract

Polymer chains with hard-core interaction on a two-dimensional lattice are modeled by directed random walks. Two models, one with intersecting walks (IW) and another with non-intersecting walks (NIW) are presented, solved and compared. The exact solution of the two models, based on a representation using Grassmann variables, leads, surprisingly, to the same analytic expression for the polymer density and identical phase diagrams. There are three different phases as a function of hopping probability and single site monomer occupancy, with a transition from the dense polymer system to a polymer liquid (A) and a transition from the liquid to an empty lattice (B). Within the liquid phase there exists a self-dual line with peculiar properties. The derivative of polymer density with respect to the single site monomer occupancy diverges at transitions A and B, but is smooth across and along the self-dual line. The density-density correlation function along the direction $x$, perpendicular to the axis of directedness has a power law decay 1/$x^2$ in the entire liquid phase, in both models. The difference between the two models shows up only in the behavior of the correlation function along the self-dual line: it decays exponentially in the IW model and as 1/$x^4$ in the NIW model.