Directed-polymer systems explored via their quantum analogs: General polymer interactions and their consequences

TL;DR

This paper investigates how various polymer-polymer interactions influence the thermodynamics, structure, and topological constraints of directed polymers in two dimensions by leveraging quantum analogs and exactly solvable models.

Contribution

It introduces a quantum-analog approach to analyze complex polymer interactions, revealing universal behaviors and response characteristics of interacting directed polymer systems.

Findings

Polymer interactions do not significantly alter the structure near topological constraints.

The free-energy cost of pinning is proportional to that of noncrossing polymers.

Generic interactions lead to a universal polymer fluid with collective excitations.

Abstract

The impact of polymer-polymer interactions of various types on the thermodynamics, structure, and accommodation of topological constraints is addressed for systems comprising many directed polymers in two spatial dimensions. The approach is predicated on the well-known equivalence between the classical equilibrium statistical mechanics of directed polymers in two spatial dimensions and the imaginary-time quantum dynamics of particles in one spatial dimension, originally exploited by P.-G. de Gennes [J.\ Chem.\ Phys.\ {\bf 48}, 2257 (1968)]. Known results concerning two exactly solvable microscopic models of quantum particles moving in one spatial dimension---the Lieb-Liniger model of contact interactions and the Calogero-Sutherland model of long-range interactions---are used to shed light on the behavior of the corresponding polymeric systems. In addition, the technique of bosonization is used to reveal how generic polymer interactions give rise to an emergent polymer fluid that has universal collective excitations. Comparison of the response to topological constraints of a fluid of simply noncrossing (i.e., noncrossing but otherwise noninteracting) directed polymers, explored in a companion Paper, to the response of a generically interacting directed polymer fluid reveals that the structure is quantitatively unchanged by the generic interactions on the line transverse to the pin, and is qualitatively unchanged by the generic interactions throughout the two dimensions of the system's extent. Furthermore, the free-energy cost associated with a pin that partitions a system having generic interactions is found to be proportional to the pin-partitioning cost for a system of simply noncrossing polymers.