Directed Polymers with Random Interaction : An Exactly Solvable Case -

TL;DR

This paper introduces an exactly solvable model of two directed polymers with random mutual interactions, analyzing phase transitions and critical behavior using renormalization group techniques, revealing disorder relevance and multicritical phenomena.

Contribution

It provides an exact renormalization group analysis of a novel directed polymer model with random interactions, including multicritical extensions and phase transition characterizations.

Findings

Disorder is marginally relevant at d=1.

A phase transition from weak to strong disorder exists for d>1.

Critical lengthscale exponent is ν=1/2|ε|.

Abstract

We propose a model for two $(d+1)$-dimensional directed polymers subjected to a mutual $\delta$-function interaction with a random coupling constant, and present an exact renormalization group study for this system. The exact $\beta$-function, evaluated through an $\epsilon(=1-d)$ expansion for second and third moments of the partition function, exhibits the marginal relevance of the disorder at $d=1$, and the presence of a phase transition from a weak to strong disorder regime for $d>1$. The lengthscale exponent for the critical point is $\nu=1/2\mid\epsilon\mid$. We give details of the renormalization. We show that higher moments do not require any new interaction, and hence the $\beta$ function remains the same for all moments. The method is extended to multicritical systems involving an $m$ chain interaction. The corresponding disorder induced phase transition for $d>d_m=1/(m-1)$ has the critical exponent ${\nu}_m=[2d(m-1)-2]^{-1}$. For both the cases, an essential singularity appears for the lengthscale right at the upper critical dimension $d_m$. We also discuss the strange behavior of an annealed system with more than two chains with pairwise random interactions among each other.