Directed polymers with random interaction: an exactly solvable case

Abstract

We propose a model for two (d+1)-dimensional directed polymers subjected to a mutual δ-function interaction with a random coupling constant, and present an exact renormalization-group study for this system. The exact β function, evaluated through an ε (=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 length-scale exponent for the critical point is ν=(2|ε|)^-1. We give details of the renormalization. We show that higher moments do not require any new interaction, and hence the β 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 v_m=[2d(m-1)-2]^-1. For both the cases, an essential singularity appears for the length scale right at the upper critical dimension d_m. We also discuss the strainge behavior of an annealed system with more than two chains with pairwise random interactions among each other.