Directed polymers with random interaction: An exactly solvable case

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>dₘ=1/(m-1) has the critical exponent νₘ=[2d(m-1)-2]^-1. For both the cases, an essential singularity appears for the length scale right at the upper critical dimension dₘ. We also discuss the strainge behavior of an annealed system with more than two chains with pairwise random interactions among each other.