Universality for interacting oriented self-avoiding walk: A transfer matrix calculation

In this paper we perform transfer matrix calculations to study the two-dimensional problem of oriented self-avoiding walks on a square lattice where nearest neighbor interactions depend on the relative orientation (parallel or antiparallel) between different parts of the path. Our main purpose is to verify a conformal field theory conjecture that the entropic exponent $\gamma$, which is related to the number of such walks, varies continuously with the energy of the parallel interaction. We find no evidence of this behavior, but see many unexpected features for the model, such as the existence of a line of $\theta$ points belonging to the universality class of polymer collapse on a Manhattan lattice. Finally we study the phase diagram in the parallel and antiparallel interaction planes and we conjecture a crucial role for the standard $\theta$ point.