Topological obstacles and the forces on them in systems of strongly interacting directed polymers

Owing to their extended structure and inpenetrability, polymers are strongly influenced by topological obstacles. To shed light on this issue we consider a model system of noncrossing directed polymers in two dimensions. As first exploited by de Gennes [1], the configurations of this system can be mapped onto the worldlines of noninteracting fermions---an analogy that enables the application to the polymer system of techniques initially developed for one-dimensional quantum systems. Via this approach, we discuss how an obstacle that forces a fixed number of polymers to pass to one side of a single topological constraint is associated with a large fluctuation of the quantum system. In addition, via the use of techniques from quantum hydrodynamics, we find that such a constraint on a system of noncrossing polymers generates an effective, long-ranged repulsion between the polymers. This repulsion causes a void to appear in the polymer fluid and generates a super-Hookean force opposing the constraint. [1] P.-G. de Gennes, J. Chem Phys. 48, 2257-2259 (1968).