DIRAC CHAINS IN THE PRESENCE OF HAIRPINS

We study semiflexible polymers of arbitrary stiffness subject to nemat ic and non-nematic elongation forces. The presence of nematic forces i s found to cause the formation of hairpins. [A hairpin is an. immediat e return (or a sharp bend) of a chain in the nematic ordering field.] An analysis of the path integrals for semiflexible (Dirac) chains with these elongational forces indicates that the distribution functions d escribing these induced hairpins satisfy the Whittaker-Hill (WH) equat ion in two dimensions. The same equation describes hairpins in three d imensions if (and only if) the Dirac monopole term is included in the corresponding path integral. The solutions of the WH equation indicate that the nonnematic stretching force can only have discrete values co rresponding to the sequential destruction of hairpins. This discretene ss disappears when the nematic force is absent, as demonstrated in pre vious work [A. Kholodenko and T. Vilgis, Phys. Rev. E 50, 1257 (1994)] . We also indicate how the hairpin problem is related to other statist ical mechanical problems of interest: commensurate-incommensurate tran sitions,quantum spin chains, Landau-Lifshitz equation, rotational Brow nian motion, strings with rigidity, etc.