Emergent tilt order in zigzagging polymer liquids

We study a liquid of zigzagging two-dimensional polymers that have bending rigidity. These are directed polymers whose conformations lie along the paths followed by pieces on a checkerboard. We observe that in the continuum limit the statistical physics of one such polymer can be described in terms of the Dirac equation for a particle having an imaginary mass. We exploit this observation to investigate a liquid of these polymers by means of quantum many-particle techniques in imaginary-time. We treat hard-core interactions between the polymers via a transmutation of quantum particle statistics, from Bose to Fermi, and we account for additional interactions between polymers by introducing two-body interactions between the fermion particles. A self-consistent approximation predicts a phase of tilted order, i.e., the polymers may develop a spontaneous preference to zig rather than zag (or vice versa). We study this behavior analytically, computing the phase diagram and response functions for the polymer liquid, and comment on the role played by fluctuations.