Frustrated Polyelectrolyte Bundles and $T=0$ Josephson-Junction Arrays

We establish a one-to-one mapping between a model for hexagonal polyelectrolyte bundles and a model for two-dimensional, frustrated Josephson-junction arrays. We find that the $T=0$ insulator-to-superconductor transition of the quantum system corresponds to a continuous liquid-to-solid transition of the condensed charge in the finite-temperature classical system. We find that the role of the vector potential in the quantum system is played by elastic strain in the classical system. Exploiting this correspondence we show that the transition is accompanied by a spontaneous breaking of a discrete symmetry associated with the chiral patterning of the array and that at the transition the polyelectrolyte bundle adopts a universal response to shear.

Figure 1

The two degenerate, “chiral-patterned” ground states of the antiferromagnetic $XY$ model on a triangular lattice are shown in (a). The helical ordering around a triangular plaquette of the bundle lattice corresponding to one of these ground states is shown in (b). The dark black lines depict the polymer backbone while the spheres depict peaks in the condensed charge density.

Figure 2

Two volume-preserving, shear deformations which lead to the relative sliding of the neighboring rods (equivalent up to a pure rotation).