Statistical mechanics of vacancy and interstitial strings in hexagonal columnar crystals

Columnar crystals contain defects in the form of vacancy-interstitial loops or strings of vacancies and interstitials bounded by column "heads" and "tails." These defect strings are oriented by the columnar lattice and can change size and shape by movement of the ends and by forming kinks along the length. Hence an analysis in terms of directed living polymers [S. A. Safran, Statistical Thermodynamics of Surfaces, Interfaces, and Membranes (Addison-Wesley, Reading, MA, 1994), Sec. 8] is appropriate to study their size and shape distribution, volume fraction, etc. If the entropy of transverse fluctuations overcomes the string line tension in the crystalline phase, a string proliferation transition occurs, leading to a supersolid phase [E. Frey, D. R. Nelson, and D. S. Fisher, Phys. Rev. B 49, 9723 (1994); see also J. Prost, Liq. Cryst. 8, 123 (1990)]. We estimate the wandering entropy and examine the behavior in the transition regime. We also calculate numerically the line tension of various species of vacancies and interstitials in a triangular lattice for power-law potentials as well as for a modified Bessel function interaction between columns such as occurs in the case of flux lines in type-II superconductors or long polyelectrolytes in an ionic solution. We find that the centered interstitial is the lowest-energy defect for a very wide range of interactions; the symmetric vacancy is preferred only for extremely short interaction ranges.