Hexatic phase

Abstract: A theory of dislocation-mediated melting in two dimensions is described in detail, with an emphasis on results for triangular lattices on both smooth and periodic substrates. The transition from solid to liquid on a smooth substrate takes place in two steps with increasing temperatures. Dissociation of dislocation pairs first drives a transition out of a low-temperature solid phase, with algebraic decay of translational order and long-range orientational order. This transition is into a "liquid-crystal" phase characterized by exponential decay of translational order, but power-law decay of sixfold orientational order. Dissociation of disclination pairs at a higher temperature then produces an isotropic fluid. The behavior of the specific heat, structure factor, and various elastic constants near these transitions is worked out. We also discuss the applicability of our results to melting on a periodic substrate. Dislocation unbinding should describe melting of a "floating" (and, in general, incommensurate) adsorbate solid into a high-temperature fluid phase. The orientation bias imposed by the substrate can alter or eliminate the disclination-unbinding transition, however. Transitions from a floating solid into a low-temperature registered or partially registered phase can also be mapped onto the dislocation-unbinding transition, but only at certain special values of the coverage. Substrate reciprocallattice vectors play the role of Burger's vectors in this case.

Abstract: The consequences of a theory of dislocation-mediated two-dimensional melting are worked out for triangular lattices. Dissociation of dislocation pairs first drives a transition into a "liquid crystal" phase with exponential decay of translational order, but power-law decay of sixfold orientational order. A subsequent dissociation of disclination pairs at a higher temperature then produces an isotropic fluid. The critical behavior, as well as the effect of a periodic substrate, is discussed.

Abstract: For a decade now the subject of the nature of the two-dimensional melting transition has remained controversial. An elegant theory based on the unbinding of pairs of crystal defects suggested that two-dimensional solids might melt by a transition sequence involving two continuous transitions separated by a novel, nearest-neighbor-bond-orientationally ordered fluid---the hexatic phase. Competing theories predict that the transition is of the usual first-order type observed in three-dimensional systems. This paper is a critical review of the current status of research into the problem of two-dimensional melting, with an emphasis on computer simulations. An attempt is made to point out unresolved issues pertaining to this fascinating and still open question.

Abstract: We study the interplay between crystalline (or hexatic) order and thermal fluctuations in membranes with vanishing surface tension. If the connectivity of the crystalline state is preserved, the membrane remains uncrumpled at low temperatures. When dislocations are allowed, however, screening of elastic stresses by buckling reduces dislocation energies, and promotes dislocation unbinding. When the resulting hexatic phase is stable, the stiffness associated with orientational correlations leads to a logarithmic enhancement of the bending rigidity which counteracts the thermal softening found in fluid surfaces. A finite temperature crumpling transition is predicted for crystalline membranes, and possibly for hexatic membranes as well.