Hexatic phase

More generally, a hexatic is any phase that contains sixfold orientational order, in analogy with the nematic phase (with twofold orientational order).

It is an anisotropic phase, since there exists a director field with sixfold symmetry.

The existence of the director field implies that an elastic modulus against drilling or torsion exists within the plane, that is usually called Frank's constant after Charles Frank in analogy to liquid crystals.

The ensemble becomes an isotropic liquid (and Frank's constant becomes zero) after the dissociation of disclinations at a higher temperature (or lower density).

The KTHNY theory of two-step melting by i) destroying positional order and ii) destroying orientational order was developed by John Michael Kosterlitz, David J. Thouless, Bertrand Halperin, David Robert Nelson and A. P. Young in theoretical studies about topological defect unbinding two dimensions.

In 2016, Kosterlitz and Thouless were awarded with the Nobel Prize in Physics (together with Duncan Haldane) for the idea that melting in 2D is mediated by topological defects.

The hexatic phase was predicted by D. Nelson and B. Halperin; it does not have a strict analogue in three dimensions.

points to a lattice site within the crystal, where the atom is allowed to fluctuate with an amplitude

The brackets denote a statistical average about all pairs of atoms with distance R. The translational correlation function decays fast, i. e. exponential, in the hexatic phase.

In a 2D crystal, the translational order is quasi-long range and the correlation function decays rather slow, i. e. algebraic; It is not perfect long range, as in three dimensions, since the displacements

diverge logarithmically with systems size at temperatures above T=0 due to the Mermin-Wagner theorem.

A disadvantage of the translational correlation function is, that it is strictly spoken only well defined within the crystal.

In the isotropic fluid, at the latest, disclinations are present and the reciprocal lattice vector is not defined any more.

The orientational order can be determined by the local director field of a particle at place

The local director field disappears for a particle with five or seven nearest neighbours, as given by dislocations and disclinations

, except a small contribution due to thermal motion.

is now defined using the local director field: Again, the brackets denote the statistical average about all pairs of particles with distance

All three thermodynamic phases can be identified with this orientational correlation function: it does not decay in the 2D crystal but takes a constant value (shown in blue in the figure).

The stiffness against local torsion is arbitrarily large, Franks's constant is infinity.

In the hexatic phase, the correlation decays with a power law (algebraic).

This gives straight lines in a log-log-plot, shown in green in the Figure.

In the isotropic phase, the correlations decay exponentially fast, this are the red curved lines in the log-log-plot (in a lin-log-plot, it would be straight lines).

The discrete structure of the atoms or particles superimposes the correlation function, given by the minima at half integral distances