Hexatic phase
The hexatic phase is a state of matter that is between the solid and the isotropic liquid phases in two dimensional systems of particles. It is characterized by two order parameters: a short-range positional and a quasi-long-range orientational (sixfold) order. More generally, a hexatic is any phase that contains sixfold orientational order, in analogy with the nematic phase (with twofold orientational order). It is a fluid phase, since the shear modulus and the Young's modulus disappear due to the dissociation of dislocations. 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 Frederick C. 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). Therefore, the hexatic phase contains dislocations but no disclinations. The 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. It is named KTHNY theory by the initial letters of the surnames of the authors. In 2016, M. Kosterlitz and D. 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. Order parameter. The hexatic phase can be described by two order parameters, where the translational order is short ranged (exponential  decay) and the orientational order is quasi-long ranged (algebraic decay). Translational order. If the position of atoms or particles is known, then the translational order can be determined with the translational correlation function formula_1 as function of the distance between lattice site at place formula_2 and the place formula_3, based on the two-dimensional density function formula_4 in reciprocal space: The vector formula_6 points to a lattice site within the crystal, where the atom is allowed to fluctuate with an amplitude formula_7 by thermal motion. formula_8 is a reciprocal vector in Fourier space. The brackets denote a statistical average  about all pairs of atoms with distance R an. 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 formula_9 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. Orientational order. The orientational order can be determined by the local director field of a particle at place formula_10, if the angles formula_11 are taken, given by the bond to the formula_12 nearest neighbours in sixfolded space, normalized with the number of nearest neighbours: formula_14 is a complex number of magnitude formula_15 and the orientation of the six-folded director is given by the phase. In a hexagonal crystal, this is nothing else but the crystal-axes. The local director field disappears for a particle with five or seven  nearest neighbours, as given by dislocations and disclinations formula_16, except a small contribution due to thermal motion. The orientational correlation function between two particles i and k at distance formula_17 is now defined using the local director field: Again, the brackets denote the statistical average about all pairs of particles with distance formula_19. 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 formula_20. Particles which are poorly correlated in position, are also poorly correlated in their director.