In statistical mechanics, the Kosterlitz–Thouless–Halperin–Nelson–Young (KTHNY) theory describes the process of melting of crystals in two dimensions (2D). The name is derived from the initials of the surnames of John Michael Kosterlitz, David J. Thouless, Bertrand Halperin, David R. Nelson, and A. Peter Young, who developed the theory in the 1970s. It is, beside the Ising model in 2D and the XY model in 2D, one of the few theories which can be solved analytically and which predicts a phase transition at a temperature T > 0 {\displaystyle T>0} .
Main idea Melting of 2D crystals is mediated by the dissociation of topological defects, which destroy the order of the crystal. In 2016, Michael Kosterlitz and David Thouless were awarded with the Nobel prize in physics for their idea, how thermally excited pairs of virtual dislocations induce a softening (described by renormalization group theory) of the crystal during heating. The shear elasticity disappears simultaneously with the dissociation of the dislocations, indicating a fluid phase. Based on this work, David Nelson and Bertrand Halperin showed that the resulting hexatic phase is not yet an isotropic fluid. Starting from a hexagonal crystal (HCP, which is the densest packed structure in 2D, along with CCP/FCC), the hexatic phase has a six-folded director field, similar to liquid crystals. Orientational order only disappears due to the dissociations of a second class of topological defects, disclinations. Peter Young calculated the critical exponent of the diverging correlations length at the transition between the crystalline and hexatic phases. KTHNY theory predicts two continuous phase transitions, thus latent heat and phase coexistence is ruled out. The thermodynamic phases can be distinguished based on discrete versus continuous translational and orientational order. One of the transitions separates a solid phase with quasi-long range translational order and perfect long ranged orientational order from the hexatic phase. The hexatic phase shows short ranged translational order and quasi-long ranged orientational order. The second phase transition separates the hexatic phase from the isotropic fluid, where both translational and orientational order is short ranged. The system is dominated by critical fluctuations, since for continuous transitions, the difference of energy between the thermodynamic phases disappears in the vicinity of the transition. This implies that ordered and disordered regions fluctuate strongly in space and time. The size of those regions grows strongly near the transitions and diverges at the transition itself. At this point, the pattern of symmetry broken versus symmetric domains is fractal. Fractals are characterized by a scaling invariance – they appear similar on an arbitrary scale or by arbitrarily zooming in (this is true on any scale larger than the atomic distance). The scale invariance is the basis to use the renormalization group theory to describe the phase transitions. Both transitions are accompanied by spontaneous symmetry breaking. Unlike for melting in three dimensions, translational and orientational symmetry breaking does not need to appear simultaneously in two dimensions, since two different types of topological defects destroy the different types of order.
Background Michael Kosterlitz and David Thouless tried to resolve a contradiction about 2D crystals: on one hand side, the Mermin-Wagner theorem claims that symmetry breaking of a continuous order-parameter cannot exist in two dimensions. This implies, that perfect long range positional order is ruled out in 2D crystals. On the other side, very early computer simulations of Berni Alder and Thomas E. Wainwright indicated crystallization in 2D. The KTHNY theory shows implicitly that periodicity is not a necessary criterion for a solid (this is already indicated by the existence of amorphous solids like glasses). Following M. Kosterlitz, a finite shear elasticity defines a 2D solid, including quasicrystals in this description.
Structure factor in 2D
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