KTHNY theory
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.
[8][9] Based on this work, David Nelson and Bertrand Halperin showed, that the resulting hexatic phase is not yet an isotropic fluid.
Orientational order only disappears due to the dissociations of a second class of topological defects, named disclinations.
Peter Young calculated the critical exponent of the diverging correlations length at the transition between crystalline and hexatic.
The thermodynamic phases can be distinguished based on discrete versus continuous translational and orientational order.
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.
On the other side, very early computer simulations of Berni Alder and Thomas E. Wainwright indicated crystallization in 2D.
Following M. Kosterlitz, a finite shear elasticity defines a 2D solid, including quasicrystals in this description.
The double sum runs over all positions of particle pairs i and j and the brackets denote an average about various configurations.
The (closed packed) crystalline phase is characterized by six-fold symmetry based on the orientational order.
The structure factor of Figure 1 is calculated from the positions of a colloidal monolayer (crosses at high intensity are artefacts from the Fourier transformation due to the finite (rectangular) field of view of the ensemble).
The second term in the brackets brings dislocations to arrange preferentially antiparallel due to energetic reasons.
The main contribution stems from the logarithmic term (the first one in the brackets) which describes, how the energy of a dislocation pair diverges with increasing distance.
To create a dislocation from an undisturbed lattice, a small displacement on a scale smaller than the average particle distance
An easy argument for the dominating logarithmic term is, that the magnitude of the strain induced by an isolated dislocation decays according to
The logarithmic distance dependence of the energy is the reason, why KTHNY-theory is one of the few theories of phase transitions which can be solved analytically: in statistical physics one has to calculate partition functions, e.g. the probability distribution for all possible configurations of dislocation pairs given by the Boltzmann distribution
For the majority of problems in statistical physics one can hardly solve the partition function due to the enormous amount of particles and degrees of freedoms.
If a 2D crystal is heated, virtual dislocation pairs will be excited due to thermal fluctuations in the vicinity of the phase transition.
Virtual means, that the average thermal energy is not large enough to overcome (two times) the core-energy and to dissociate (unbind) dislocation pairs.
Nonetheless, dislocation pairs can appear locally on very short time scales due to thermal fluctuations, before they annihilate again.
The principle is completely analogue to calculating the bare charge of the electron in quantum electrodynamics (QED).
Roughly spoken one can summarize: If the crystal is softened due to the presence of virtual pairs of dislocation, the probability (fugacity)
David Nelson, Bertrand Halperin and independently Peter Young formulated this in a mathematically precise way, using renormalization group theory for the fugacity and the elasticity: In the vicinity of the continuous phase transition, the system becomes critical – this means that it becomes self-similar on all length scales
The softening of the system after a length scale transformation (zooming out to visualize a larger area implies to count more dislocations) is now covered in a renormalized (reduced) elasticity.
The blue curve is from computer simulations and shows a reduced elasticity due to lattice vibrations at
Turquoise symbols are from measurements of elasticity in a colloidal monolayer, and confirm the melting point at
The squared distance of two disclinations can be calculated the same way, as for dislocations, only the prefactor, denoting the coupling constant, has to be changed accordingly.
Typically, Kosterlitz–Thouless transitions have a continuum of critical points which can be characterised by self-similar grains of disordered and ordered regions.
In second order phase transitions, the correlation length measuring the size of those regions diverges algebraically: Here,
The increasing shielding of orientational stiffness due to disclinations has not to be taken into account – this is already done by dislocations which are frequently present at
