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. Although they annihilate, they have a detectable impact on elasticity: they soften the crystal. The principle is completely analogue to calculating the bare charge of the electron in quantum electrodynamics (QED). In QED, the charge of the electron is shielded due to virtual electron-positron pairs due to quantum fluctuations of the vacuum. Roughly spoken one can summarize: If the crystal is softened due to the presence of virtual pairs of dislocation, the probability (fugacity) y for creating additional virtual dislocations is enhanced, proportional to the Boltzmann factor of the core-energy of a dislocation y = e E C k B T . If additional (virtual) dislocations are present, the crystal will get additionally softer. If the crystal is additionally softer, the fugacity will increase further... and so on and so forth. 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 ≫ ≫ a . Executing a transformation of all length scales by a factor of l , the energy E → → E ( l ) and fugacity y → → y ( l ) will depend on this factor, but the system has to appear identically, simultaneously due to the self similarity. Especially the energy function (Hamiltonian) of the dislocations have to be invariant in structure. 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 recursion relation for elasticity and fugacity are: