Long-range translational order is forbidden in low dimensional systems with short-range interactions: solid phases only have quasi long-range translational order. A mechanism for the transition from solid to liquid led by the dissociation of dislocation pairs was proposed by Kosterlitz & Thouless in their 1972 & 1973 Nobel prize papers. Knowing that long-range orientational order is possible in two dimensions, Nelson, Halperin and Young argued that the transition actually occurs in two steps, the second one being triggered by the unbinding of disclinations. In this picture the intermediate phase keeps quasi long-range orientational order, and the two transitions are of infinite order. Based on simulations of hard disks performed with advanced numerical methods, the latter picture was recently questioned. In this talk I will review the current understanding of melting in two dimensions and I will extend its analysis to systems made of self-propelled particles, the constituents of active matter, a new kind of soft matter relevant to describe numerous biological problems.

A detailed description of this problem can be found in Phases of planar active matter, L. F. Cugliandolo and G. Gonnella, Les Houches Lecture Notes (Oxford University Press) arXiv:1810.11833 See also "Full Phase Diagram of Active Brownian disks: from melting to motility-induced phase separation", P. Digregorio et al. Phys. Rev. Lett. 121, 098003 (2018).