Melting Transition of Hard Disks : the Solution of a Fiftyyear Old Puzzle

For fifty years, the issue has raged about the nature of the melting transition in two-dimensional systems of hard disks : is it like in three dimensions, by an abrupt transition (first order), or by a mechanism of Kosterlitz –Thouless type with continuous changes between the liquid and a state called "hexatic" and between the hexatic and solid ? Etienne P. Bernard and Werner Krauth have shown with a new simulation algorithm that the melting transition follows neither one of the two scenarios studied for such a long time : it indeed proceeds through a hexatic phase, but the liquid-hexatic transition is first order, while the hexatic-solid transition is continuous.

The hard sphere model plays a fundamental role in statistical physics. Introduced by D. Bernoulli in 1738, it simply consists in impenetrable disks of equal size filling a finite fraction of the two-dimensional space, and it has been studied by leading physicists such as Boltzmann. In 1962, numerical simulations by Alder and Wainwright showed to everyone’s surprise that hard disks underwent a phase transition between a liquid (low density) and a two-dimensional solid (high density) although in two dimensions, no crystalline state can exist. For fifty years, the issue has raged about how exactly the melting transition was taking place in this generic system : was it merely, just like in three dimensions, by an abrupt transition (first order), or by a typical two-dimensional mechanism of Kosterlitz Thouless-type with continuous changes between the liquid and a state called "hexatic" and then between the hexatic state and the solid ? Hundreds of articles have been written, but the researchers could not agree on one or the other scenarios ...

In 2011, Etienne P. Bernard and Werner Krauth, in the Statistical Physics Laboratory at Ecole normale supérieure (CNRS-ENS-UPMC) showed (with a new simulation algorithm) that this venerable model was actually hiding a big secret :

its melting transition did not follow any of the two scenarios studied so long : the fusion of hard disks indeed proceeds through a hexatic phase, however the liquid-hexatic transition is a first order one, while the hexatic-solid transition is continuous. Before accepting this stunning solution, it was still necessary to test it independently. This is now achieved with a collaboration involving researchers on three continents (University of Michigan, MIT, Nagoya Institute of Technology, Japan, ENS). Using three completely independent methods (including the Metropolis algorithm on massively parallel graphic cards and an ultra-fast algorithm for molecular dynamics), this collaboration was able to confirm all the results of 2011. With this fundamental result and its clear confirmation, we finally understand the fusion of the simplest two-dimensional system, and hold in hand a key to understanding the melting transition in films, thin layers, and interfaces.

Original article (2011) :
E. P. Bernard, W. Krauth "First-order liquid-hexatic transition in hard
disks" Physical Review Letters 107, 155704 (2011)

M. Engel, J. A. Anderson, S. C. Glotzer, M. Isobe, E. P. Bernard, W. Krauth « Hard-disk equation of state : First-order liquid-hexatic transition in two dimensions with three simulation methods » Phys. Rev. E 87, 042134 (2013)

Contact : Werner KRAUTH +33 6 76 32 10 61
werner.krauth@ens.fr

Web page : http://www.lps.ens.fr/~krauth/ (contains related articles).