Long-range bond-orientational order in 2D particle systems
In plain words
In two dimensions thermal vibrations destroy perfect positional order, but the directions of the bonds between neighbors should still line up across the whole sample. This is proved only for simplified spring models.
Precise statement
For a 2D classical particle system with a smooth pair potential whose ground state is triangular, prove at low T a Gibbs state in which the bond-orientation field $\psi_6(x)=\sum \text{ over neighbors of } \exp(6 i \theta)$ has nonzero mean, $\theta$ the bond angle. Merkl, Rolles (2009), Heydenreich, Merkl, Rolles (2014) and Aumann (2015) prove orientational order for harmonic-spring models on a fixed triangulation, with limited or arbitrary defects; the case where particles move freely and neighbors change is open.
What would settle it
A proof of nonzero mean bond orientation for a genuine 2D continuum Gibbs point process at low temperature.
Status in the literature
Spring models with fixed connectivity are settled (Aumann 2015); continuum particle systems are not.