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Accepted · first revision · record

Title: Long-range bond-orientational order in 2D particle systems
Plain: 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: 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.
Settled by: A proof of nonzero mean bond orientation for a genuine 2D continuum Gibbs point process at low temperature.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
Status note: Spring models with fixed connectivity are settled (Aumann 2015); continuum particle systems are not.
Posed since: 1968