MATHPH In the literature: open

Broken translation symmetry for a 3D continuum particle system

In plain words

Prove that some realistic system of particles in continuous 3D space has, at low temperature, a thermal equilibrium state in which particles keep preferred lattice positions over infinite distances.

Precise statement

For classical particles in $R^3$ with a stable, regular pair potential $V$ (for example Lennard-Jones, or a smooth short-range potential with a crystalline ground state), prove that for large $\beta = 1/T$ at suitable density there is an infinite-volume Gibbs point process that is not translation invariant, with density correlations showing long-range periodic order. One such potential suffices.

What would settle it

A construction, for example by a multiscale or Pirogov-Sinai-type expansion around the crystal, of a non-translation-invariant Gibbs measure for one continuum potential in 3D.

Status in the literature

Unverified note

Proved for lattice models via Pirogov-Sinai theory, and positional order is ruled out for 2D continuum systems (Richthammer 2007); as of 2026 no continuum system in $d \ge 3$ has a proof of a crystalline Gibbs state.