Crystalline order at positive temperature

5 problems

• Broken translation symmetry for a 3D continuum particle system

  Open

  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.

• Crystalline phase of hard spheres at high packing fraction

  Open

  Hard balls with no attraction freeze into a crystal when packed densely enough, as computer simulations show above about 55 percent filling. Prove that such a crystalline equilibrium state exists.

• Existence of a hexatic phase between 2D solid and liquid

  Open

  Theory predicts that 2D crystals melt in two steps through an intermediate hexatic phase, in which bond directions stay aligned while positions are disordered as in a liquid. Prove such a phase exists for some particle system.

• Long-range bond-orientational order in 2D particle systems

  Open

  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.

• Power-law decay of positional correlations in 2D crystals

  Open

  Theory predicts that in a 2D crystal at low temperature positional order fades slowly, as a power of distance, instead of quickly. Prove this power law for a particle system.