MATHPH
In the literature: open
Power-law decay of positional correlations in 2D crystals
In plain words
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.
Precise statement
For a 2D continuum particle system with triangular ground state at low T, prove that the translational correlation $\langle \rho_G(x) \rho_G(0)*\rangle$, $G$ a reciprocal lattice vector of the triangular lattice, decays as $\mid x\mid^{-\eta_G(T)}$ with $\eta_G(T)\to 0$ as $T\to 0$ (quasi-long-range order), with matching upper and lower power-law bounds.
What would settle it
Upper and lower power-law bounds on positional correlations with exponent vanishing as $T \to 0$.
Status in the literature
Absence of true positional order for 2D continuum particles is proved (Richthammer 2007).