Crystalline and fluid order on a random topography

Abstract

The statistical mechanics of particles embedded in a surface with quenched fluctuations in its topography is considered. If the fluctuations are not too violent, stable crystalline phases are possible at finite temperatures, with elastic constants renormalised from their flat-space values. Point dislocations and disclinations couple only to the intrinsic gaussian curvature of the surface. Other effects of the surface can be gauged away, just as in Mattis models of spin glasses. At sufficiently low temperatures, crystalline arrays must melt re-entrantly via a dislocation instability. The resulting hexatic phase is also unstable at low temperatures. The random-topography problem is similar in many respects to that of particles in flat space disrupted by a quenched random array of impurities.