Quantum fluid-solid transition in a simple variational approach

A determinant (permanent) of plane waves for N fermions (bosons) in their lowest-energy state, and a determinant (permanent) of spatially-localized, nonoverlapping, single-particle functions are used to determine conditions on the short-ranged, square-well repulsive plus Yukawa attractive, two-body potential for which the spatially homogeneous or periodic particle-density states are energetically preferred. A considerable region in the coupling parameter-density plane is found where “crystalline” solutions are preferred. In particular, for purely repulsive forces, the low-density region for which the “fluid” solution is lowest shrinks to zero in the limit of infinitely repulsive cores, but faster for bosons than fermions. Inclusion of attractive forces is seen not to alter qualitatively the main results.