Eur. Phys. J. B 31, 401-412 (2003)
https://doi.org/10.1140/epjb/e2003-00048-0

Andreev-Lifshitz supersolid revisited for a few electrons on a square lattice. I

CEA/DSM, Service de Physique de l'État Condensé, Centre d'Études de Saclay, 91191, Gif-sur-Yvette Cedex, France

Corresponding author: ^a jpichard@cea.fr

Received: 21 June 2002
Published online: 14 February 2003

Abstract

In 1969, Andreev and Lifshitz have conjectured the existence of a supersolid phase taking place at zero temperature between the quantum liquid and the solid. In this and a succeeding paper, we re-visit this issue for a few polarized electrons (spinless fermions) interacting via a U/r Coulomb repulsion on a two dimensional square lattice with periodic boundary conditions and nearest neighbor hopping t. This paper is restricted to the magic number of particles N=4 for which a square Wigner molecule is formed when U increases and to the size L=6 suitable for exact numerical diagonalizations. When the Coulomb energy to kinetic energy ratio reaches a value , there is a level crossing between ground states of different momenta. Above , the mesoscopic crystallization proceeds through an intermediate regime () where unpaired fermions with a reduced Fermi energy co-exist with a strongly paired, nearly solid assembly. We suggest that this is the mesoscopic trace of the supersolid proposed by Andreev and Lifshitz. When a random substrate is included, the level crossing at  is avoided and gives rise to a lower threshold where two usual approximations break down: the Wigner surmise for the distribution of the first energy excitation and the Hartree-Fock approximation for the ground state.