Andreev-Lifshitz supersolid revisited for a few electrons on a square lattice. I
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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 L×L 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 rs = UL/(2t\(\)) reaches a value rsF ≈ 10, there is a level crossing between ground states of different momenta. Above rsF, the mesoscopic crystallization proceeds through an intermediate regime ( rsF < rs < rsW ≈ 28) 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 rsF is avoided and gives rise to a lower threshold rsF(W) < rsF 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.
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