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arXiv:cond-mat/0203477 · Europhysics Letters (EPL)

Ground state of a partially melted Wigner molecule

Z. Á Németh, J.‐L. Pichard

Published Jun 1, 2002• 3 citations
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Abstract

We consider three spinless fermions free to move on 2 d square lattice with periodic boundary conditions and interacting via a U / r Coulomb repulsion. When the Coulomb energy to kinetic energy ratio r s is large, a rigid Wigner molecule is formed. As r s decreases, we show that melting proceeds via an intermediate regime where a floppy two-particle molecule coexists with a partially delocalized particle. A simple ansatz is given to describe the ground state of this mesoscopic solid-liquid regime.

DOI
10.1209/epl/i2002-00412-2

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We consider three spinless fermions free to move on 2 d square lattice with periodic boundary conditions and interacting via a U / r Coulomb repulsion. When the Coulomb energy to kinetic energy ratio r s is large, a rigid Wigner molecule is formed. As r s decreases, we show that melting proceeds via an intermediate regime where a floppy two-particle molecule coexists with a partially delocalized particle. A simple ansatz is given to describe the ground state of this mesoscopic solid-liquid regime.
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For intermediate values of the Coulomb energy to Fermi energy ratio rsr_srs​, the ground state of a few spinless fermions confined on a two dimensional torus is the quantum superposition of a floppy Wigner molecule with delocalized vacancies and of a Fermi liquid of intersticial particles. This raises the question of the existence of an unnoticed liquid-solid phase between the Fermi liquid and the Wigner crystal for fermionic systems in two dimensions.
The quantum-classical crossover from the Fermi liquid towards the Wigner solid is numerically revisited, considering small square lattice models where electrons interact via a Coulomb U/rU/rU/r potential. The studies of models without disorder and spin and including disorder and spin show that the electron solid is formed in two stages, giving rise to an intriguing solid-liquid regime at intermediate couplings
Using a few electrons interacting via a U/r Coulomb repulsion on two dimensional L ×L square lattices with periodic boundary conditions and nearest neighbor hopping t, I review on results of systematic exact finite size numerical studies of the ground state as a function of the Coulomb energy to kinetic energy ratio rs=UL/(2t√πN). Studies of the persistent currents, of the magnetization, of the inter-particle spacing distributions, of a simple analytical ansatz converge towards the same conclusion: there is a new regime for intermediate values rs, where delocalized electrons co-exist with a solid Wigner molecule. The study of this new regime, first done in the presence of random potentials, is extended to the clean lattice limit. The results are consistent, for a finite size lattice, with a possibility proposed by Andreev and Lifshitz for the continuous thermodynamic limit: a quantum crystal may have delocalized defects without melting, the number of sites of the crystalline array being smaller than the total number of particles.
For intermediate Coulomb energy to Fermi energy ratios rsr_srs​, spinless fermions in a two-dimensional random potential form a new quantum phase, different from the Fermi glass (weakly interacting Anderson localized states) and the Wigner crystal (regular array of charges pinned by the disorder). The intermediate phase is characterized by an ordered flow of persistent currents with a typical value decreasing with excitation energy. Extending finite size scaling analysis to the many body ground state, we find that electron-electron interactions can drive the Fermi glass towards an intermediate metallic phase (Coulomb metal).
We have investigated the behavior and stability of a Wigner crystal of rotating dipolar fermions in two dimensions. Using an ansatz wave function for the ground state of rotating two-dimensional dipolar fermions, which occupy only partially the lowest Landau level, we study the correlation energy, elastic moduli and collective modes of Wigner crystals in the lowest Landau level. We then calculate the mean square of the displacement vector of Wigner crystals. The critical filling factor, below which the crystalline state is expected, is evaluated at absolute zero by use of the Lindeman's criterion. We find that the particle (hole) crystal is locally stable for filling factor is less than 1/15 (between filling factors 14/15 and 1), where the stable regime of the crystal is much narrower than the result from Baranov, Fehrmann and Lewenstein, [Phys. Rev. Lett. 100, 200402 (2008)].