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arXiv:cond-mat/0305619

Effect of a lattice upon an interacting system of electrons: Breakdown of scaling and decay of persistent currents

Houman Falakshahi, Zoltán Németh, Jean‐Louis Pichard

Published May 27, 2003• 0 citations
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Abstract

For an interacting system of N electrons, we study the conditions under which a lattice model of size L with nearest neighbor hopping t and U/r Coulomb repulsion has the same ground state as a continuum model. For a fixed value of N, one gets identical results when the inter-electron spacing to the Bohr radius ratio r_s < r_s^*. Above r_s^*, the persistent current created by an enclosed flux begins to decay and r_s ceases to be the scaling parameter. Three criteria giving similar r_s^* are proposed and checked using square lattices.

DOI
10.48550/arxiv.cond-mat/0305619

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For an interacting system of N electrons, we study the conditions under which a lattice model of size L with nearest neighbor hopping t and U/r Coulomb repulsion has the same ground state as a continuum model. For a fixed value of N, one gets identical results when the inter-electron spacing to the Bohr radius ratio r_s < r_s^. Above r_s^, the persistent current created by an enclosed flux begins to decay and r_s ceases to be the scaling parameter. Three criteria giving similar r_s^* are proposed and checked using square lattices.
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.
We study the properties of a quarter-filled system of electrons on a square lattice interacting through a local repulsion U and a nearest-neighbor repulsion V in the limit V=+\ensuremath{\infty}. We identify the ground state for U large enough and show that domain walls appear below a critical value Uc{\mathit{U}}_{\mathit{c}}Uc​=4t/\ensuremath{\pi}. We argue that this corresponds to a metal-insulator transition due to Mott localization.
An exact-diagonalization technique on small clusters is used to study the ground states and single-particle excitations of the Hubbard model with on-site (U) and nearest-neighbor (V) Coulombic repulsive interactions. It is shown that the long-range charge-density-wave state realized in a half-filled two-dimensional square lattice for 4V\ensuremath{\gtrsim}U persists up to quarter filling, and that the doped carriers form a small Fermi surface that encloses an area corresponding to the number of doped carriers rather than the total number of electrons involved in the system. The present result offers a well-defined counterexample in the discussion of the Fermi-surface dimensions of strongly correlated electron systems, and may provide a complementary understanding of the large Fermi surface believed to exist in the ordinary Hubbard model.
We propose an effective description of the interaction between the nearest-neighboring particles in a continuum theory. The contributions of the electron-electron interaction to the persistent current in 1D strongly correlating mesoscopic rings with or without impurities are analyzed. It is shown that the nearest-neighborhood int eraction gives significant contributions to the current and correlation functions. The enhance of the theoretical value of current magnitude is observed at finite temperature in the presence of the impurity scattering. The statistical property of the persistent current over random impurity distribution is also discussed. It is found that the exponential law of the persistent current for a non-interacting system will remain in an interacting one, as long as the interactions between nonnearest-neighborhoods are excluded.
We consider 2d gas of spinless fermions with the Coulomb interaction on a lattice at T=0 and at different values of the hopping amplitude J. At small J electrons form a periodic structure. At filling factor ν=1/6 this structure melts at J as low as 0.02--0.03 in units of the nearest-neighbor Coulomb energy. We argue that this transition is connected to the dielectric-metal and dielectric-superconductor transitions. To demonstrate this point we perform computer modeling of the systems 6x6 with 6 and 7 electrons and 6x12 with 12 electrons. By sweeping J we compute simultaneously persistent current and structural characteristics of the electron distribution.