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arXiv:cond-mat/0104125

The simulation of the spin ground states of the coulomb clusters in a broad 2D parabolic well

B. Abdullaev, A. D. Kidisyuk, M.M. Musakhanov

Published Apr 7, 2001• 0 citations

Abstract

By variational Monte-Carlo method developed Ceperley et al. for the simulation of fermi systems in macroscopic confining potential well we simulate various spin ground states of the coulomb clusters with 2,3 and 4 particles in a broad two-dimensional (2D) parabolic well. In this method quantum state numbers determining the variational wave function are not the numbers of well quantum states but numbers of the equilibrium spatial positions of particles that give a minimum of the system potential energy. The ground states with parallel, antiparallel spins and as well, as bose state are simulated. For the cluster with three particles it is also simulated the state when two particles have one direction of spin and third opposite. The simulation shows that clusters with parallel spins have lower ground state energy than clusters with other spin configurations and bose state. That reminds a Hund's rule in atomic physics when in not full filled atomic shells electrons prefer to have a state with parallel spins.

DOI: 10.48550/arxiv.cond-mat/0104125

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The ground state of a two-dimensional charged quantum system is studied using the method of correlated-basis functions. For particles obeying the Bose-Einstein statistics we compute the ground-state energy variationally for all particle densities using the Bijl-Dingle-Jastrow trial wave function. The Bose ground-state energy and wave function so determined are used next to study the Fermi system. We compute the Fermi ground-state energy using a cluster expansion approach, which is again valid for all particle densities, and find that the ground state is paramagnetic. While these calculations provide variational upper bounds to the ground-state energies, we also establish independent lower bounds to the ground-state energies.

The Pad\'e approximant technique and the variational Monte Carlo method are applied to determine the ground-state energy of a finite number of charged bosons in two dimensions confined by a parabolic trap. The particles interact repulsively through a Coulombic

1/r

potential. Analytic expressions for the ground-state energy are obtained. The convergence of the Pad\'e sequence and comparison with the Monte Carlo results show that the error of the Pad\'e estimate is less than 4% at any boson density and is exact in the extreme situations of very dilute and high density.

The ground-state properties of a system with a small number of interacting bosons over a wide range of densities are investigated. The system is confined in a two-dimensional isotropic harmonic trap, where the interaction between bosons is treated as a hard-core potential. By using variational Monte Carlo method, we diagonalize the one-body density matrix of the system to obtain the ground-state energy, condensate wavefunction and the condensate fraction. We find that in the dilute limit the depletion of central condensate in the 2D system is larger than in a 3D system for the same interaction strength; however as the density increases, the depletion at the centre of 2D trap will be equal to or even lower than that at the centre of 3D trap, which is in agreement with the anticipated in Thomas–Fermi approximation. In addition, in the 2D system the total condensate depletion is still larger than in a 3D system for the same scattering length.

The ground-state properties of a single-component one-dimensional Coulomb gas are investigated. We use Bose-Fermi mapping for the ground-state wave function which permits solution of the Fermi sign problem in the following respects: (i) the nodal surface is known, permitting exact calculations; and (ii) evaluation of determinants is avoided, reducing the numerical complexity to that of a bosonic system and, thus, allowing simulation of a large number of fermions. Due to the mapping, the energy and local properties in one-dimensional Coulomb systems are exactly the same for Bose-Einstein and Fermi-Dirac statistics. The exact ground-state energy is calculated in homogeneous and trapped geometries using the diffusion Monte Carlo method. We show that in the low-density Wigner crystal limit an elementary low-lying excitation is a plasmon, which is to be contrasted with the high-density ideal Fermi gas/Tonks-Girardeau limit, where low-lying excitations are phonons. Exact density profiles are compared to the ones calculated within the local density approximation, which predicts a change from a semicircular to an inverted parabolic shape of the density profile as the value of the charge is increased.