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arXiv:cond-mat/0511075 · New Journal of Physics

Interacting electrons on a quantum ring: exact and variational approach

S S Gylfadottir, Ari Harju, Teppo T. Jouttenus, C Webb

Published Sep 27, 2006• 16 citations
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Abstract

We study a system of interacting electrons on a one-dimensional (1D) quantum ring using exact diagonalization (ED) and the variational quantum Monte Carlo (VMC) method. We examine the accuracy of the Slater–Jastrow-type many-body wavefunction and compare energies and pair distribution functions obtained from the two approaches. Our results show that this wavefunction captures most correlation effects. We then study the smooth transition to a regime where the electrons localize in the rotating frame, which for the ultra-thin quantum ring system happens at quite high electron density.

DOI
10.1088/1367-2630/8/9/211

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We study a system of interacting electrons on a one-dimensional (1D) quantum ring using exact diagonalization (ED) and the variational quantum Monte Carlo (VMC) method. We examine the accuracy of the Slater–Jastrow-type many-body wavefunction and compare energies and pair distribution functions obtained from the two approaches. Our results show that this wavefunction captures most correlation effects. We then study the smooth transition to a regime where the electrons localize in the rotating frame, which for the ultra-thin quantum ring system happens at quite high electron density.
We study a system of interacting electrons on a one-dimensional quantum ring using exact diagonalization and the variational quantum Monte Carlo method. We examine the accuracy of the Slater-Jastrow -type many-body wave function and compare energies and pair distribution functions obtained from the two approaches. Our results show that this wave function captures most correlation effects. We then study the smooth transition to a regime where the electrons localize in the rotating frame, which for the ultrathin quantum ring system happens at quite high electron density.
Variational quantum Monte Carlo (VMC) method applied to interacting electron systems has played a pivotal role for the studies of electronic states of atoms, molecules and condensed matters. First, the many-body wave functions optimized by the VMC method provide good guiding functions in more accurate, diffusion quantum Monte Carlo (DMC) calculations for identical systems. Second, the VMC calculations themselves can give a good upper bound for the exact ground-state energy of many-electron systems when an appropriate form for the variational wave function is employed. In addition to a conventional form for the variational wave function that is expressed by the product of the Jastrow factor and the single Slater determinant, recent investigations have found that the expression in terms of many Slater determinants can provide a very accurate description of electronic states, especially for intrinsically multi-reference systems, which corresponds to the configuration interaction (CI) expansion method in conventional quantum chemistry. Although the CI method needs an extremely large number of expansion coefficients as variational parameters, the number of parameters can be reduced efficiently by using an idea of (molecular) orbital pair correlations. The purpose of the present paper is to examine the feasibility and usefulness of this idea within the framework of VMC method with the Jastrow–Slater wave function, where the ground state of a water molecule is investigated as a test case. In this study we rely on a variational wave function composed of a product of the Jastrow and Slater parts as 1⁄4 J S; ð1Þ
We consider few-electron concentric double quantum rings with parabolic confining potential and compare the ground-state energies calculated by exact diagonalization of the Hamiltonian, accurate quantum Monte Carlo, and local spin-density functional approaches. Electronic localization in one of the rings and the formation of rotating Wigner molecules is shown, respectively, from the one-body and the two-body density operators. As the confinement strength of the outer ring is finely increased, the circularly symmetric electron density exhibits a radial crossover from this ring to the inner one without altering the angular character of the system.
Abstract In this work, the computational method of Variational Monte Carlo is used to study the ground state properties of electron-hole (e-h) bi-wire systems. We have calculated the pair correlation function and static structure factor of a coupled infinitely thin e-h wire system of electronic density (rs= 1) and inter-wire distance (d = 1). The numerical calculation is carried out using the many-body trial wave function of the Slater-Jastrow backflow type. It is observed that the inter-wire correlations become quite important in the case of the mass-symmetric electron-hole bi-wire system due to the coulomb attractive nature of the bi-wire system.
We develop a variational Monte Carlo (VMC) method for electron-phonon coupled systems. The VMC method has been extensively used for investigating strongly correlated electrons over the last decades. However, its applications to electron-phonon coupled systems have been severely restricted because of its large Hilbert space. Here, we propose a variational wave function with a large number of variational parameters, which is suitable and tractable for systems with electron-phonon coupling. In the proposed wave function, we implement an unexplored electron-phonon correlation factor, which takes into account the effect of the entanglement between electrons and phonons. The method is applied to systems with diagonal electron-phonon interactions, i.e., interactions between charge densities and lattice displacements (phonons). As benchmarks, we compare VMC results with previous results obtained by the exact diagonalization, the Green function Monte Carlo method and the density matrix renormalization group for the Holstein and Holstein-Hubbard model. From these benchmarks, we show that the present method offers an efficient way to treat strongly coupled electron-phonon systems.
We present a controlled method for computing the exchange coupling in correlated one-dimensional electron systems based on the relation between the exchange constant and the pair-correlation function of spinless electrons. This relation is valid in several independent asymptotic regimes, including the low-electron-density case, under the general condition of a strong spin-charge separation. Explicit formulas for the exchange constant are obtained for thin quantum rings and wires with realistic Coulomb interactions by calculating the pair-correlation function via a many-body instanton approach. A remarkably smooth interpolation between high- and low-electron-density results is shown to be possible. These results are applicable to the case of one-dimensional wires of intermediate width as well. Our method can be easily generalized to other interaction laws, such as the inverse distance squared one of the Calogero-Sutherland-Moser model. We demonstrate excellent agreement with the known exact results for the latter model and show that they are relevant for a realistic experimental setup in which the bare Coulomb interaction is screened by an edge of a two-dimensional electron gas.