Interacting electrons on a quantum ring: exact and variational approach

Gylfadóttir, Sigríður Sif; Harju, Ari; Jouttenus, T.; Webb, C. E.

doi:10.1088/1367-2630/8/9/211

Cited by 18 publications

(18 citation statements)

References 25 publications

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“…2 It is also routinely used to obtain spectra of model quantum dots, see for example Refs. 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18. The method is based on a projection of the model Hamiltonian onto a finite-dimensional subspace of the many-body Hilbert space in question, hence the method is an instance of the Rayleigh-Ritz method.…”

Section: Introductionmentioning

confidence: 99%

Effective interactions, large-scale diagonalization, and one-dimensional quantum dots

2007

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The widely used large-scale diagonalization method using harmonic oscillator basis functions (an instance of the Rayleigh-Ritz method, also called a spectral method, configuration-interaction method, or "exact diagonalization" method) is systematically analyzed using results for the convergence of Hermite function series. We apply this theory to a Hamiltonian for a one-dimensional model of a quantum dot. The method is shown to converge slowly, and the non-smooth character of the interaction potential is identified as the main problem with the chosen basis, while on the other hand its important advantages are pointed out. An effective interaction obtained by a similarity transformation is proposed for improving the convergence of the diagonalization scheme, and numerical experiments are performed to demonstrate the improvement. Generalizations to more particles and dimensions are discussed. PACS numbers: 73.21.La, 31.15.Pf

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Section: Introductionmentioning

confidence: 99%

Effective interactions, large-scale diagonalization, and one-dimensional quantum dots

2007

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“…Modern microfabrication technology has yielded InGaAs and GaAlAs/GaAs QRs that bind only a few electrons, 24,25 in contrast with the mesoscopic rings on GaAs which hold much larger numbers of electrons. 26 These low-dimensional systems are the subject of considerable scientific interest and have been intensively studied, both experimentally [22][23][24][25][26][27][28][29] and theoretically, 13,19,[30][31][32][33][34][35][36][37][38][39] mainly because of the observation of Aharonov-Bohm oscillations. 35,[40][41][42] As a first approximation, QRs can be modelled by electrons confined to a perfect ring (i.e., δ = 0).…”

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confidence: 99%

Uniform electron gases. I. Electrons on a ring

Loos

Gill

2013

The Journal of Chemical Physics

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We introduce a new paradigm for one-dimensional uniform electron gases (UEGs). In this model, n electrons are confined to a ring and interact via a bare Coulomb operator. We use Rayleigh-Schrödinger perturbation theory to show that, in the high-density regime, the ground-state reduced (i.e., per electron) energy can be expanded as ε(r(s),n)=ε0(n)r(s)(-2)+ε1(n)r(s)(-1)+ε2(n)+ε3(n)r(s+)⋯ , where r(s) is the Seitz radius. We use strong-coupling perturbation theory and show that, in the low-density regime, the reduced energy can be expanded as ε(r(s),n)=η0(n)r(s)(-1)+η1(n)r(s)(-3/2)+η2(n)r(s)(-2)+⋯ . We report explicit expressions for ε0(n), ε1(n), ε2(n), ε3(n), η0(n), and η1(n) and derive the thermodynamic (large-n) limits of each of these. Finally, we perform numerical studies of UEGs with n = 2, 3, [ellipsis (horizontal)], 10, using Hylleraas-type and quantum Monte Carlo methods, and combine these with the perturbative results to obtain a picture of the behavior of the new model over the full range of n and r(s) values.

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“…Many-electron QRs have been investigated theoretically using various methods, such as model Hamiltonian [11][12][13], exact diagonalization [14,15], quantum Monte Carlo [15,16], and density-functional theory [17][18][19][20] (DFT). Accurate numerical calculations on two-electron QRs have been reported in Ref.…”

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confidence: 99%

Exact Wave Functions of Two-Electron Quantum Rings

Loos

Gill

2012

Phys. Rev. Lett.

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We demonstrate that the Schrödinger equation for two electrons on a ring, which is the usual paradigm to model quantum rings, is solvable in closed form for particular values of the radius. We show that both polynomial and irrational solutions can be found for any value of the angular momentum and that the singlet and triplet manifolds, which are degenerate, have distinct geometric phases. We also study the nodal structure associated with these two-electron states. Quantum rings are usually modelled by electrons confined to a strict-or quasi-one-dimensional circular space interacting via a short-ranged or Coulomb operator. In this Letter, we focus on the simple system in which two electrons are confined to a ring of radius R and interact via a Coulomb operator. This choice has often been avoided in the literature due to the divergence of the Coulomb interaction at small interelectronic distances.Contrary to frequent claims, systems with two electrons do not inevitably have intractable Schrödinger equations and we show here that, for each electronic state of a two-electron QR, the Schrödinger equation can be solved exactly for a countably infinite set of R values, yielding both polynomial and irrational solutions in terms of the interelectronic distance. Quantum mechanical systems whose Schrödinger equations can be solved in this way, such as the Hooke's law [22] or spherium [23,24] atoms, have ongoing value both for illuminating more complicated systems [25,26] and for testing and developing theoretical approaches, such as DFT [27][28][29][30] and explicitly correlated methods [31].In atomic units ( = m = e = 1), the Hamiltonian of two electrons on a ring of radius R iŝ

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Interacting electrons on a quantum ring: exact and variational approach

Cited by 18 publications

References 25 publications

Effective interactions, large-scale diagonalization, and one-dimensional quantum dots

Effective interactions, large-scale diagonalization, and one-dimensional quantum dots

Uniform electron gases. I. Electrons on a ring

Exact Wave Functions of Two-Electron Quantum Rings

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