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arXiv:0712.3819 · Physical Review B

Entanglement and density-functional theory: Testing approximations on Hooke’s atom

J. P. Coe, Anthony Sudbery, Irene D’Amico

Published May 22, 2008• 98 citations
arXivHTMLPDF

Abstract

We present two methods of calculating the spatial entanglement of an interacting electron system within the framework of density-functional theory. These methods are tested on the model system of Hooke's atom for which the spatial entanglement can be exactly calculated. We analyze how the strength of the confining potential affects the spatial entanglement and how accurately the methods that we introduced reproduce the exact trends. We also compare the results to the outcomes of standard first-order perturbation methods. The accuracies of energies and densities when using these methods are also considered.

DOI
10.1103/physrevb.77.205122

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We present two methods of calculating the spatial entanglement of an interacting electron system within the framework of density-functional theory. These methods are tested on the model system of Hooke's atom for which the spatial entanglement can be exactly calculated. We analyze how the strength of the confining potential affects the spatial entanglement and how accurately the methods that we introduced reproduce the exact trends. We also compare the results to the outcomes of standard first-order perturbation methods. The accuracies of energies and densities when using these methods are also considered.
In this chapter we discuss methods to calculate the entanglement of a system using density-functional theory. We firstly introduce density-functional theory and the local-density approximation (LDA). We then discuss the concept of the 'interacting LDA system'. This is characterised by an interacting many-body Hamiltonian which reproduces, uniquely and exactly, the ground state density obtained from the single-particle Kohn-Sham equations of density-functional theory when the local-density approximation is used. We motivate why this idea can be useful for appraising the local-density approximation in many-body physics particularly with regards to entanglement and related quantum information applications. Using an iterative scheme, we find the Hamiltonian characterising the interacting LDA system in relation to the test systems of Hooke's atom and helium-like atoms. The interacting LDA system ground state wavefunction is then used to calculate the spatial entanglement and the results are compared and contrasted with the exact entanglement for the two test systems. For Hooke's atom we also compare the entanglement to our previous estimates of an LDA entanglement. These were obtained using a combination of evolutionary algorithm and gradient descent, and using an LDA-based perturbative approach. We finally discuss if the position-space information entropy of the density—which can be obtained directly from the system density and hence easily from density-functional theory methods—can be considered as a proxy measure for the spatial entanglement for the test systems.
We calculate the 'exact' potential corresponding to a one-dimensional interacting system of two electrons with a specific, tailored density. We use one-dimensional density-functional theory with a local-density approximation (LDA) on the same system and calculate densities and energies, which are compared with the 'exact' ones. The 'interacting-LDA system'[4] corresponding to the LDA density is then found and its potential compared with the original one. Finally we calculate and compare the spatial entanglement of the electronic systems corresponding to the interacting-LDA and original interacting system.
The system of two interacting electrons in one-dimensional harmonic potential or Hooke’s atom is considered, again. On one hand, it appears as a model for quantum dots in a strong confinement regime, and on the other hand, it provides us with a hard test bench for new methods with the “space splitting” arising from the one-dimensional Coulomb potential. Here, we complete the numerous previous studies of the ground state of Hooke’s atom by including the excited states and dynamics, not considered earlier. With the perturbation theory, we reach essentially exact eigenstate energies and wave functions for the strong confinement regime as novel results. We also consider external perturbation induced quantum dynamics in a simple separable case. Finally, we test our novel numerical approach based on real-time path integrals (RTPIs) in reproducing the above. The RTPI turns out to be a straightforward approach with exact account of electronic correlations for solving the eigenstates and dynamics without the conventional restrictions of electronic structure methods.
We explore the main entanglement properties exhibited by the eigenfunctions of two exactly soluble two-electron models, the Crandall atom and the Hooke atom, and compare them with the entanglement features of helium-like systems. We compute the amount of entanglement associated with the wavefunctions corresponding to the fundamental and first few excited states of these models. We investigate the dependence of the entanglement on the parameters of the models and on the quantum numbers of the eigenstates. It is found that the amount of entanglement of the system tends to increase with energy in both models. In addition, we study the entanglement of a few states of helium-like systems, which we compute using high-quality Kinoshita-like eigenfunctions. The dependence of the entanglement of helium-like atoms on the nuclear charge and on energy is found to be consistent with the trends observed in the previous two model systems.
Providing sufficient accuracy at low-computational cost, not surprisingly, density-functional theory ( DFT ) is a standard tool in theoretical material science. Similarly, quantum-electrodynamical density-functional theory ( QEDFT ) provides a promising route to describe experiments at the interface of quantum optics and quantum chemistry. Formally, in the Kohn-Sham ( KS ) formulation of DFT and QEDFT the exact many-particle Schrödinger equation is replaced by a set of coupled non-linear one-particle differential equations. The electron-electron, the electron-ion and the electron-photon interactions of the interacting system are mimicked by their corresponding effective KS potential with an, in principle unknown exchange-correlation ( xc ) term. In practice, DFT and QEDFT require approximations for this xc potential. The quality of which fully determines how much the DFT and QEDFT observables deviate from the exact ones. Understanding the shortcomings and improving the quality of approximate functionals makes the study of the exact functional of model systems indispensable. Here, we employ exact diagonalization for the Hamiltonian of different types of real-space lattice models in one spatial dimension to access the ground-state wave function, and thus to explicitly construct the exact functionals of DFT and QEDFT . To study the functional behavior of purely electronic systems, we consider two singlet-state electrons moving in an external potential with two connected atomic wells of variable depth and distance. To study coupled electron-photon systems, we consider the Rabi Hamiltonian with variable electron-photon coupling strength. By varying the model parameters, we systematically construct the functionals for systems from weak to strong coupling between the building blocks of the global system. Using the concept of the effective density-to- potential map, we discuss the intrasystem steepening which is an exact condition on the xc functional that captures the static correlation present in the system and carve out how it relates to the well-known intersystem derivative discontinuity. We further construct arbitrary observables as functionals of the internal variables which are directly affected by such features of the effective density-to-potential map. By constructing the reduced dipole map in the four-particle case of coupled electron-photon system, we show that the density-to-potential map helps to identify and understand recently discovered physical effects in the field of strongly coupled electron-photon systems. This work provides a contribution to the development of approximate function- als required in practical density-functional calculations of arbitrary experimentally accessible observables. Both the intrasystem steepening and the intersystem derivative discontinuity play a crucial role for systems with highly-localized electrons. Including both features in approximate functionals will improve the accu- racy of DFT evaluation of observables for such systems. Those results provide novel insight into the properties of the exact xc potentials in QEDFT . This work opens a new route to construct new functionals using coarse-grained or embed- ding approaches within the general and exact frameworks of DFT and QEDFT.
There are very few systems of interacting particles (with continuous variables) for which the entanglement of the concomitant eigenfunctions can be computed in an exact, analytical way. Here we present analytical calculations of the amount of entanglement exhibited by s -states of spherium . This is a system of two particles (electrons) interacting via a Coulomb potential and confined to a ( d −1)-sphere (that is, to the surface of a d -dimensional ball). We investigate the dependence of entanglement on the radius R of the system, on the spatial dimensionality d , and on energy. We find that entanglement increases monotonically with R , decreases with d , and also tends to increase with the energy of the eigenstates. These trends are discussed and compared with those observed in other two-electron atomic-like models where entanglement has been investigated.