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Journal of Physics Conference Series

Approximation of the entanglement in quantum dot chains using Hubbard models

J. P. Coe, V. V. França, Irene D’Amico

Published Mar 1, 2011• 3 citations
Publisher

Abstract

We investigate the Hubbard model, and the extended Hubbard model with and without correlated hopping, as an approximation to the average single-site (or local) entanglement of two-electron one-dimensional quantum dot chains with long range interactions. We focus on the effect of the chain length on the approximation accuracy. We also consider the particle-particle spatial entanglement of the three Hubbard models using the technique of Ref. [3] and compare the accuracy of this approximation to the numerically exact results of the nanostructure system.

DOI
10.1088/1742-6596/286/1/012048

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•Jan 1, 2018•0 citations
We investigate the Hubbard model, and the extended Hubbard model with and without correlated hopping, as an approximation to the average single-site (or local) entanglement of two-electron one-dimensional quantum dot chains with long range interactions. We focus on the effect of the chain length on the approximation accuracy. We also consider the particle-particle spatial entanglement of the three Hubbard models using the technique of Ref. [3] and compare the accuracy of this approximation to the numerically exact results of the nanostructure system.
We investigate the extended Hubbard model as an approximation to the local and spatial entanglement of a one-dimensional chain of nanostructures where the particles interact via a long-range interaction represented by a "soft" Coulomb potential. In the process we design a protocol to calculate the particle-particle spatial entanglement for the Hubbard model and show that, in striking contrast with the loss of spatial degrees of freedom, the predictions are reasonably accurate. We also compare results for the local entanglement with previous results found using a contact interaction ( Coe J. P. et al ., Phys. Rev. A , 81 (2010) 052321 ) and show that while the extended Hubbard model recovers a better agreement with the entanglement of a long-range interacting system, there remain realistic parameter regions where it fails to predict the quantitative and qualitative behaviour of the entanglement in the nanostructure system.
We investigate how well the one-dimensional Hubbard model describes the entanglement of particles trapped in a string of quantum wells. We calculate the average single-site entanglement for two particles interacting via a contact interaction and consider the effect of varying the interaction strength and the interwell distance. We compare the results with the ones obtained within the one-dimensional Hubbard model with on-site interaction. We suggest an upper bound for the average single-site entanglement for two electrons in MMM wells and discuss analytical limits for very large repulsive and attractive interactions. We investigate how the interplay between interaction and potential shape in the quantum-well system dictates the position and size of the entanglement maxima and the agreement with the theoretical limits. Finally, we calculate the spatial entanglement for the quantum-well system and compare it to its average single-site entanglement.
We consider the position-space information and linear entropies as proxy measures to the average single-site entanglement–quantified using the von Neumann entropy–of the one-dimensional Hubbard model and of a one-dimensional nanostructure system comprised of an array of quantum-dots. Spatial entanglement in the quantum-dot system is also investigated via the three entropies. We appraise the use of the possible proxy measures in the Hubbard model as an approximation to their use for the nanostructure system.
The dimerized one-dimensional Hubbard model is studied in the framework of lattice density-functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable. The corresponding interaction-energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is defined by Levy's constrained search. Exact numerical results are obtained for $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23}),$ where ${\ensuremath{\gamma}}_{12}={\ensuremath{\gamma}}_{i,i+1}$ for odd i and ${\ensuremath{\gamma}}_{23}={\ensuremath{\gamma}}_{i,i+1}$ for even i are the nearest-neighbor density-matrix elements along the chain. The domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}}$ and the functional dependence of $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ are analyzed. A simple, explicit approximation to $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ is proposed, which is derived from scaling properties of W, exact dimer results, and known limits. Using this approximation, LDFT is applied to determine ground-state properties and charge-excitation gaps of finite and infinite dimerized chains as a function of the Coulomb-repulsion strength U/tU/tU/t and of the alternation $\ensuremath{\delta}t$ of the hopping integrals tij{t}_{\mathrm{ij}}tij​ ${(t}_{\mathrm{ij}}=t\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}t).$ The accuracy of the method is demonstrated by comparison with available exact solutions and accurate numerical calculations. Goals and limitations of the present approach are discussed particularly concerning its ability to describe the crossover from weak to strong electron correlations.
We investigate the entanglement properties in semiconductor quantum dot systems modeled by the extended Hubbard model, focusing on the impacts of potential energy variations and electron interactions within a four-site quantum dot spin chain. Our study explores local and pairwise entanglement across configurations with electron counts N=4 and N=6, under different potential energy settings. By adjusting the potential energy in specific dots and examining the entanglement across various interaction regimes, we identify significant variations in the ground states of quantum dots. We extend this analysis to larger systems with L=6 and L=8, comparing electron counts N=L and N=L+2, revealing sharper entanglement transitions and reduced finite-size effects as the system size increases. Our results show that local potential shifts and the Coulomb interaction strength lead to notable redistributions of the electron configurations in the quantum dot spin chain, significantly affecting the entanglement properties. These changes are depicted in phase diagrams that highlight entanglements' dependencies on the interaction strengths and potential energy adjustments, illustrating complex entanglement dynamics shifts triggered by interdot interactions.
Describing many-body quantum systems has been an analytically and computationally challenging task since the advent of quantum mechanics. However, in the past 50 years as a result of our technological advancement and the emergence of methods such as density-functional theory (DFT), we have taken crucial steps forward regarding our ability to study and understand large quantum systems. In this work,we have studied the extended Hubbard model using a mean-field approximation where we have tested an approach to the long-ranged electronic interaction that is not entirely local. We have developed a near-local approximation (NLA) for the model under study, where the exact non-local electronic interaction at site i is approximated using the densities at a pair of sites neighbouring i in addition to the density at i. As preliminary to a discussion of NLA, we showed results for the density of states of systems with a three-site supercell, thus providing a simple characterization of the mean-field treatment (in the case of local interactions only). When it comes to the main results of our work, i.e. a derivation and the testing of the NLA, our findings can be summarised as follows: we have found three NLA variants, namely a left site approximation (LSA), a right site approximation (RSA) and a center site approximation (CSA). Furthermore, we have found that CSA performs better than a local approximation both under the effect of a parabolic external potential and a distorted one. We have also found that the performance of LSA and RSA are scarce in general. Albeit this work (being based on the extended Hubbard model under a mean-field approximation) does not address exchange and correlation effects directly, it provides a first step towards future work where a near-local treatment is carried out on more accurate grounds.