Feasibility of approximating spatial and local entanglement in long-range interacting systems using the extended Hubbard model
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Feasibility of approximating spatial and local entanglement in long-range interactingsystems using the extended Hubbard model
J. P. Coe , ∗ V. V. Fran¸ca , † and I. D’Amico ‡ The Department of Physics, The University of York, Heslington, York, UK. Physikalisches Institut, Albert-Ludwigs-Universit¨at,Hermann-Herder-Straße 3, D-79104 Freiburg, Germany.
We investigate the extended Hubbard model as an approximation to the local and spatial entan-glement of a one-dimensional chain of nanostructures where the particles interact via a long rangeinteraction represented by a ‘soft’ Coulomb potential. In the process we design a protocol to cal-culate the particle-particle spatial entanglement for the Hubbard model and show that, in strikingcontrast with the loss of spatial degrees of freedom, the predictions are reasonably accurate. Wealso compare results for the local entanglement with previous results found using a contact inter-action [1] and show that while the extended Hubbard model recovers a better agreement with theentanglement of a long-range interacting system, there remain realistic parameter regions where itfails to predict the quantitative and qualitative behaviour of the entanglement in the nanostructuresystem.
PACS numbers: 03.67.Bg, 71.10.Fd, 73.21.La
I. INTRODUCTION
The properties of quantum dot-based nanostructuresmake them possible candidates as hardware for the ma-nipulation of quantum information [2–6]. Chains of quan-tum dots (QDs) have been proposed to transfer quan-tum information [7], and to generate, distribute, andfreeze entanglement [8, 9]. Entanglement is in fact con-sidered a fundamental resource for quantum informationprocesses and is increasingly gaining the attention of thecondensed-matter community as a probe for delicate phe-nomena, such as quantum phase transitions [10].The Hubbard model [11] is a simplified model of itin-erant interacting fermions with positions discretized toa lattice, and usually only includes interaction within alattice site, while extended Hubbard models may includelong-range interactions. The calculation of the entan-glement in a realistic system of many fermions, such aselectrons trapped in a nanostructure, is usually compu-tationally too demanding. In this respect, proving theHubbard model – or a variant of the Hubbard model –accurate as an approximation to the entanglement of arealistic many-fermion system would open the possibilityof using density-functional theory techniques to calculatethe entanglement of these complex systems: a powerfullocal-density approximation approach is in fact availablefor calculating the entanglement of the Hubbard model[12].In previous work [1] the accuracy of the 1D Hubbardmodel (HM) as an approximation to the average local en-tanglement of two-electrons trapped in a chain of quan-tum dots (QDs) was considered. In [1] we considered a ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] contact interaction between particles and found the HMto be accurate for single-site entanglement calculations,but the spatial entanglement could not be estimated asthere was no scheme for such entanglement measurementin the HM.In this paper we consider a similar system but witha more realistic long range particle-particle interaction,which we model as a ‘soft’ Coulomb potential. The accu-racy of the local entanglement of the 1D extended Hub-bard model (EHM) [13, 14] and the EHM with correlatedhopping (EHM+CH) as an approximation to that of theQD structure is then appraised. In addition we propose amethod to calculate the spatial entanglement of the HMand EHM and compare this somewhat severe approxi-mation with the spatial entanglement of the QD system.Surprisingly our results show that even if in the HM andEHM cases the number of spatial degrees of freedom arereduced to just a few lattice sites, the HM and EHM spa-tial entanglement still capture most of the features andtrends of the spatial entanglement of the quantum dotstructure, and it does so for both attractive and repul-sive interactions.
II. THE QD-CHAIN TWO-PARTICLE SYSTEM
We consider a system of two fermions trapped withina one-dimensional QD-chain. In effective atomic units,the Hamiltonian is H QD = X s = a,b (cid:18) − d dx s + v ( x s ) (cid:19) + C U f ( | x a − x b | ) . (1)Here v ( x s ) is the potential used to represent an array ofregularly spaced, identical square wells, each well repre-senting a QD. The chain is symmetric around the origin,and defined by the quantities: M the number of wells, d the barrier width between two consecutive wells, and w and v the width and depth of each well respectively. Wedefine and vary the interaction strength C U to facilitatecomparison of the QD system with the HM. The inter-action type we consider is either a contact interaction, f ( | x a − x b | ) = δ ( x a − x b ), or a long range interaction ofthe form (see e.g. refs. [15, 16]) f ( | x a − x b | ) = 1 p ( l + ( x a − x b ) ) . (2)This is often referred to as a ‘soft’ Coulomb potential.Here we use l = 1 a , a the effective Bohr radius. Inthis work we focus on a system of four wells as previ-ous results [1] showed that this was the smallest numberfor which the average local entanglement exhibits a non-trivial dependence on C U . We calculate the solution ofthe time-independent Schr¨odinger equation correspond-ing to eq. (1) by using ‘exact’ diagonalization with a basisformed by the eigenfunctions of the non-interacting sys-tem. We note that the system ground-state is a singletdue to the choice of zero magnetisation. III. AVERAGE SINGLE-SITE (OR LOCAL)ENTANGLEMENT
Here we consider the average single-site (or local) en-tanglement of the system ground state. This type ofentanglement is relevant for systems of indistinguishablefermions [17, 18]. To this aim we divide our QD systeminto contiguous ‘sites’, each site centred around a singlewell. The entanglement entropy S of the system is givenby S = 1 M M X i S i , (3)with S i = − T rρ red ,i log ρ red ,i the i -site von Neu-mann entropy of the reduced density matrix ρ red ,i .By dividing the system into sites and moving to asite-occupation basis the reduced density matrix be-comes the 4 × ρ red ,i =diag [ P i ( ↑↓ ) , P i ( ↑ ) , P i ( ↓ ) , P i (0)] , with P i ( α ) the probabil-ity of double ( α = ↑↓ ), single ( α = ↑ or ↓ ), or zero ( α = 0)electronic occupation at site i .We calculate the ground-state wavefunction, for aneven number M of wells, and from that obtain the occu-pation probabilities, as described in detail in ref. [1]. IV. COMPARISON WITH HUBBARD MODELVARIANTS
We consider the Hamiltonian H = X i,σ (cid:16) ˆ c † i,σ ˆ c i +1 ,σ + h.c (cid:17) (cid:2) ˜ t ′ (ˆ n i, − σ + ˆ n i +1 , − σ ) − t (cid:3) + ˜ U X i ˆ n i, ↑ ˆ n i, ↓ + ˜ U ′ X i,σ,σ ′ ˆ n i,σ ˆ n i +1 ,σ ′ , (4) where ˜ U and ˜ U ′ are respectively the on-site and inter-sites interaction strength, t is the hopping parameter and˜ t ′ is the correlated hopping term. ˆ c † i,σ (ˆ c i,σ ) are the cre-ation (annihilation) operators for a fermionic particle ofspin σ at site i , and ˆ n i,σ = ˆ c † i,σ ˆ c i,σ is the particle numberoperator. To solve eq. (4) we use exact diagonalization inthe single-site occupation basis {|↑↓i , |↑i , |↓i , | i} withopen boundary conditions and an average particle den-sity of n = n ↓ + n ↑ = 2 /M . Here n σ is the average densityof the σ -spin component. Again we calculate the averagesingle-site entanglement according to eq. (3) [19–21]. Wenote that for ˜ t ′ = 0, eq.(4) describes the EHM while for˜ U ′ = ˜ t ′ = 0 , the HM is recovered.In order to calculate the equivalent of t for the QDsystem, t QD , we use the same procedure employed in [1]for the contact interaction. The corresponding on-site,inter-site and correlated hopping for the QD model arecalculated from I hijk = C U Z φ h ( x a ) φ i ( x b ) φ j ( x a ) φ k ( x b ) p ( l + ( x a − x b ) ) dx a dx b , (5)with ˜ U QD = I LLLL , ˜ U ′ QD = I LRLR , ˜ t ′ QD = I LLLR and φ L ( R ) the single-particle ground state of the finite singlesquare well potential, but positioned in the left ( φ L ) orright ( φ R ) well.Usually the hopping parameter t is used to rescale ˜ U ,˜ U ′ and ˜ t ′ , giving the dimensionless interactions U = ˜ U /t , U ′ = ˜ U ′ /t and t ′ = ˜ t ′ /t . For the QD system we obtain U = ˜ U QD /t QD = 17865 C U , U ′ = 0 . U and t ′ = 2 × − U for d = w = 2 a , and v = 10 Hartree.We first consider the HM. Fig. 1 shows that the HM isnot as good an approximation when long range interac-tions are used. In the QD system with d = 2 a , the ‘longrange’ average single-site entanglement for 0 < U . U QD and t QD usingthe single-particle single square well ground-state may beexpected to be a more severe approximation for the longrange than for the contact interaction case. This does notappear to be the main reason for the difference shown infig. 1 though, as a scaling of U by fitting ˜ U QD / ( t QD C U )cannot rectify this difference. This suggests that thereare significant contributions to the interaction beyondthe on-site repulsion.We also investigate the long range interaction casewhen the wells are further apart ( d = 4 a ): this wouldbe expected to reduce the electron density in the barrierregion, and make the sites better defined and hence thesystem more similar to the HM. We note that our methodfor calculating t QD becomes in this case too susceptibleto the noise in the tail of the numerical wavefunction, asthere is an almost negligible overlap between the single-particle wavefunctions centred in the right and left well.So this method fails to give a reliable t QD value. Hencefor d = 4 a we use t QD as a parameter to generate a S U Contact d=2 aLong range d=2 aLong range d=4 aHM
FIG. 1: Average single-site entanglement vs interactionstrength U for M = 4. HM (dotted line) and QD system with:contact interaction, U = 12344 C U and d = w = 2 a (solidline); long-range interaction, U = 17865 C U and d = w = 2 a (thick dashed line); long-range interaction, U = 8 . × C U , d = 4 a and w = 2 a (thin dashed line). For all QD systems v = 10 Hartree. QD entanglement curve closer to the one representingthe HM system. With this procedure, the average single-site entanglement’s maximum is closer to that of the HMthan for d = 2 a , but the overall shape of the entan-glement curve resembles the d = 2 a case, as shown infig. 1. It therefore strongly suggests that much of thedissimilarity with the HM is caused by the long range in-teraction rather than the proximity of the wells. In fact,unlike a contact interaction, a long range interaction doesnot require the particles’ density to overlap. To test thishypothesis we consider next the EHM and EHM+CHin which interactions between particles on neighbouringsites are included.For d = 2 a the entanglement results for the EHM andEHM+CH are indistinguishable on the scale of the plots(see fig. 2). We see in the upper panel of fig. 2 that theEHM reproduces better the behaviour of the entangle-ment when long range interactions are considered, withthe most appropriate U ′ seemingly residing somewherebetween 0 . U and 0 . U . This difference with our es-timated value of U ′ = 0 . U could arise from the useof the non-interacting single square well solutions in ourcalculation of U ′ .For attractive interactions, the lower panel of fig. 2shows that the EHM reproduces the single-site entan-glement of the long-range QD system fairly well at all U values, as well as the HM reproduces the single-siteentanglement of the QD with contact interaction. Forrepulsive interactions, although the contact interactionand the HM embody some of the features of the long-range interaction system for the parameters chosen, theaverage single-site entanglement arising from long rangeinteractions is lower for U &
8, significantly so for verylarge U . This is due to the long range repulsion forcingnearly all of the particle density into the outer wells: theparticle density becomes significantly different from zeroonly in the outer wells where though only single occu-pation remains non-negligible. For this case the entan- d=2 a S EHM+CH U’=0.28Long range d=2 aEHM U’=0.15EHM U’=0.28EHM U’=0.2 0.6 0.8 1 1.2 1.4 1.6−100 −50 0 50 100 150 S U HMContact d=2 aLong range EHMEHM+CH
FIG. 2: Average single-site entanglement vs interactionstrength U for M = 4. Upper panel: long range interaction, d = w = 2 a , U = 17865 C U (thick dashed line); EHM andEHM+CH (circles) with: U ′ = 0 . U (solid line); U ′ = 0 . U (dotted line); U ′ = 0 . U (thin dashed line). Lower panel:contact interaction, d = w = 2 a , U = 12344 C U (solidline); HM (dotted line); long-range interaction, d = w = 2 a , U = 17865 C U (thick dashed line); EHM (thin dashedline) and EHM+CH (circles) with U ′ = 0 . U . For all QDsystems v = 10 Hartree. glement would be expected to be bounded from belowby 2 /M . Fig. 2 shows that in the long range case theentanglement does indeed come close to this limit. Thisdensity configuration is not accessible with the contact-type interaction: this interaction could affect the densityprofile to some extent, but could not make the outer wellsmore favourable than the inner wells (see [1] and fig. 5).The EHM reproduces fairly well the effect on the den-sity of long-range interactions for U .
10, and we seethat the related entanglement saturates at a value lowerthan the HM for higher values of U . However this satu-ration value is still much higher than the entanglementvalues reached by the QD system with long range inter-actions, which do not yet saturate even for U as large as150. On the other hand, the greater effect on the densitydistribution means that the QD system with long rangeinteraction comes closer to achieving the theoretical max-imum entanglement for M = 4, S thmax = 1 .
623 (see tableI in ref. [1]): for long-range interactions S max = 1 . S max = 1 .
550 for the contact interaction(see fig. 1).We see in fig. 3 that in the limiting case of d = 0, forwhich our QD model reduces to a single dot of width M w , the EHM is fairly accurate for the entanglement ofthe QD system when | U | is small. However the maximumand U ∼ d = 2 a :compare the upper panels of figs. 2 and 3. We note thatfor d = 0 the HM and EHM are also substantially lessaccurate for U < d = 2 a case (comparelower panels of figs. 2 and 3). Including the correlatedhopping term modifies only slightly the entanglement ofthe EHM for U >
0, but has a noticeable effect for
U < S U EHM+CHHMLong range 1.38 1.42 1.46 1.5 1.54 1.58 0 2 4 6 8 10HMLong rangeEHM+CHEHM S FIG. 3: Average single-site entanglement vs interactionstrength U for M = 4. Upper panel: HM (dotted line); longrange interaction, d = 0, w = 2 a , U = 1 . C U (thick dashedline); EHM with U ′ = 0 . U (thin dashed line); EHM+CHwith U ′ = 0 . U , t ′ = 0 . U (circles). For all QD systems v = 10 Hartree. Lower panel: same as upper panel but for agreater range of interaction strengths. The Hubbard model with
U < U (up to ∼ U in a quantumdot system, we may use effective atomic units and theobservation that a system with C U = K , well depth v ,well width w , barrier width d and minimum interactionlength l is equivalent to a system with C U = 1, well depth v /K , well width Kw , barrier width Kd and minimuminteraction length Kl . Under these conditions, given aspecific QD system and the well parameters, the on-site U can be estimated, as shown in table I. We then note thatsome typical QD parameters such as in [24] and [25] cor-respond to values of U for which the EHM and EHM+CHare unable to reproduce the QD entanglement.The results in table I suggest that for a long rangeinteraction the HM, EHM and EHM+CH are not at allsuitable to accurately model the average single-site en-tanglement of realisable quantum dots when d is compa- rable or larger than w . However they all appear to bea fair approximation to the average single-site entangle-ment in the limiting case of d = 0 and small U suggestingthat the HM, EHM and EHM+CH are useful in mod-elling systems of dots when d is much smaller than w assuch systems correspond to small U . Interestingly, thismay include the case of a single dot when modelled as a(small) set of finite partitions (system GaAs (3) in tableI). V. PARTICLE-PARTICLE SPATIALENTANGLEMENT
We now investigate spatial entanglement [26], i.e. theparticle-particle entanglement related to the spatial de-grees of freedom. For the QD system the related reduceddensity matrix is infinite dimensional and given by ρ red,spatial ( x, x ′ ) = Z Ψ ⋆ QD ( x, x τ )Ψ QD ( x ′ , x τ ) dx τ (6)where Ψ QD is the QD system wavefunction.The von Neumann entropy is then S spatial = − T rρ red,spatial log ρ red,spatial . We consider the QDsystem with both contact and long-range interactions,and compare it with the HM and the EHM (with andwithout the CH term) with M sites. To calculate theHM (EHM) spatial entanglement we transform the HM(EHM) wavefunction from the site occupation basis tothe particle basis to give Ψ HM (EHM) . We move fromhaving a superposition of the tensor products of M sites, where each site can be in one of four states, to asuperposition of the tensor products of two particles,each of which can be in one of M sites. This mapping,for the two -electron system under consideration, is givenby | i . . . |↑↓i i . . . | i M → | x i,a i | x i,b i⊗ √ (cid:2) |↑ a i |↓ b i − |↓ a i |↑ b i (cid:3) , (7)for states’ components with doubly occupied sites, andby | i . . . |↑i i . . . |↓i j . . . | i M → √ (cid:2) | x i,a i | x j,b i |↑ a i |↓ b i− | x j,a i | x i,b i |↓ a i |↑ b i (cid:3) , (8)for states’ components with single-occupied sites, whereantisymmetry in particle exchange has been imposed.Here { x i,s } , with i = 1 , . . . M , and s = a, b , repre-sent the discrete M possible coordinates (site positions)for particle s . The spin part of the resulting ground-state wavefunction factorises into a singlet, so its entan-glement is constant and, as such, not of interest here.The spatial part of the wavefunction Ψ HM (EHM) can nowbe used to calculate the M × M reduced density matrix ρ HM ( EHM )red,spatial = T r A (cid:12)(cid:12) Ψ HM (EHM) (cid:11) (cid:10) Ψ HM (EHM) (cid:12)(cid:12) where we
TABLE I: Estimate of the on-site interaction strength U for different QD systems: GaAs-type systems with reduced mass m eff = 0 . m e and dielectric constant ǫ = 10 . (1) to GaAs (3) ); GaAs-based system with m eff = 0 . m e and ǫ = 12 . (4) [24]); and CdSe-based system with m eff = 0 . m e and ǫ = 9 . d = 0 limiting case, the system isphysically a single dot of width 44nm but modelled as four partitions of width w = 11 nm. System QD Parameters Corresponding Long-range model parameters v (eV) w (nm) d (nm) l (nm) v (Hartree) w ( a ) d ( a ) l ( a ) ∼ U GaAs (1) (2) (3) . (4) [24] 1.0 5.0 5.0 3.4 10 1.48 1.48 1 500CdSe [25] 0.6 2.0 2.0 1.7 10 1.19 1.19 1 1000 trace out the subsystem A corresponding to one of theparticles. The reduced density matrix is then used in theevaluation of the von Neumann entropy S spatial .Due to the lattice discretization, as an approximationto Ψ QD , Ψ HM (EHM) retains very few of the spatial de-grees of freedom, as few as 4 in the system at hand. Yet,our results in fig. 4 show that the agreement between theEHM spatial and the QD system spatial entanglement issurprisingly good, especially for attractive interactions.We also find that the HM spatial entanglement is an ex-cellent approximation to the QD system with a contactinteraction for both attractive and repulsive interactions(fig. 4). Interestingly, the QD system with long rangeinteraction has a lower entanglement for large positive U than the QD system with contact interaction, even if theformer has stronger, Coulomb-dependent, correlations.That occurs because the long range interaction shapesthe particle density substantially more, which means thatfor large repulsion almost all of the density resides inthe outer wells, see fig. 5. This results in a triplet-typestate [16] for which the spatial entanglement is equivalentto the entanglement of a maximally entangled two-qubitstate, i.e. , unity. Our data confirm this picture. Interest-ingly in this regime the EHM is actually a poorer approx-imation than the HM to the long range QD system. TheEHM has increased Coulomb correlations in respect tothe HM and hence displays a higher entanglement. How-ever the EHM fails to effectively exclude the particlesfrom the inner wells and to reproduce the triplet-typestate (with its lower entanglement) which characterisesthe QD system when U >>
10. In fact at U = 100the single-particle occupation at the outer (inner) wellsis 0 .
96 (0 . .
74 and at the inner sitesit is still as high as 0 .
26. This implies that the rank ofthe EHM density matrix does not reduce to the one of atwo-qubit density matrix.
VI. CONCLUSION
We studied the Hubbard, extended Hubbard, and ex-tended Hubbard with correlated hopping models as an s p a ti a l S U Contact d=2 aLong range d=2 aEHMEHM+CHHM
FIG. 4: Spatial entanglement vs interaction strength U : con-tact interaction, d = w = 2 a , U = 12344 C U (solid line);HM (dotted line); long-range interaction, d = w = 2 a , U = 17865 C U (thick dashed line); EHM (thin dashed line),EHM+CH (circles) with U ′ = 0 . U . v = 10 Hartree for allQD systems. d=2 a d=2 a − D e n s it y ( a ) FIG. 5: Density profile for the QD system at U = 100 with M = 4, d = w = 2 a , v = 10 Hartree and a contact (solidline) or long range (thick dashed line) interaction. approximation to the local and spatial entanglementof a one-dimensional chain of nanostructures, in whichtrapped particles are interacting via long-range interac-tions. We focused on a system comprised of 4 quan-tum dots, though the typical trends we found apply tolonger chains (not shown), as the long-range interactionwill eventually force the density into the outer wells.For well-separated QDs the EHM reproduces fairly wellthe average-single site entanglement for attractive long-range interactions and 0 < U .
10, though with a max-imum positioned at a weaker interaction strength. Incontrast with the contact interaction case [1], for longrange interactions and U &
10 the EHM completely failsto reproduce the single-site entanglement, with an errorwhich rapidly increases up to ∼ U = 500 (notshown). We emphasise that U -values as high as 5700do indeed correspond to some of the typical parameterranges for QDs, so care must be taken when trying tomodel nanostructure systems with the HM and EHM.We note that ref. [27] found the HM not to be anaccurate approximation when considering the exchangecoupling in two-electron double quantum-dots modelledwith a parabolic confining potential and a Coulomb in-teraction in two dimensions or an effective model in onedimension. However the HM did at least qualitativelyreproduce the behaviour of the exchange coupling as theinter-dot distance was varied, while, interestingly, theEHM did not. In contrast we found the EHM to be abetter approximation than the HM to the average single-site entanglement of the QD-chain system, though withthe limitations described above.We considered the limiting case of interdot barrier d = 0. In this case our model describes a single QD,which is divided into an arbitrary number M of parti-tions. We then considered the average entanglement ofone of these partitions with the others. We found thatthis corresponds to small U for physically realisable dotsand in this region the HM and EHM accuracy as an ap-proximation to the entanglement were fairly good.We also studied the case of attractive interaction: forwell-separated QDs the average single-site entanglementwas well approximated by both the HM and the EHM forany interaction strength. However as the interdot barrierwidth decreases, the HM and EHM fail to reproduce theentanglement of the QD system at medium and largeinteraction strength, quickly reaching a discrepancy of ∼
40% for the limiting case d = 0.We designed a procedure to calculate the spatial en-tanglement from the HM and found that the spatial en-tanglement associated with the HM was a very good ap-proximation to the spatial entanglement of the four QDchain when particle-particle contact interaction was con- sidered. This was surprising as we lose most of the spatialdegrees of freedom when using the HM to calculate thespatial entanglement: we consider one position per siteonly – four positions in total for the specific case anal-ysed in this paper – compared with an infinite numberfor any QD system. Yet our results show that the perti-nent information seems to be retained. The QD systemwith long range, repulsive interaction was less well ap-proximated by the HM, as could be expected, but inter-estingly the EHM performed worse still. However theirerrors at large U , ∼
13% and ∼
33% respectively, wereconsiderably smaller than for the average single-site en-tanglement of the long-range QD system. We attributedthe worse performance of the EHM to the long range in-teraction causing most of the electron density to residein the two outer wells so essentially giving the maximumentanglement of a two-qubit system. The EHM fails tomodify the density to this extent, and hence to lowerthe entanglement when
U >>
10. We find that for theanalysed QD system the correlated hopping term is verysmall and produces appreciable effects on the entangle-ment only in the limiting case of d = 0 with attractive interactions.Our results show that when dealing with nanostruc-tures care must be taken in assessing the validity of theHM or EHM as an approximation to the system be-haviour, as for some of the experimentally relevant pa-rameter range these approximations fail badly to repro-duce the local entanglement. Surprisingly, our calcula-tions show that for QD chains the essential properties ofthe spatial entanglement are retained when discretizingthe spatial degrees of freedom to a handful of points, andin this case the HM and EHM remain fair approximationsat all parameter values. Acknowledgments
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