Electronic entanglement in late transition metal oxides

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by  Patrik Thunström

Electronic entanglement in late transition metal oxides
Patrik Thunstr¨om, Igor Di Marco, and Olle Eriksson
Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120, Uppsala, Sweden
(Dated: February 20, 2012)
Here we present a study of the entanglement in the electronic structure of the late transition
metal monoxides - MnO, FeO, CoO, and NiO - obtained by means of density-functional theory in
the local density approximation combined with dynamical mean-field theory (LDA+DMFT). The
impurity problem is solved through Exact Diagonalization (ED), which grants full access to the
thermally mixed many-body ground state density operator. The quality of the electronic structure
2 is affirmed through a direct comparison between the calculated electronic excitation spectrum and
1 photoemission experiments. Our treatment allows for a quantitative investigation of the entangle-
0 mentin theelectronic structure. Two main sources of entanglement are explicitly resolved through
2 theuse of a fidelity based geometrical entanglement measure, and additional information is gained
b from a complementary entropic entanglement measure. We show that the interplay of crystal field
e effects and Coulomb interaction causes the entanglement in CoO to take a particularly intricate
F form.

7 PACSnumbers:03.67.Mn,71.27.+a,71.15.-m,71.20.Be
1

] Entanglementisafundamentalaspectofquantumme- access to the localmany-body groundstate of the impu-
l
e chanics, responsible for a large range of complex phe- rity problem to analyse the entanglement in detail.
-
r nomena not present in a classical setting. The entangle- TheLDA+DMFTschemeisbuiltaroundthemapping
t
s ment of distinguishable particles was historically seen as ofthelocallatticeproblemtoaneffectiveimpurityprob-
t. something spooky, but is now considered a valuable re- lem,whichisthensolvedthroughvarioustechniques.The
a
source.It playsan essentialrolein quantum information ED solver is a natural extension of the Hubbard-I Ap-
m
theory, and has been studied in great detail both the- proximation (HIA)[8–10]. In the latter the hybridization
-
d oretically and experimentally[1]. Entanglement of indis- between the impurity and the self-consistent bath is ne-
n tinguishableparticleshasreceivedlessexplicitattention, glected, leading to an atomic-like system described only
o but it has been studied indirectly in both the quantum interms ofthe localprojectedLDA HamiltonianHˆLDA,
c chemistry and condensed matter communities. Describ- a double counting term HˆDC, and the on-site Coulomb
[
ing the electronic structure ofmaterials with a pure sep- interaction Uˆ. The Hamiltonian of the finite system is

1 arable state, represented as a single Slater determinant, diagonalized numerically to produce an analytical self-
v is at the very heart of the modern computational ap- energy. Such an approach gives excellent results for the
5
proaches, and the inability to do so is known under the spectral properties of actinides and lanthanides[9, 10].
7
9 term ‘correlation’. The term encompasses both classical However, in the TMOs the hybridization between the
3 andquantumcorrelations(entanglement),wherethefor- TM-3d and O-2p orbitals is too strong to be neglected.
. mer results in mixed states and the latter in entangled WhileretainingtheadvantagesoftheHIA,theEDsolver
2
0 states. The presence of entanglement is generally seen alsotakesintoaccountasignificantpartofthehybridiza-
2 as a computational complication as it prevents the use tion. In ED additionalauxiliarybath states are included
1 of these standard approaches. The development of reli- in the local Hamiltonian, giving
:
v ablecomputationalmethodsandentanglementmeasures
1
i are of key importance to turn also the entanglement of HˆED = H˜LDA H˜DC cˆ†cˆ + U˜ ˆc†ˆc†cˆ cˆ
X indistinguishableparticlesfromacomplicationintoapo- ij − ij i j 2 ijkl i j l k
r Xij (cid:16) (cid:17) Xijkl
tential resource.
a
+ V˜ ˆc†ˆc +H.c. + E˜ ˆc† cˆ , (1)
Well known examples of strongly correlated materi- im i m m m m
als are the late transition metal monoxides (TMO) – Xim (cid:16) (cid:17) Xm
MnO, FeO, CoO and NiO – which have been under in- where the indices i,j,k,l run over the local correlated
tense experimental and theoretical attention for a long orbitals and m runs over the auxiliary bath states.
time[2–4]. Here we report on a theoretical description of The energies E and the hybridization strength V
m im
the late TMOs using the Local Density Approximation of the auxiliary bath states are determined from a fit-
plus Dynamical Mean Field Theory (LDA+DMFT)[5] ting of the impurity hybridization function ∆(ω). The
where the effective impurity model is solved by Exact calculations were carried out in the paramagnetic (PM)
Diagonalization(ED)[6]. The quality of the results is as- phase, using a finite temperature (β =0.00173Ry) fully
sessed through a direct comparison between the com- charge self-consistent LDA+DMFT implementation[10–
puted density of states (DOS) and photoemission exper- 12].FurthertechnicaldetailscanbefoundintheSupple-
iments(XPS/BIS).Wethentakeadvantageofthe direct mental Material[13].
2
4 44
MnO FeO
3 33
nit) nit)
u u
b. b.
y (ar2 y (ar22
nsit nsit
e e
nt nt
I I
1 11
0 00
-10 -5 0 5 -10 -5 00 5
Energy (eV) Energy (eV)
4 5
CoO NiO
4
3
nit) nit)
b. u b. u3
y (ar2 y (ar
nsit nsit2
e e
nt nt
I I
1
1
0 0
-10 -5 0 5 -10 -5 0 5
Energy (eV) Energy (eV)
Figure1.(Colouron-line)PDOSoftheTM3dstates(thickblacklines)andO2pstates(dashedredlines)inMnO,FeO,CoO,
and NiO, and corresponding XPS/BIS data (black circles) [7]. The Fermi level is at zero energy.
In Fig. 1 the calculated projected density of states gradualdecreasewithamarkedshoulderat-3.5eV.The
(PDOS) of the TM-3d and O-2p states are compared to CoO 3d PDOS has two sharp main peaks at -1.5 and
experimental XPS (electron removal) and BIS (electron -3.5 eV, and an almost constant density from -4 to -10.5
addition) data[7] showing mainly the contributionof the eV. The BIS spectrum shows a broad peak with a max-
TM-3d states, due to the photon energies used. imum at 4 eV, which is located about 1 eV higher than
Both the occupied part of the MnO PDOS and the the theoretical peak. The PDOS also contains a smaller
XPS data have a gradual onset developing into a broad peak at5.5 eV, not visible in the BIS peak, likely due to
structure between -1 and -8 eV, with the main peak at the limited experimental resolution.
around -3.5 eV. The shoulder at -1 eV is well described
The NiO 3d PDOS resembles the CoO 3d PDOS in
by contributions from both TM 3d and O 2p states, and
that it has two main peaks at -2.0 and -3.5 eV, and a
the sudden drop in intensity at about -8 eV is clearly
rather constant density from -5 to -10 eV, in excellent
reproduced.Noticethatalsominorexperimentalfeatures
agreementwith the XPS spectrum. It is evenpossible to
like the satellite at -11 eV are found in our theory. The
assign the small features in the XPS spectrum at -5.5,
unoccupied part contains a single large peak at 5.5 eV,
-6.5,and-8.5eV,to their respectivepeaks inthe PDOS.
in excellent agreement with the BIS spectrum.
Thesinglepeakat4.0eVinthePDOSisalsoinexcellent
The FeO 3d PDOS contains peaks at 1.5, -1.5, -4.0,
agreement with the experimental data.
-6.0, and -7.5 eV, all in acceptable agreement with the
XPS spectrum[7]. The peaks at 3.5 and -10 eV are not Giventhesatisfactorycomparisonofthespectralprop-
evident in the experimental data, which may be related ertieswithexperimentaldata,wenowturnourattention
to the fact that the experiment was performed with a to the entanglement in the thermal many-body ground
non-stoichiometric sample (Fe0.95O), in contrast to the state of the impurity problem, described by the density
conventional FeO unit cell used in the calculation. operator ρˆT = e−βHED/Tr(e−βHED). There exist sev-
TheCoOXPSspectrumhasastrongonsetatlowbind- eral ways to measure the entanglement in an N-electron
ing energies, with a maximum at -1.5 eV, followed by a many-body state[1]. In the case of a pure state Ψ , we
| i
3
first look at the geometric entanglement measure[14] Sˆ Ψs ′ , where Ψs ′ is the closest separable state to
λm| i,si | i,si
Ψs . Hence the separable state
E [Ψ Ψ]=1 max Ψ′ Ψ 2, (2) | i,si
G
| ih | − |Ψ′i |h | i| 1 s
ρˆ′ = Sˆ Ψs ′ Ψs ′ Sˆ† , (6)
where Ψ′ isrestrictedtobe pureandseparable.Froma i 2s+1m=−s λm| i,sih i,s| λm
| i X
computationalpointofviewthismeasureisnaturalsince
Ψ′ corresponds to the best possible single Slater deter- gives a local minimum of EG[ρˆTi ]. Moreover each term
| i ρˆλm contains the same amount of entanglement, origi-
minantdescriptionofthesystem.Thesearchfortheopti- i
mal Ψ′ isperformedthroughascanofthecorresponding nating from the material specific |Ψsi,si and the action
Hilb|ertispacebyrecursivelyremovingoneelectronatthe of the renormalization operator Nˆz. Therefore we con-
time from the natural orbitals[13]. Although this proce- jecture that the local minimum given by ρˆ′ is in fact a
i
dureisingeneralnotguaranteedtofindtheoptimal Ψ′ , global minimum. The geometric entanglement measure
itisexactforseparablestatesandforpure2-electron|sysi- for the PM state ρˆT can then be written as[13]
tems, where it gives E [Ψ Ψ] = 1 p with p
the largest eigenvalue oGf|thiehco|rrespo−ndinmgaxone-partmicalxe 1 E [ρˆT] s 2s 2
redInucoerddedrentositaynamlyasteriρxˆTρ˜iijt=is hnΨec|ce†jsscai|rΨyit.o generalize the EG[ρˆT]=1− 22−s(2sG+1s)"mX=−ss(cid:18)s−m(cid:19)# , (7)
entanglement measure EG in Eq. (2) to mixed states. where ρˆTs = ipi|Ψsi,sihΨsi,s|. Even when EG[ρˆTs] is zero
This can be achieved by replacing the overlap with the thereisstillanon-trivialtermremaininginEq.(7).This
fidelity[15] between ρˆT and any separable state ρˆ′, generic sourcPe of entanglement is very robust as it does
not depend on the details of the electronic structure of
2 the system but only on Sˆ2 .
EG[ρˆT]=1−mρˆa′xTr ρˆTρˆ′ ρˆT , (3) As a complement to thhe gieometric entanglement mea-
(cid:20)q (cid:21)
p p sure we have used an entropic measure, based on the
where ρˆ′ = ipi|Ψ′iihΨ′i|, ipi = 1, and |Ψ′ii are sepa- reductionofthe many-bodydensitymatrix ρˆto the one-
rable states. Performing the restrained maximization in particle reduced density matrix ρ˜ =Tr(ρˆc†c ). This re-
P P ij j i
Eq.(3)isingeneralaformidabletask,asonehastocon- ductionconvertstheentanglementinρˆtoentropy,which
sider the effect of mixing several non-orthogonalsepara- canbe measureddirectly[1,16], e.g.in formof linearen-
blestates.However,theproblemcanbesimplifiedbynot- tropy S [ρ˜]= Tr[ρ˜(1˜ ρ˜)]. However, quantifying entan-
L
ing that HˆED of a PM system with negligible spin-orbit glementin formofent−ropyrequirescare,as any classical
interaction commutes with Sˆ2, Sˆz and its ladder opera- correlation in ρˆis converted to entropy as well. We pro-
torsSˆ±.Thisimpliesthatthereisacommoneigenbasisin pose the following entropic entanglement measure
whichthemany-bodyeigenstates Ψs canbeindexed
by s(s+1)= Sˆ2 , and m = Sˆ | ,ia,mnsdithat the eigen- EL[ρˆ]=SL[ρ˜] min Tr[ρ˜],Tr[1˜ ρ˜] SL[ρˆ], (8)
s z − −
h i h i
values Es are independent of m . The thermal density
i s based on the fact that the e(cid:0)ntropy gained f(cid:1)rom the clas-
matrix is therefore a mixture of several spin-degenerate
sical correlations is less than the number of electrons or
eigenstates
holes times the entropy in ρˆ[13]. Removing this upper
bound of the classical contribution simplifies the evalua-
s
p
ρˆT = piρˆTi = 2s+i 1 |Ψsi,msihΨsi,ms|, (4) tion of EL, but at the same time makes it less sensitive
Xi Xi mXs=−s to detect entanglement in mixed states.
Apart from the entanglement measures proposed
where we have restricted ourselves to a single many- above,otherchoicesarepossible.Anintermediatescheme
body spin quantum number s. We also observe that is to evaluate the geometric entanglement by using the
the spin-coherent operator Sˆλ = exp(λSˆ−)(1+ λ2)−Sˆz relative von Neumann entropy[14]. However,the current
| |
conserves the entanglement[13] of the maximally spin- lack of an efficient minimization procedure for the rela-
polarized state |Ψsi,si. Setting λm = exp(2iπm/(2s+1)) tiveentropywithrespecttothe separablereferencestate
yields ρˆ′ reduces its applicability to setting upper bounds on
the entanglement[17].
ρˆT = s ρˆλim s NˆzSˆλm|Ψsi,sihΨsi,s|Sˆ†λmNˆ†z, The many-body eigenstate decomposition of ρˆT from
i 2s+1 ≡ 2s+1 the ED solver is shown in Table I. The auxiliary bath
m=−s m=−s
X X orbitalsareassignedaspin,butzeroorbitalangularmo-
(5)
mentum. The ground state of NiO contains two sets of
where Nˆ = 2s/ 2s (2s+1). When applying the
z s−Sˆz near degenerate states, with an energy difference of 80
proposed Slater seqar(cid:0)ch alg(cid:1)orithm to the pure state ρˆλim meV. Although the energy difference is small, the low
oftheTMOs,themaximaloverlapisalwaysobtainedfor temperature strongly favours the eigenstates with the
4
Table I. Entanglement of the thermal ground state at T = 273K. The values of s is defined by s(s+1) = hSˆ2i, and the
degenerate eigenstate |Ψmsi corresponds to ms = −s,..,s for each value of hLˆzi. E (eV) is the relative energy of |Ψmsi, and
EG and EL are definedin thetext.
TMO EG[ρˆTs] EG[ρˆT] EL[ρˆT] E (eV) s hLˆzi EG[|ΨsmsihΨsms|] EL[|ΨsmsihΨsms|]
ms= 0 ±1/2 ±1 ±3/2 ±2 ±5/2 0 ±1/2 ±1 ±3/2 ±2 ±5/2
MnO 0.00 0.15 -1.67 0.00 5/2 0.00 – 0.90 – 0.80 – 0.00 – 2.40 – 1.60 – 0.00
FeO 0.00 0.11 -1.40 0.00 2 0.00 0.83 – 0.75 – 0.00 – 2.01 – 1.51 – 0.01 –
0.00 2 ±0.98 0.83 – 0.75 – 0.00 – 2.01 – 1.51 – 0.01 –
CoO 0.02 0.09 -0.81 0.00 3/2 0.00 – 0.72 – 0.16 – – – 1.64 – 0.55 – –
0.00 3/2 ±1.45 – 0.71 – 0.12 – – – 1.57 – 0.43 – –
NiO 0.00 0.03 -0.35 0.00 1 0.00 0.50 – 0.00 – – – 1.00 – 0.00 – – –
0.08 1 0.00 0.50 – 0.00 – – – 1.00 – 0.00 – – –
0.08 1 ±0.49 0.63 – 0.25 – – – 1.38 – 0.75 – – –
lowest energy. In all the studied materials, the ground an extra electron in the TM-3d orbitals. As a result
state configurations maximize Sˆ2 due to the strong the strength of the pair-hopping becomes very small,
h i
Coulomb interaction. and it gives rise only to a very weak entanglement with
We start by looking at the entanglement in the Sˆ E [Ψs Ψs ]=0.003 and E [Ψs Ψs ]=0.012.
h zi G | sih s| L | sih s|
andhLˆziresolvedcomponents|ΨsmsihΨsms|ofρˆT.Asseen The MnO ground state is dominated by the Mn ↑23↓00
inTableI,bothEGandELincreaseinmagnitudeas ms configuration.Evenwhenoneextraelectronisintroduced
| |
becomessmaller.Thistrendisadirectconsequenceofthe from the bath, the Coulomb interaction cannot induce
paramagnetism and is associated to the number of ways any pair hopping, which makes ρˆT fully separable.
s
itispossibletodistributetheelectronsaccordingtotheir TheCoulombinteractiongivesanadditionalcontribu-
spin. For such components EG takes the form of tion to the entanglement when two electrons are trans-
ferred from the bath to the TM-3d orbitals due to
E [Ψs Ψs ]=1 1 E [Ψs Ψs ] / 2s . (9)
G | msih ms| − − G | sih s| s−ms the quadratic increase in repulsion energy. However, for
Let us now focus on (cid:0)the entanglement(cid:1)in(cid:0)the (cid:1)maxi- TMOs this contribution is of the order of 10−4 for the
mally spin-polarized eigenstate Ψs . It depends mainly geometric measure, i.e. too small to be seen in Table I.
| si
on the complicated interplay between the Coulomb in- Let us now consider the entanglement in the thermal
teraction and the crystal field energies. The maximally PMgroundstateρˆT.ForNiO,FeOandMnOtheEG[ρˆTs]
spin-polarized ground state of NiO is dominated by a term in Eq. (7) is zero, and only the generic part con-
Ni d8 high spin configuration ( 2 0 where the subscript tributes to the entanglement. A single Slater determi-
↑3↓3
and the superscript stand for the number of t and e nant method can therefore in principle obtain the max-
2g g
electrons respectively). The on-site Coulomb interaction imally spin-polarized states Ψs , and then reconstruct
| i,si
must preserve both Sˆ and Lˆ which implies that it the groundstates throughEq. (5). However,suchan ap-
z z
cannotcouplethissthateitoanyhothierstate.TheCoulomb proach would not be adequate for CoO, as EG[ρˆTs] is
interactionisthereforeeffectivelyreducedittoaHartree- non-zero in this case. The generic contribution causes
Fock term, which makes the eigenstate |Ψssi separable. EG[ρˆT] to increase monotonically from NiO to MnO. A
The eigenstates at 80 meV have mainly 2 1 character. similartrendcanbe seeninthe relativeentropybetween
↑3↓2
Here the Coulomb interaction is allowed to transform the locally projected[13] thermally mixed density oper-
these states to 2 2 through pair-hopping. Nevertheless, ators in LDA and LDA+DMFT[17]. However, due to a
when the maxim↑3a↓l1ly spin-polarized states are combined strong reduction of the large number of available many-
intothedensitymatrixρˆT theentanglementinthestates body states in the LDA solution to only a few in the
s
cancels out, giving E [ρˆT]=0.00. LDA+DMFTsolution,thesizeoftherelativeentropyre-
G s
The ground state of CoO has mainly a Co 2 0 con- flects mainly the different degree of classical correlation
↑3↓2
figurationthat couples to 2 1 through the pair-hopping inthetwocalculations.Thistrendintherelativeentropy
↑3↓1
induced by the Coulomb interaction. The pair-hopping occurs in fact even when the LDA+DMFT ground state
gives rise to a strong entanglement in Ψs , and in con- isreplacedbytheseparablegroundstateofacorrespond-
| si
trastto NiO,the entanglementisstill presentinρˆT,giv- ing Hartree-Fock-like LDA+U simulation.
s
ing E [ρˆT]=0.02. The proposed E measure was not able to resolve the
G s L
The ground state of FeO has primarily a Fe 2 0 con- entanglement in the PM phase, as shown by the nega-
↑3↓1
figuration. The Coulomb interaction is again reduced to tivevaluesinTableI.Theconditionusedtosubtractthe
a Hartree-Fock term, except when the bath introduces classicalcontributionprovedtoostrongforthesesystems,
5
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4
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We are grateful to the Swedish Research Council, [11] A. Grechnev, I. Di Marco, M. I. Katsnelson, A. I.
Energimyndigheten (STEM), the KAW foundation and Lichtenstein, J. Wills, and O. Eriksson, Phys. Rev.B
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have been performed at the Swedish national computer
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centers UPPMAX and NSC. We like to thank S. Gar-
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nerone and M. I. Katsnelson for useful discussions.
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(2009)A. Fujimori, F. Minami, and S. Sugano, ibid. 29,
5225 (1984)R. Eder, ibid. 78, 115111 (2008)A. Kutepov,
K.Haule,S.Y.Savrasov,andG.Kotliar,ibid.82,045105
(2010)
6
Supplemental material with a moderate number of spin-orbitals (K = 30) and
closetocompletefilling (N =0.8K =24),the sizeofthe
Inthe followingformulaswehavefollowedthe conven- Hilbert space becomes too large to handle in practice.
tion of using hats for many-body operators, e.g. Uˆ, and However often the system under analysis possesses use-
tildes for one-particle operators, e.g. ∆˜. A many-body ful symmetries, which can be used to find some criteria
operatorBˆ canbe obtainedfromaone-particleoperator to a priori identify ablockstructure inthe Hamiltonian,
B˜ as and treat these blocks separately.
A general way to obtain these criteria is to define the
Bˆ = B˜ ˆc†ˆc . (10) many-body basis vectors as the eigenvectors of a set of
ij i j
ij some commuting observables Aˆk , and then determine
X { }
how the states mix under the actionof the Hamiltonian.
Since the basis vectors should be representable by sin-
Exact Diagonalization gleSlaterdeterminants,onlycommutingone-electronob-
servables
Block diagonalization
Aˆk = A˜kcˆ†cˆ , (12)
ij i j
The Exact Diagonalization (ED) method solves the ij
X
Andersonimpurityproblembymappingittoafinitesize
need to be considered. Furthermore the additional con-
problem, defined by the Hamiltonian
dition that the observables should commute with Uˆ al-
HˆED =Hˆ0+Uˆ, (11) lows to keep the calculation of the mixing rather simple.
Thesetworestrictionsreducethelistofpotentialobserv-
where Hˆ0 contains the one-electron terms, including the ables Aˆk for the correlated orbitals to Sˆz and Lˆz (or
{ }
hybridization with a few auxiliary bath orbitals, and Uˆ alternatively Sˆx and Lˆx and Sˆy and Lˆy). The auxiliary
describes the full Coulomb interaction between the elec- bathorbitalsarenotdirectlyaffectedbyUˆ,soeachbath
trons in the correlated orbitals[6]. spin-orbitalmcanbeassignedanobservablenˆm =cˆ†mˆcm
An important technical issue in the implementation that measures its occupation.
of ED is constructing the matrix representation of the Eachspin-orbitalj canbe transformedintoacommon
Hamiltonian with respect to the many-body basis. This eigenstate of all these observables and assigned a vector
stepisgreatlysimplifiedifthebasisvectorsaredefinedas of eigenvalues ~aj. A many-body basis vector ΨN , de-
| i i
singleSlaterdeterminantsinsomeone-particlebasis.The finedasasingleSlaterdeterminantwithrespecttothese
Slater determinants can be visualized in the occupation orbitals, is trivially an eigenstate of the observables in
numberformalism.Forexample5electronsina10orbital Aˆk , with an eigenvalue vector
{ }
manifold can form the following many-body states:
K
|Ψ51i=|1111100000i, A~i = hΨNi |ˆc†jcˆj|ΨNi i~aj. (13)
Ψ5 = 1111010000 , Xj=1
| 2i | i
Ψ5 = 1111001000 , Obtaining these eigenvalue vectors is not enough to de-
| 3i | i
. termine the block structure of HˆED, as the one-electron
.
. Hamiltonian Hˆ0 does not in general commute with the
Ψ5 = 0000101111 , observables in Aˆk . In particular, the hybridization
| MΨ−51i=|0000011111i. with the bath or{bita}ls rarely commute with L˜z. The off-
| Mi | i diagonal elements of H˜0 determine how the blocks will
Here M is the number of possible many-body states, i.e. form. Each non-zero off-diagonal element H˜0 ˆc† cˆ al-
mn m n
the binomial coefficient of 10 over 5. When the creation lows a many-body basis vector Ψ to couple to a basis
i
| i
and annihilation operators are given in the same one- vector Ψ if the following condition is fulfilled
j
particle basis as ΨN , then their action becomes a sim- | i
| i i
ple remapping of the indices N N 1,i j and a A~j A~i =~am ~an. (14)
→ ± → − −
multiplication of a phase factor ( 1).
±
In a system with a large Hubbard U and bath ener- The problem of generating a block structure can now be
gies E far from the Fermi energy, it is usually sufficient transformed into finding the connected components of
to consider the ground-state configurations with N elec- an undirected graph, where the vertices are defined by
trons, and the excited configurations with N 1 elec- A~i and the edges by ~am ~an,0 through Eq. (14).
± { } { − }
trons. The size of the many-body basis is then equal to Thisproblemcanbe solvedefficiently throughthe useof
allpossible configurationsofN,and N 1 electrons dis- sparselogicalsquarematrixmultiplication,inthefollow-
±
tributed in K one-electron spin-orbitals. However, even ing steps:
7
1. Map each vertex to a matrix index, and each edge implementation[11, 12] contains two choices of orbitals
to a true element in the logical matrix T. thatmeettheseconditionstoalargeextent,theheadsof
theLMTO’s(MT)andLo¨wdin’snaturalorbitals(ORT).
2. Multiply T with itself, until it remains constant.
The MT orbitals are given by
This procedure makes the connected components
complete. ψMT(~r)=φ (r)θ(S r)Y (rˆ), (17)
lm l − lm
3. ThetrueelementsinaroworcolumninT givesthe where θ is a step function, Y is a spherical har-
lm
indicesofalltheverticesofaconnectedcomponent. monic function, and φ (r) is the solution to the radial
l
Schrodingerequationwithin the muffin-tin sphere ofra-
This algorithm can be improved by contracting dense
dius S. The MT orbitals are both localized and eigen-
regions of the graph before T is defined. Furthermore,
states of L˜ by construction. The ORT orbitals at a site
the second step can be performed even more efficiently z
R~ are defined as
through the use of an update matrix V:
ψORT(~r)= ei~k·R~ψLMTO(~r)[O−1/2] , (18)
1. V T limi ~kj k ji
← ~kX∈BZXj
2. While V =0
6 where ψLMTO(~r) is an LMTO orbital, O is the LMTO
~kj
(a) T T .or. V
overlap matrix, the index j runs over all the LMTO or-
←
(b) V (.not. T) .and. T.V bitals, and ~k runs over all the k-points in the Brillouin
←
zone. The ORT orbitals are in general less localized and
Finally the Hamiltonian can is ready to be block diago- only approximately eigenstates to L˜ , due to the inclu-
z
nalizedbygroupingallthemany-bodybasisvectorswith
sionof the LMTO tail functions andthe mixing through
quantum number vectors A~i matching a given con- O−1/2. However, when ψORT(~r) correspond to localized
nected component. { } limi
states,both choicesleadto similarorbitals.As anexam-
ple,intheupperpanelFig.2theNi-3dprojecteddensity
ofstatesofNiOinLDAisreported.Thedensitiesforthe
Correlated orbitals and the hybridization function
MT and ORT bases are practically identical. Neverthe-
less, the small differences in the projections do have a
TheEDmethodisbuiltaroundfittingafewhybridiza- large impact on ∆˜. In the lower panel of Fig. 2 we show
tion parameters V˜ and E˜ to the hybridization function
the corresponding spectral density of the hybridization
function
−1
∆˜(ω)=ω1˜ H˜LDA−DC G˜0(ω) , (15)
− − Nhyb(E)= 1 (Tr[∆˜(E+iδ)]), (19)
h i
−πℑ
whereG˜0(ω)isthebathgreen’sfunctionandH˜LDA−DC
istheprojectedLDAHamiltonianwiththedoublecount- in ORT and MT. A striking difference between them is
ing removed. In our implementation the hybridization thattheMTorbitalsgiverisetoahugehigh-energypeak
function is given on the Matsubara axis ω = iω = structure starting at 10 eV, while this is nearly absent
n
iπT(2n+1), and the fitting is performed by minimizing whentheORTorbitalsareused.Suchfeatureiscommon
the cost function to all the transition metals oxides of the present study -
NiO, CoO, FeO, MnO - and to many other systems as
2
F(V˜,E˜)= W V˜†[iω 1˜ E˜]−1V˜ ∆˜(ω) , well. The strong high-energy hybridization given by the
n n
− − F MT orbitals is very detrimental to the fitting process,
Xn (cid:13) (cid:13)
(cid:13) (cid:13) (16) sinceitattractthepolesofthefittedfunctioninEq.(16).
(cid:13) (cid:13)
through the conjugate gradient method. Here W is a ThereforeeventhoughtheORTorbitalsarelesslocalized
n
set of weights and A˜ 2 = Tr(A˜†A˜) is the {Frob}enius and require the use of a simplified basis[11, 12], they are
k kF
norm. For the ED method to give physically relevant re- stilltobepreferred.TheuseoftheORTorbitalsimproves
sults itisimportantthatthe originalhybridizationfunc- the situation, but it does not completely eliminate the
tion can indeed be approximated by a small number of high-energy hybridization.
peaks. The weights W can be chosen to make the cost func-
n
The choice of correlated orbitals determines the map- tion focus more on the physically relevant energy scale.
ping from the lattice problem to the effective impurity Inthecalculationspresentedinthenextsection,W was
n
model, and therefore the value of ∆˜(ω). In order for the set to sample a logarithmic mesh from 7 to 200 eV. The
mapping to be realistic, the orbitals should be localized initialtightspacingofthemeshpointsgivesmoreweight
andnearlydispersionless.Inthe EDsolverit is advanta- totheresiduesontheeVscale,whilethesparselyspaced
geousiftheorbitalsareeigenstatesoftheoperatorL˜ ,as high-energypoints make the cost function smoother and
z
explained in previous section.Our currentLDA+DMFT increase the stability of the procedure. The Matsubara
8
8 improvesit closerto the realenergyaxis.Itis important
thatareasonabledescriptionoftheoriginalhybridization
isobtainedforbothaxes,andsuchsituationisusuallyre-
6 alized when the discardedstates are well separated from
nit) the selected bath states.
u
b.
y (ar4
nsit Computational details
e
nt
I
2
The TMOs were all set up in the NaCl crystal struc-
ture. The lattice constants were taken from experimen-
tal data, and the Brillouin zone was sampled through
0
-20 0 20 40
Energy (eV) a conventional Monkhorst-Pack mesh of 12 x 12 x 12 k-
points.Thecalculationswerecarriedoutintheparamag-
netic phase, using a finite temperature fully charge self-
4 consistent LDA+DMFT implementation[10, 11] based
on the full-potential linear muffin-tin orbitals (LMTO)
nit) method[12]. The double counting correction was defined
u
b. as HˆDC = µDCNˆ, where Nˆ gives the number of elec-
ar
y ( trons in the correlated orbitals. µDC was treated as a
nsit2 free parameter to ensure that the system has the cor-
e
Int rectnumberofelectronsintheground-stateandthatthe
chemicalpotentialisplacedinthebandgapaccordingto
the experimental photoemission spectra.
The Coulomb interaction was parametrized by the
0-10 -5 0 5 10 SlaterparametersF0,F2,andF4.Thelattertwowerere-
Energy (eV)
calculatedateveryiterationfromradialintegrationofthe
Figure 2. (Colour online) Upper panel: The Ni-3d projected bare Coulomb interaction and then multiplied with the
density of states of NiO in LDA. The black lines correspond screeningfactors 0.82and 0.88,respectively.The screen-
to theMT orbitals and thered lines to theORTorbitals de- ingfactorsweresettoreproducetheRPAscreenedSlater
finedinthetext.Lowerpanel:Correspondingspectraldensity parametersfor NiO inRef. 3.The parameterF0, i.e.the
of the Ni-3d hybridization function in the same notation as Hubbard U, could not be treated in the same way as it
above.
istoostronglyaffectedbythe screening,butwassettoa
fixedvaluefromthestart.Thefinalself-consistentvalues
frequencies smaller than 7 eV were excluded since they of the Slater parameters are shown in Table II.
biased the fitting procedure in favour of a detailed de- The ED calculations of NiO, CoO and FeO were per-
scriptioncloseto the Fermienergy,atthe expense ofthe formed with 10 TM 3d spin-orbitals and 20 auxiliary
overalldescription on the eV scale. bath spin-orbitals. For MnO only 10 bath spin-orbitals
The weights improve the fitting of the hybridization were used, due to the greater computational effort. The
function on the Matsubara axis, but further refinement final energy and hybridization strength parameters for
of the fitting procedure is necessary to obtain a good the bath states are reported in Table II. Note that the
physicaldescriptionforrealenergyvalues.Tothisendwe values are given in the crystal field basis, e and t , in
g 2g
usedsixbathstatesperspin-orbitalinEq.(16),butonly which the hybridization function is diagonal. The auxil-
the physicallyrelevantelements with the largestvalueof iary bath states between -17.9 and -19.5 eV correspond
the utility function closelytothe O2sorbitals,whilethe resthavemainlyO
2p character.
E˜
M(V˜,E˜)= V˜V˜†, (20)
−1˜+3E˜ 3
| |
wereincluded inthe finalEDcalculation.Here M(V˜,E˜) Entanglement
is given in a basis where E˜ is diagonal. The utility
functionM(V˜,E˜)excludescontributionsfromthe broad Correlation and projection
high-energystructure describedabove,andimprovesthe
stability with respect to an initially metallic density. The term correlation has come to represent all elec-
The post-selection procedure on the basis of the util- tronic or quasi-particle interactions that go beyond a
ity function worsens the fit on the Matsubara axis, but simple single Slater determinant description. It can be
9
where
TableII.Self-consistentSlaterparametersandauxiliarybath
state fitting parameters. F0 was held fixed at the tabulated = Aˆ† 0 ;Aˆ† ′ ,
value, while F2, F4, and the bath state parameters (Bath A | i ∈A
n o
par.) were recalculated at each newiteration. Thebath state = Bˆ† 0 ;Bˆ† ′ .
parameters are given in the crystal field basis eg and t2g in B | i ∈B
n o
which the hybridization function is diagonal. All values are The localprojectionof a pure state Ψ upon A
given in eV. | i∈A⊗B
is obtained by taking the partial trace over :
B
Slater par. Bath par. (eg) Bath par. (t2g)
TMO F0 F2 F4 E V E V E V E V ρˆA = ρˆAijAˆ†i|0ih0|Aˆj, (21)
ij
MnO 6.0 9.0 6.1 -5.1 2.1 – – -6.2 1.4 – – X
FeO 6.5 9.2 6.2 -4.4 1.9 -17.9 2.1 -8.9 0.7 -4.6 1.2 where ρˆA = α α∗ . The locally projected state ρˆA
ij k ik jk
CoO 7.0 10.0 6.7 -4.6 1.8 -17.9 2.0 -7.5 0.9 -4.0 1.0 is in general a mixed state, unless ρˆA is idempotent.
P ij
NiO 7.5 10.1 6.7 -5.7 1.9 -19.5 2.0 -8.8 0.9 -5.1 0.8 Itbecomeshardertodistinguishtwostatesafteralocal
projection since some information is lost in the partial
trace over . For example, the highly mixed state ρˆ, the
divided into two parts, classical and quantum correla- B
pureandseparablestateρˆ′,andthepureentangledstate
tion. The former gives rise to mixed states, and the lat-
ρˆ′′ given by
tertoentangledstates.Thefourdifferentcombinationsof
1
classical and quantum correlation result in the following ρˆ= ˆc†ˆc† 0 0ˆc ˆc +ˆc†ˆc† 0 0ˆc ˆc +
classifications of an N-electron state in terms of wave- 4 1 2| ih | 2 1 1 3| ih | 3 1
(cid:16)
functions and density operators cˆ†cˆ† 0 0ˆc cˆ +cˆ†ˆc† 0 0ˆc cˆ ,
2 4| ih | 4 2 3 4| ih | 4 3
N cˆ† +cˆ† ˆc† +ˆc† ˆc +cˆ cˆ +cˆ(cid:17)
Ψ′ = ˆc† 0 pure separable Uncorr. ρˆ′ = 1 3 2 4 0 0 2 4 1 3,
| ii ip| i ) √2 √2 | ih | √2 √2
p
Y N ˆc† +ˆc† ˆc† ˆc†
|Ψii = aij ˆc†jp|0i pure entangled  ρˆ′′ = ˆc†1 2√2 3 + 2√−2 3ˆc†4!|0i
j p
ρˆ′ = XXi p′i|ΨY′iihΨ′i| mixed separable  Corr. h0|(cid:18)ˆc2√+2cˆ3ˆc1+ˆc4ˆc2√−2ˆc3(cid:19),
ρˆ = p Ψ Ψ mixed entangled
i i i become indistinguable after a local projection onto the
where 0 Xisi the| vaichuum| state. Here p andp′ are sets oprobsistiabllse1toanddra2w(a{n1y,2c}oAn)c.luItsiiosntshaebreofuotrethine gcelansesriaclalnoort
ofprob|abiilities normalizedto one,andai areeixpansion quantum correlations in the original unprojected state
ij
from its local projection.
coefficients.Thevectorialindicesiandj arecomposedof
Quantumcorrelationscanstemfromphysicalelectron-
the ordered components i and j referring to a generic
p p
one-particle basis. For example the state cˆ†cˆ†ˆc† 0 cor- electron interaction, e.g. Coulomb interaction, or be in-
1 3 7| i duced by a non-local projection. A non-local projection
responds to a many body index i=(1,3,7).
isqualitativelydifferentfromalocalprojectioninthatit
Classical correlations can derive from physical pro-
projectsonto a setof many-bodystates,andnot a setof
cesses, e.g. thermal decoherence, but also be induced by
orbitals.Theentanglementoftheprojectedstatemaybe
a local projection upon some set of orbitals, e.g. the d-
larger than that of the original state. A trivial example
or f-orbitals of single atom in a solid. To see this, let us
is when the non-local projection is defined with respect
consider a set orbitals A labelled by the indices j ,
p A
{ } to a single entangled state Ψ , as any separable state
and define the operator subspaces | i
with a non-zero overlap with Ψ becomes entangled af-
| i
Nj ter the projection. A less trivial example is given by the
A′ =Aˆ† = aj ˆc†jp;jp ∈{jp}A, |aj|2 =1, projection operator
 Xj Yp Xj  Pˆ =cˆ† cˆ† 0 0ˆc cˆ +cˆ† ˆc† 0 0ˆc cˆ . (22)
 Nj  1↑ 2↓| ih | 2↓ 1↑ 1↓ 2↑| ih | 2↑ 1↓
B′ =Bˆ† = bj cˆ†jp;jp ∈/ {jp}A, |bj|2 =1. The projection with Pˆ makes the pure separable spin-
 Xj Yp Xj  coherent state (1/2)(ˆc†1↑ +cˆ†1↓)(ˆc†2↑ +ˆc†2↓)|0i entangled.
In contrast to the first example it should be noted that
The full Hilbert space H can be partitioned as  Pˆ projects onto two separablestates. Moreover,the pro-
jection itself corresponds to a simple Stern-Gerlach ex-
= α Aˆ†Bˆ† 0 ;Aˆ† ′,Bˆ† ′ , periment of a spin-1 particle with a single slit to remove
H A⊗B ≡ ij i j| i i ∈A j ∈B 
Xij  the spin ±1 components.
 
10
an overlap equal to one. The latter case can be proved
function F is: by noting that the first electron removal leads to a state
input: pure state |Ψi, current maximal overlap O′
max with one single electron, which is necessarily separable.
1. #Measurement optimization: Hence, no intensity will be lost in the second measure-
Diagonalize the one-particle density matrix ment, which implies that the maximal overlap is equal
ρ˜=V˜D˜V˜†, where ρ˜ij =hΨ|ˆc†jˆci|Ψi, such that the to the maximal probability of successfully removing the
eigenvalues D˜ii are sorted from largest tosmallest. first electron.
2. #Detect the vacuum state: The search space of the algorithm can be extended
if (D˜11 =0) return hΨ|Ψi further,e.g.by shifting the orderof the electronremoval
3. #Perform the measurement and add another measurements or converting electrons to holes, but such
one-particle detector: development is beyond the scope of this article.
for i=1,2,··· , numberof orbitals
if (D˜ii>Om′ ax) then
Entropic entanglement measure
Om′ ax ←max(cid:16)Om′ ax,FhPjˆcjV˜ji|Ψi,Om′ axi(cid:17)
end if
Forageneralmany-bodydensityoperatorρˆ,thelinear
done
entropy
4. return O′
max
end F SL[ρˆ]=Tr[ρˆ(1ˆ ρˆ)]
−
measures how far such a state is from being pure. It has
Figure3.Apseudocoderepresentation oftherecursivedepth
first search function F[|Ψi,O′ ]. Notethat thestate |Ψi is beenknownforalongtime[16]thatρˆcanberepresented
max
notrenormalizedaftereachmeasurement,andthattheinitial by a single Slater determinant, i.e. a pure and separable
valueofO′ canbesettozero.Commentsaregiveninitalic. state, if and only if the one-particle reduced density ma-
max
trix ρ˜ =Tr(ρˆcˆ†ˆc ) is idempotent. The linear entropyof
ij j i
the one-particle reduced density matrix
Pure separable state search algorithm
S [ρ˜]=Tr[ρ˜(1˜ ρ˜)]
L
−
The problemoffinding the maximaloverlapO be-
max
measures the deviation from idempotency, which makes
tweenanN-electronpure state Ψ andany pure separa-
| i it sensitive to both classical correlations and entangle-
ble state can be written as
ment in ρˆ. A lower bound on the contribution from
K 2 the entanglement can be obtained by removing the
O [Ψ ]=max 0 ˆc V˜ ˆc V˜ Ψ , maximal contribution from a mixed but separable en-
max | i V˜ (cid:12)(cid:12)(cid:12)h |i1X···iN iN iNN··· i1 i11| i(cid:12)(cid:12)(cid:12) sSem[ρˆb′l]e=ρˆS′ =[ρˆ] Qia=n1dp′iN|Ψ=′iihTΨr′i[|ρ˜,]u=ndTerr[ρ˜′t]h.eLceotnusstracionnstihdaert
(cid:12) (cid:12) L L
where K is the num(cid:12) ber of orbitals and V˜ is a unit(cid:12)ary a fictitious exPtended system, with N electrons and QN
matrix. A direct numerical optimization of the overlap, orbitals, for which we define
with respect to the parameters in V˜, becomes quickly
verycumbersomeasthenumberoforbitalsandelectrons Q
ρ˜′ = p′ˆc† ˆc† 0 0ˆc ˆc .
increase. A practical alternative to a brute force numer- e i Ni··· 1+N(i−1)| ih | 1+N(i−1)··· Ni
ical optimization is the search function F, schematically Xi
presented in pseudocode form in Fig. 3. The algorithm
The concavity of S implies that
L
mimics an experiment where a sequence of one-electron
removal measurements are performed. It is easy to visu- S [ρ˜′] S [ρ˜′].
alize the searchforthe optimalV˜ asanextremizationof L ≤ L e
the intermediate intensities at each step in the measure- Evaluating the previous expression finally leads to
ment process. As the intensity monotonically decreases
at each step, it is advantageous to cast the algorithm SL[ρ˜′] Tr[ρ˜]SL[ρˆ]
≤
as a recursive depth first search function F[Ψ ,O′ ],
| i max An analagous construction can be made for
where O′ is the currently known maximal overlap.
max M =Tr[1˜] N holes and QM orbitals, giving that
Although F [Ψ ,0] may not return the maximal over- −
| i
lap for a general state, it is guaranteed to be exact if S [ρ˜′] Tr[1˜ ρ˜]S [ρˆ].
L L
Ψ is separable or representsa 2-electronsystem. In the ≤ −
| i
former case Ψ can be represented as a single Slater de- Hence, the state ρˆis necessarily entangled if
| i
terminant. Its natural orbitals are therefore either com-
pletely occupied or empty, and the algorithmwill return E [ρˆ]=S [ρ˜] min(Tr[ρ˜],Tr[1˜ ρ˜])S [ρˆ]>0. (23)
L L L
− −