Reading entanglement in terms of spin configurations in quantum magnets

0.34 MB·English

by  Andrea Fubini

EPJ manuscript No.
(will be inserted by the editor)
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Reading entanglement in terms of spin configurations in
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a quantum magnets
J

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2 Andrea Fubini1,2, Tommaso Roscilde3, Valerio Tognetti2,4,6, Matteo Tusa2, and Paola Verrucchi4,5

] 1 MATIS CNR-INFM & Dipartimento di Metodologie Fisiche e Chimiche - Universit`a di Catania, V.le A. Doria 6, I-95125
h
Catania, Italy
c 2 Dipartimento diFisica dell’Universit`a di Firenze, Via G. Sansone 1, I-50019 Sesto F.no (FI), Italy
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m 3 Department of Physics and Astronomy,Universityof Southern California, Los Angeles, CA 90089-0484
4 CNR-INFM,UdRFirenze, Via G. Sansone 1, I-50019 Sesto F.no (FI), Italy
t- 5 Istituto deiSistemi Complessi - CNR, Sez. diFirenze via Madonna del Piano, I-50019 Sesto F.no (FI), Italy
a 6 Istituto Nazionale diFisica Nucleare, Sez. diFirenze, Via G. Sansone 1, I-50019 Sesto F.no (FI),Italy
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s
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t Received: date/ Revised version: date
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m
Abstract. We consider a quantum many-body system made of N interacting S=1/2 spins on a lattice,
-
and develop a formalism which allows to extract, out of conventional magnetic observables, the quantum
d
n probabilities for any selected spin pair to be in maximally entangled or factorized two-spin states. This
o resultisusedinordertocapturethemeaningofentanglementpropertiesintermsofmagneticbehavior.In
c particular,weconsidertheconcurrencebetweentwospinsandshowhowitsexpressionextractsinformation
[ on the presence of bipartite entanglement out of the probability distributions relative to specific sets of

two-spin quantum states. We apply the above findings to the antiferromagnetic Heisenberg model in a
2
uniformmagneticfield,bothonachainandonatwo-legladder.UsingQuantumMonteCarlosimulations,
v
we obtain the above probability distributions and theassociated entanglement, discussing their evolution
0
underapplication of the field.
8
2
5 PACS. 03.67.Mn Entanglement production, characterization, and manipulation – 75.10.Jm Quantized
0 spin models – 05.30.-d Quantum statistical mechanics
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0
/ 1 Introduction is of crucial relevance in developing possible solid-state
t
a devices for quantum computation.
m
In this context a privileged role is played by the con-
- Entanglement properties have recently entered the tool currence C, which measures the entanglement of forma-
d
kit for studying magnetic systems, thanks to the insight tionbetweentwoq-bitsbyanexpressionwhichisvalidnot
n
they provide on aspects which are not directly accessi- only for pure states but also for mixed ones[8,9]. In the
o
c blethroughtheanalysisofstandardmagneticobservables framework of interacting spin systems, exploiting differ-
: [1,2,3,4,5,6,7]. The analysis of entanglement properties ent symmetries of such systems the concurrence has been
v
is particularlyindicated whenever purely quantum effects relatedtospin-spincorrelatorsandtomagnetizations.[10,
i
X come into play, as in the case of quantum phase tran- 11,12,13]However,C hasnotyetbeengivenageneralin-
r sitions. However, in order to gain a deeper insight into terpretation from the magnetic point of view, and a gen-
a quantumcriticality,as wellas into other phenomenasuch uinely physicalunderstandingofits expressionis stillelu-
as field-induced factorization [7] and saturation, the con- sive.
nection between magnetic observables and entanglement Scope ofthis paperis thereforethatofgivinga simple
estimators should be made clearer, a goal we aim at in physicalinterpretationofbipartiteentanglementofforma-
this paper. On the other hand, most entanglement esti- tion, building a direct connection between entanglement
mators,asdefinedfor quantummagneticsystems,areex- estimatorsandoccupationprobabilitiesoftwo-spinstates
pressed in terms of magnetizations and spin correlation in aninteracting spin system.To this purpose we develop
functions.Itcomesthereforenaturaltowonderwhere,in- ageneralformalismforanalyzingthespinconfigurationof
side the standard magnetic observables, the information the system, so as to directly relate it with the expression
about entanglement is actually stored, and how entangle- ofthe concurrence.The resulting equations are then used
ment estimators can extract it. Quite clearly, by posing toreadourdatarelativetotheS =1/2antiferromagnetic
thisquestion,onedoesalsoaddresstheproblemoffinding Heisenberg model in a uniform magnetic field, both on a
a possible experimental measure of entanglement, which chainandonatwo-legladder.Themodelisacornerstone
2 AndreaFubiniet al.: Readingentanglement in terms of spin configurations in quantummagnets
inthestudyofmagneticsystems,extensivelyinvestigated The averages represent expectation values over the
h · i
and quite understood in the zero-field case. When a uni- ground state for T = 0, and thermodynamic averages for
form magnetic field is applied the behavior of the system T >0.
is enriched, gradually transforming its ground state and We now show that the above single-spin and two-spin
thermodynamic behavior. The analysis of entanglement quantitiesprovideadirectinformationonthespecificquan-
properties in this model, and in particular that referring tum state of any two spins of the system. Let us consider
to the range of pairwise entanglement as field increases, the T = 0 case first: For a lighter notation we drop site-
sheds new light not only onthe physicalmechanism lead- indexes,allowingtheirappearancewheneverneeded.After
ing to magnetic saturation in low-dimensional quantum selectingtwospins,sittingonsiteslandm,anypurestate
systems,but alsoonthe nature of someT =0 transitions of the system may be written as
observed in bosonic and fermionic systems, such as that
of hard-core bosons with Coulomb interaction, and that Ψ = ν c Γ , (3)
| i | i νΓ| i
described by the bond-charge extended Hubbard model,
ν Γ
X∈S X∈R
respectively. In the former case, the connection between
magneticandbosonicmodelisobtainedbyanexactmap- where is an orthonormal basis for the 4-dimensional
S
pingthatallowsastraightforwardgeneralizationofourre- Hilbert space of the selected spin pair, while is an or-
sultstothediscussionofthephasediagramofthestrongly thonormalbasisforthe2N−2-dimensionalHilbRertspaceof
interacting boson-Hubbard model [14]. In the more com- the rest of the system. Moreover,in order to simplify the
plexcaseofthethebond-chargeextendedHubbardmodel, notation,weunderstandproductsofketsrelativeto(oper-
adirectconnectionbetweenthe Heisenbergantiferromag- ators acting on) different spins as tensor products, mean-
net in a field is not formally available, but a recent work whiledroppingthecorrespondingsymbol .Thequantum
⊗
by Anfossi et al.[15] has shown that some of the T = 0 probability for the spin pair to be in the state ν , being
transitions observed in the system are characterized by the system in the pure state |Ψi, is pν ≡ Γ |c|νΓi|2, and
loobnsge-rrvaenignedoupramirwagisneeteinctmanogdleelmaetnstaotufrtahteionsa.me type we theWnoermcoanlsizidaetirotnhcroenedpiatirotnicuhΨla|rΨbia=se1sifmorptPliheesspiνnppνa=ir:1.
P
Ourdataresultfromstochasticseriesexpansion(SSE)
u , u , u , u , (4)
quantum Monte Carlo simulations based on the directed- S1≡{| Ii | IIi | IIIi | IVi}
loop algorithm[16]. The calculations were carried on a 2 e1 , e2 , e3 , e4 , (5)
S ≡{| i | i | i | i}
chainwithsizeL=64andonaL 2ladderwithL=40. u , u , e , e , (6)
× S3≡{| Ii | IIi | 3i | 4i}
In order to capture the ground-state behavior we have
considered inverse temperatures β =2L. with
InSec.2wedefinethemagneticobservableswereferto,
u , u ,
and develop the formalism which allows us to write them | Ii≡|↑il|↑im | IIi≡|↓il|↓im
in terms of probabilities for two spins to be in specific u , u ,
| IIIi≡|↑il|↓im | IVi≡|↓il|↑im
states, both at zero and at finite temperature. In Sec. 3 e = 1 (u + u ) , e = 1 (u u ) ,
weshowhowconcurrenceextracts,outoftheaboveproba- | 1i √2 | Ii | IIi | 2i √2 | Ii−| IIi
bilities,thespecificinformationonbipartiteentanglement e = 1 (u + u ) , e = 1 (u u ) , (7)
| 3i √2 | IIIi | IVi | 4i √2 | IIIi−| IVi
offormation.InSecs.4and5wepresentourSSEdatafor
the antiferromagnetic Heisenberg model on a chain and where ( ) are eigenstates of Sz with eigen-
| ↑il,m | ↓il,m l,m
on a square ladder respectively, and read them in light of value +1( 1). For the coefficients entering Eq. (3), and
the discussion of Secs. 2 and 3. Conclusions are drawn in for each2st−at2e Γ, the following relations hold
Sec. 6.
c = 1 (c +c ),c = 1 (c c ) , (8)
1Γ √2 IΓ IIΓ 2Γ √2 IΓ − IIΓ
c = 1 (c +c ),c = 1 (c c ) , (9)
2 From magnetic observables to spin 3Γ √2 IIIΓ IVΓ 4Γ √2 IIIΓ − IVΓ
configurations meaning also
c 2+ c 2 = c 2+ c 2 , (10)
We study a magnetic system made of N spins S = 1/2 | 1Γ| | 2Γ| | IΓ| | IIΓ|
sitting on a lattice. Each spin is described by a quantum c 2 c 2 = c c cos(ϕΓ ϕΓ) , (11)
operator S , with [Sα,Sβ] = iδ ε Sγ, l and m being | 1Γ| −| 2Γ| | IΓ IIΓ| I − II
l l m lm αβγ l c 2+ c 2 = c 2+ c 2 , (12)
the site-indexes. | 3Γ| | 4Γ| | IIIΓ| | IVΓ|
The magnetic observables we consider are the local c 2 c 2 = c c cos(ϕΓ ϕΓ) , (13)
| 3Γ| −| 4Γ| | IIIΓ IVΓ| III − IV
magnetization along the quantization axis:
Mz Sz , (1) wheArecccoνrΓdi≡ng|ctνoΓ|ethiϕeΓν.usual nomenclature and are
l ≡h li S1 S2
the standard and Bell bases, respectively,while is here
3
S
andthecorrelationfunctionsbetweentwospinssittingon calledthemixedbasis.Suchbasesarecharacterizedbythe
sites l and m: fact that states corresponding to parallel and antiparallel
gαα SαSα . (2) spinsdonotmixwitheachother.Itthereforemakessense
lm ≡h l mi
AndreaFubiniet al.: Readingentanglement in terms of spin configurations in quantummagnets 3
toreferto u , u , e ,and e asparallel states,andto
| Ii | IIi | 1i | 2i
u , u , e , and e as antiparallel states. The proba-
| IIIi | IVi | 3i | 4i
bilities specifically related with the elements of will be
1
S
hereafter indicated by p ,p ,p , and p while p ,p ,p ,
I II III IV 1 2 3
and p will be used for those relative to the elements of
4
. From the normalization conditions
2
S
p +p +p +p =1 (14)
I II III IV
p +p +p +p =1 (15)
1 2 3 4
p +p +p +p =1 , (16)
I II 3 4
or equivalently from Eqs. (10) and (12), follows p +p =
I II
p +p , and p +p = p +p , representing the prob-
1 2 III IV 3 4
ability for the two spins to be parallel and antiparallel,
respectively.Wedoalsonoticethattheelementsof are
1
S
factorized states, while those of are maximally entan-
2
S
gled ones.
The above description is easily translated in terms of
the two-site reduced density matrix
ρ= Γ Ψ Ψ Γ = ν λ c c , (17)
h | ih | i | ih | νΓ ∗µΓ
Γ νλ Γ
X X X
whose diagonal elements are the probabilities for the ele-
mentsofthebasischosenforwritingρ.Thenormalization
conditions Eqs. (14-16) translate into Tr (ρ)=1.
Thanks to the above parametrization, the magnetic
observables(1)and(2)aredirectlyconnectedtotheprob-
abilities of the two spins being in one of the states (7). In
fact it is
4 AndreaFubiniet al.: Readingentanglement in terms of spin configurations in quantummagnets
2(gxx+gyy)=hΨ|Sl+Sm−+Sl−Sm+|Ψi=
=hΨ|Sl+Sm−+Sl−Sm+| |e3i c3Γ|Γi+|e4i c4Γ|Γi!=
Γ Γ
X X
= Ψ e c Γ e c Γ
h | | 3i 3Γ| i−| 4i 4Γ| i!
Γ Γ
X X
=(p p ) , (18)
3 4
−
AndreaFubiniet al.: Readingentanglement in terms of spin configurations in quantummagnets 5
and similarly to the notation of the previous section, it is easily shown
that
2(gxx gyy)=(p p ) , (19)
1 2
gzz =−21(pI +pII)−−14 = 12(p1+p2)− 41 , (20) C(|φi)= (c21 −c22)−(c23 −c24) =2|cIcII −cIIIcIV| , (30)
M 1(Mz+Mz)=(p p ) , (21) (cid:12) (cid:12)
z ≡ 2 l m I − II whereEqs.(cid:12)(8)-(9)havebeenuse(cid:12)d,withindexΓ obviously
suppressed. The above expression shows that C extracts
whereall aresuitabletocalculategzz,while(gxx gyy)
i the information about the entanglement between the two
S ±
and M specifically require and , respectively. After
z 2 3 spins by combining probabilities and phases relative to
S S
Eqs. (18)-(21), one finds
specific two-spin state.
In fact, one should notice that a finite probability for
p = 1 +gzz+M , (22)
I 4 z two spins to be in a maximally entangled state does not
p = 1 +gzz M , (23) guarantee per se the existence of entanglement between
II 4 − z
p = 1 +gxx gyy+gzz , (24) them, since this probability may be finite even if the two
1 4 − spins are in a separable state.[18] In a system with de-
p2 = 41 −gxx+gyy+gzz , (25) caying correlations,at infinite separation all probabilities
p = 1 +gxx+gyy gzz , (26) associated to Bell states attain the value of 1/4, but this
3 4 − of course tells nothing about the entanglement between
p = 1 gxx gyy gzz . (27)
4 4 − − − them, which is clearly vanishing. It is therefore expected
that differences between such probabilities, rather than
It is to be noticed that the probabilities relative to the
the probabilities themselves give insight in the presence
Bell states do not depend on the magnetization.
or absence of entanglement.
Inthethefinitetemperaturecase,thegeneralizationis
When the many-body case is tackled, the mixed-state
straightforwardly obtained by writing each of the Hamil-
concurrenceofthe selected spinpair has an involveddefi-
tonian eigenstates, numbered by the index n, in the form
nition in terms of the reduced two-spindensity matrix.[9]
(3), so that
However,possiblesymmetriesoftheHamiltonian greatly
H
simplifytheproblemtothe extentthatC resultsasimple
ρ(T)= |νihµ| e−En/T cνΓ,nc∗µΓ,n . (28) function of the probabilities (22)-(27) only. We here as-
νµ n Γ sume that is real, has parity symmetry (meaning that
X X X H
either leaves the z component of the total magnetic
Intermsofprobabilitiestheaboveexpressionsimplymeans momenHt unchanged, or changes it in steps of 2), and is
thatthepurelyquantumpµ shallbereplacedbythequan- further characterized by translational and site-inversion
tum statistical probabilities invariance. The two latter properties implies Mz as de-
l
fined in Eq. (1) to coincide with the uniform magnetiza-
pµ(T)≡ e−En/T |cνΓ,n|2 . (29) tion Mz ≡ lhSlzi/N, and the probabilities pIII = pIV,
n Γ respectively.
X X P
Under these assumptions, the concurrence for a given
Therefore,apartfrom the further complicationof the for- spin pair is[13]
malism,thediscussiondevelopedforpurestatesstayssub-
stantially unchanged when T >0. C 2max 0,C ,C , (31)
Equations (22)-(27) show that magnetic observables (r) ≡ { (′r) (′′r)}
allow a certain insight into the spin configuration of the C gxx +gyy 1 +gzz 2 M2 , (32)
system, as they give, when properly combined, the prob- (′r) ≡| (r) (r)|− 4 (r) − z
r
abilities for any selected spin pair to be in some spe- C gxx gyy 1 +(cid:16)gzz , (cid:17) (33)
cificquantumstate.However,themereknowledgeofsuch (′′r) ≡| (r)− (r)|− 4 (r)
probabilities is not sufficient to appreciate the quantum
where r is the distance in lattice units between the two
characteroftheglobalstate,andmorespecificallytoquan-
selectedspins.Despitebeingsimple combinationsofmag-
tify its entanglement properties.
netic observables, the physical content of the above ex-
pressions is not straightforward. However, by using the
expressionfoundinSection2,onecanwriteEqs.(32)and
3 From spin configurations to entanglement (33)intermsoftheprobabilitiesforthetwospinstobein
properties maximally entangled or factorized states, thus finding, in
some sense, an expression which is analogous to Eq. (30)
for the case of mixed states. In fact, from Eqs. (18)-(19),
We here analyze the entanglement of formation[17,8,9]
it follows
betweentwospins,quantifiedbytheconcurrenceC.Inthe
simplestcaseoftwoisolatedspinsinthepurestate φ the
concurrencemaybewrittenasC =| iα2i|,where|αii are 22CC′ ==|pp3−pp4|−(21√pIppII , p )= (34)
thecoefficientsenteringthedecompositionof φ uponthe ′′ 1 2 1 2
| − |− − −
magic basis {|e1i,i|e2i,i|e3i,|e4i}. HPowever,|ifione refers =|p1−p2|−2√pIIIpIV , (35)
6 AndreaFubiniet al.: Readingentanglement in terms of spin configurations in quantummagnets
where we have used pIII=pIV and hence p3 +p4 = 2pIII = 0.5 Mz
2√pIIIpIV. The expression for C′′ may be written in the 0.4 g(1)xx
particularly simple form 0.3 g(1)zz
g xx
2C′′ =2max{p1,p2}−1 , (36) 0.2 g(2z)z
(2)
0.1
telling us that, in order for C to be positive, it must be
′′
either p > 1/2 or p > 1/2. This means that one of the 0
1 2
two parallel Bell states needs to saturate at least half of
-0.1
the probability, which implies that it is by far the state
where the spin pair is most likely to be found. -0.2 0 0.5 1 1.5 2
Despite the apparently similar structure of Eqs. (34) h
and (35), understanding C′ is more involved, due to the Fig.1.Magnetizationandcorrelatorsversusthemagneticfield
fact that √pIpII cannot be further simplified unless pI = h for the chain Eq. (40). The dashed lines mark the value of
pII.The markeddifference betweenC′ andC′′ reflects the thefield where pII =0 (see text).
differentmechanismthroughwhichparallelandantiparal-
lel entanglement is generated when time reversal symme-
tryisbroken,meaningp =p andhenceM =0.Infact, case in which the two spins are not entangled with the
I 6 II z 6
in the zeromagnetizationcase,it is p =p =(p +p )/2 rest of the system. By using Eq. (11), one can select two
II I 1 2
and hence particular situations all leading to p1 =p2:
2C =2max p ,p 1 , (37) (i) c or c vanishes Γ, meaning that Ψ does not
′ { 3 4}− contaIΓin statIeIΓs where the∀two selected spins| aire parallel
which is fully analogous to Eq. (36), so that the above
and entangled;
analysis can be repeated by simply replacing p and p
1 2 (ii) for each Γ such that both c and c are non-
with p and p . | IΓ| | IIΓ|
3 4 zero,itisϕΓ ϕΓ =π/2.Thus,whatevertheantiparallel
ForM =0,the structureofEq.(37)issomehowkept I − II
z 6 components are, the parallel terms of Ψ appear in the
by introducing the quantity | i
form (αe +α e ).
1 ∗ 2
| i | i
The above analysis suggests the first term in C (C )
∆2 (√p √p )2 , (38) ′′ ′
≡ I − II to distill, out of all possible parallel (antiparallel) spin
configurations, those which are specifically related with
so that
entangledparallel(antiparallel)states.These characteris-
2C =2max p ,p (1 ∆2) , (39)
′ { 3 4}− − tics reinforce the meaning of what we have called parallel
meaning that the presence of a magnetic field favors bi- and antiparallel entanglement.
partiteentanglementassociatedtoantiparallelBellstates,
e and e .Infact,whentimereversalsymmetryisbro-
3 4
| i | i
ken the concurrence can be finite even if p , p <1/2. 4 Chain
3 4
From Eqs. (36) and (39) one can conclude that, de-
pending on C being finite due to C′ or C′′, the entangle- We consider the isotropic Heisenberg antiferromagnetic
ment of formation originates from finite probabilities for chain in a uniform magnetic field, described by
the two selected spins to be parallel or antiparallel, re-
spectively. In this sense we will speak about parallel and H = S S hSz , (40)
antiparallel entanglement. J i· i+1− i
i
Moreover,fromEqs.(34)-(35)we notice that,inorder X
for parallel (antiparallel) entanglement to be present in wheretheexchangeintegralJ ispositive,andthereduced
the system,the probabilitiesforthe twoparallel(antipar- magnetic field h gµ H/J is assumed uniform.
B
≡
allel) Bell states must be not only finite but also different This model is characterized by the rotational symme-
from each other. Thus, the Bell states e and e (e try on the xy plane, as well as by the existence of a satu-
1 2 3
| i | i | i
and e ) result mutually exclusive in the formation of ration field h =2, such that for h h the ground state
4 s s
| i ≥
entanglement between two spins in the system, the latter is the factorized ferromagnetic one, with all spins aligned
beingpresentonlyifoneoftheBellstateismoreprobable along the field direction. Moreover, Eq. (40) has all the
than the others. The case p =p = 1/2 (p = p =1/2) necessary symmetries for Eqs. (31)-(33) to hold.
1 2 3 4
corresponds in turn to an incoherent mixture of e and Due to the rotational symmetry on the xy plane, it is
1
e (e and e ). | i gxx =gyy,meaningp =p = 1+gzz 1/2,accordingto
| 2i | 3i | 4i 1 2 4 ≤
In fact, the occurrence of the differences p p and Eqs. (24) and (25), and hence null parallel entanglement
1 2
| − |
p p is intriguing.Let uscommenton p p ,as the (C 0) between any two spins along the chain, no mat-
3 4 1 2 ′′
| − | | − | ≤
samekindofanalysisholdsfor p p .Inthegeneralcase ter the field, the temperature, and the distance between
3 4
| − |
thedifferencep p canvanishbecauseofgenuinemany- them.
1 2
−
bodyeffects whicharenotdirectlyreadableintermsof2- In Fig. 1 we show the T = 0 correlation functions for
spinentangledorseparablestates.It is easierto interpret nearest neighboring (n.n.) and next-nearest neighboring
Eq. (35) [Eq. (34)], if one restricts the possibilities to the (n.n.n.) spins, together with the uniform magnetization,
AndreaFubiniet al.: Readingentanglement in terms of spin configurations in quantummagnets 7
1 1
0.75 p4 p = 0II 0.75 pI
0.5 0.5
C
(1) p
3
0.25 p 0.25 p
I II C
p p (2)
3 4
0 pII 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
h h
Fig. 2. Concurrence and related probabilities of the mixed Fig. 3. Concurrence and related probabilities of the mixed
basis states 3 [Eq. (6)] for n.n. sites of thechain Eq. (40) basis states 3 [Eq. (6)] for n.n.n.sites of the chain Eq. (40).
S S
0.4
asthe fieldis varied.Beyondthe overallregularbehavior,
we notice that there exists a value of the magnetic field 0.3 8 R
whereonesimultaneouslyobservesgzz =0andM =1/4
(indicated by the dashed lines). Ac(c1o)rding to Eqzs. (20) C(r)0.2 6
and(21)thisimpliesnullprobabilityp foradjacentspins 4
II
to be parallel in the direction opposite to the field. This 0.1
2
means that the ground-stateconfigurationis a superposi- 1.4 1.6 1.8 2
tionofspinconfigurationsentirelymadeofstableclusters
0
ofspinsparalleltothefieldseparatedbyN´eel-likestrings. 0 0.5 1 1.5 2
h
In Fig. 2 we show the probabilities for n.n. spins to
Fig.4.Fromtheuppertothelowercurves:concurrencesfrom
be in the states ofthe mixed basis,together with the n.n. thenearest-uptothe5th-neighborsofthechainEq.(40).The
concurrence:Thevalueofthen.n.concurrenceforh=0is inset shows the divergence of the range of the concurrence as
in agreement with the exact resut in the thermodynamic h hs, theline shows the(h hs)−1/2 behavior.
limit.[12] In presence of an external magnetic field, C → −
(1)
is found positive h, meaning that, no matter the value
∀
of the field, the probabilities p3 and p4 for adjacent spins mum,respectively,inthe fieldregionwhereC(2) getspos-
arealwaysdifferentfromeachother.The probabilitiesfor itive (as from the comparison between Fig. 1 and 3).
the triplet states e , u , and u are equal for h = 0
| 3i | IIi | Ii Regarding the probabilities, one finds that, although
anddepart fromeachother whenthe field is switchedon.
themostlikelystateisalways u ,p issurprisinglylarge,
The singlet e evidently dominates the ground state up | Ii 3
4 andalmostequaltop ,asfarash<1.Moreover,bothp
to a field wh|ichi roughly corresponds to the value where I 3
and p have a non monotonic behavior and increase with
4
pII vanishes. huptothefieldwherewesimultaneouslyobservegxx and
(2)
Asfortheconcurrence,despitetheground-statestruc- gyy attaining their extreme values, p exceeding 1/2, p
(2) I 4
ture evidently changes as the field increases, C stays
(1) getting larger than p , and C switching on.
substantially constant up to a large value of the field, II (2)
As observed in the n.n. case, when p for n.n.n. spins
mainly due to the fact that not only p but also p de- II
4 3 vanishes C switches on. Let us further comment upon
creases with the field. This behavior mimics the one oc- (3)
C , C , and C . Given the fact that only antiparallel
curring in a spin dimer, whose ground state is the singlet (1) (2) (3)
entanglement may exist in this chain, it is not surprising
state e up to h = 1 where, after a level crossing, u
becom|es4ienergetically favored. However, in a spin cha|inIi, that C(1) > 0 and C(2) = 0 at low fields, as n.n. spins
belong to different sublattices, while n.n.n. spins belong
many-body effects smear the sharp behavior of the dimer
to the same sublattice. However, the fact that C be-
duetothelevelcrossing.WedoalsonoticethatC starts (2)
(1) comes finite indicates a ground-state evolution from the
to decrease as soon as the total probability for parallel
N´eel-like to the ferromagnetic state such that the system
spins (p +p ) gets larger than that for antiparallel spins
I II enters a region where quantum fluctuations increase the
(p +p ). The further reduction of C is mainly driven
3 4 (1) totalprobabilityforspinsbelongingtothesamesublattice
by p starting to rapidly increase.
I tobeantiparallelandentangled.Theoppositeeffectisun-
In the same field region where a substantial change derstoodwhenC isconsidered:inordertokeepC =0
(3) (3)
in the n.n. configuration occurs, the n.n.n. concurrence almost up to the saturation field, quantum fluctuations
C switches on. This is seen in Fig. 3, where the proba- must reduce the total probability for spins belonging to
(2)
bilities for n.n.n. spins are showntogetherwith the corre- different sublattices to be antiparallel and entangled.
spondingconcurrence.Infact,whenconsideringthen.n.n. TheabovecommentsuponC andC maybegener-
(2) (3)
quantities, we notice that both gxx and gzz have a non- alizedto C withevenandoddn,respectively.InFig.4
(2) (2) (n)
monotonic behavior, displaying a maximum and a mini- we in fact show C up to n = 5. The concurrence for in-
n
8 AndreaFubiniet al.: Readingentanglement in terms of spin configurations in quantummagnets
creasing distance between the two spins gets finite for a
p
big enough field resembling the phenomenology of finite 0.6 I 0.25
spin clusters.[1] Moreover, combining the exact results of C(1)
0.5
Refs.[19] and [20], we find that the range of the concur- 0.2
renceforthemodel(40),namelythedistanceRsuchthat 0.4
p
C vanishes for r >R, is 4 0.15
(r)
0.3 C
(2)
θ 0.1
ρ 0.2
R= , (41)
(cid:12)√π(2+4Mz)(12 −Mz)1/2(cid:12) 0.1 p3 p 0.05
(cid:12) (cid:12) II
(cid:12) (cid:12)
with the constan(cid:12)t ρ = 0.924.... When h →(cid:12)hs, it is Mz ≃ 00 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 0
1 √2√h h and θ 2 2√2√h h, and the range k T / J k T / J
2 − π s− ≃ − π s− B B
of the concurrence is seen to diverge according to R
ρ√2(h h) 1/2. In other terms, approaching the satura≃- Fig. 5. Left panel: n.n. probabilities versus temperature at
32 s− − h=1.8. Rightpanel:n.n.andn.n.n.concurrencesversustem-
tion field, all C become finite of order O(1/N), con-
(n) peratureat h=1.8.
sistently with the occurrence of a W state[21]. For
N
| i
such state the entanglementis maximally bipartite in the
sense of the Coffman-Kundu-Wootters inequality.[21,22,
Themodel(42)isknowntodescribecupratecompounds
23] This scenario is consistent with our numerical data.
likeSrCu O andithasbeenextensivelystudiedforzero[24,
As shown in Fig.4 up to n = 5, for any C it exists a 2 3
(n) 25] and finite field[26]. The system shows a gap ∆ in the
field h > h such that C is positive for h [h ,2),
with hn 2n−fo1r n . Th(ne)divergence of the∈rannge of excitationspectrumthat can be interpretedessentiallyas
n → → ∞ duetotheenergycostforproducingatripletexcitationon
the concurrenceforh h is showninthe insetofFig.4.
→ s arung [24]. The systemreachesfull polarization[26],with
Although the correct power-law behavior shows up, the
all spins aligned along the field direction, for h>γ+2.
precisionof the numerical data is not sufficient to get the
correctmultiplicative constant. In fact, the above expres- Inthefollowingwewillspecificallyconsidertheisotropic
case γ =1, which is characterizedby a gap∆ 0.5J,[24]
sion (41) is derived from asymptotic exact results, valid
≈
andbyasaturationfieldh =3.As inthe chaincase,due
only for r 1, when C becomes too small to resolve it s
(r)
≫ to the rotational invariance on the xy plane, parallel en-
numerically.
tanglementcannotdevelopintheisotropicladder.Onthe
Theformalismintroducedintheprevioussectionsworks
other hand, antiparallel bipartite entanglement can here
also in the finite temperature case, where it describes the
developbetweenspinsbelongingtothesameleg,ortothe
effects of thermal fluctuations on quantum coherence. In
samerung,or to a differentrung andleg.Two-spinquan-
Fig. 5, the temperature dependence of probabilities and
tities will be hereafter pinpointed by the two-component
concurrences, for h = 1.8, shows how thermal fluctua-
vector(r ,r )joiningthetwoselectedspins,thefirstcom-
tions progressively drive the system towards an incoher- i α
ponentreferringtothedirectionofthelegs,andthesecond
entmixtureofstates.IncreasingT theconcurrences(right
panel)areprogressivelysuppressedandabovek T 0.8J onetothatoftherungs.Theindexes(01),(10),(11),(20)
B ∼ will therefore indicate n.n. spins on the same rung, n.n.
alsothen.n.concurrencevanishes.Athighertemperatures
along one leg, n.n.n. on adjacent rungs, and n.n.n. along
noneofthespinpairsinthesystemisentangledandquan-
the same leg, respectively.
tum coherence is lost. The temperature behavior of the
probabilities (left panel) is non monotonic, signaling the Our SSE data in Fig. 6 for the uniform magnetiza-
relative weight of the different states in the energy spec- tion and the n.n. correlationfunctions gαα , gαα confirm
(01) (10)
trum of the system. Eventually, at high T all the proba- the description given in the previous paragraph: Before
bilities tends to the asymptotic value pν =1/4. the Zeeman interaction fills the energy gap at the critical
value h 0.5, the ground-state configuration is frozen
c
≃
andcharacterizedbythesinglet e beingbyfarthemost
4
| i
5 Two-Leg Ladder likely state for each rung.
The use of the formalism developed in Sec. 2 gives a
The above picture further enriches when considering the direct information on the physics of the system: In Fig. 7
two-leg isotropic ladder, described by weseethatthesingletprobabilitiesp relativeton.n.spins
4
onarungandalongoneleg,asfunctionsofthefield,share
HJ = (Si,α·Si+1,α−hSiz,α)+γSi,0·Si,1 , (42) awhsiemreilapr bfoehranv.ino.rsepvienrsywsihtetirnegbuotnatthethseacmrietircualngfiesldhohwcs,
i α=0,1 4
X X
up a kink that is not present in the singlet probability
where the index i runs on both the right (α = 0) and alongthe leg. This qualitativelydifferent behavior clearly
left (α = 1) leg. The first term is the Heisenberg Hamil- reflects the nature of the energy gap that closes at h .
c
tonian(40) for the right and left legs, while the last term The sharp decrease of p in favor of p on the rung just
4 I
describes the exchange interaction between spins of the above h testifies that, even in the case, here considered,
c
same rung, whose relative weight is γ. of equal exchange interaction along the legs and on the
AndreaFubiniet al.: Readingentanglement in terms of spin configurations in quantummagnets 9
0.5
g same leg 0.4 same leg
xx
0.4 gxxopposite leg 0.3
g same leg
zz
g opposite leg 0.2
0.3 zz
M
z 0.1
Mz 0.2
, gαα 0.04 opposite leg
0.1 p , p340.3
0 p , II0.2
p , I0.1
-0.1
0
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3 h
h
Fig.8. Fromtheuppertothelowercurves:concurrencesrela-
Fig.6.Magnetizationandcorrelatorsversusthemagneticfield
tivetospinsbelongingtothesame(upperpanel)anddifferent
h for the ladderEq. (42).
legs (lower panel) versus field for the isotropic ladder up to
r=4.
1
0.15 Finally, we apply the formalism of Sec. 3 to extract
featuresof the groundstate fromthe concurrences.Fig.8
0.75 0.1 shows the concurrences C(ri,rα) up to the distance r ≡
r +r =4forspinssittingonthesame(upperpanel)and
i α
p , p34 pI obnetwdieffeenretnwtolsepgisn.sTsihtetinbgipaatrtaitgeivaenntidpiastraanllceel ernitsainnggleemneernatl
p , p , III 0.5 p lCargAersaonenxdpdCeiffcteerde,nttotwhleeagrsfid,eseldve,eancahfbteeoyrthocnelrdo.stiQnhugeitnteh.neu.ngceaaxsppe,.epctuesdhleys,
4 (01) (10)
however, this evolution includes a region where n.n. con-
0.25 currencealongthe leg,C ,increases.Itisinterestingto
(10)
notice that C switches on at h 1.8, where p and
pII p3 pI for n.n. sp(in11s)are seen to cross e≃ach other in F4ig. 6,
signaling the crossover from an antiferromagnetic to a
0
0 0.5 1 1.5 2 2.5 3 ferromagnetic-like configuration of the n.n. spins.
h
Fig. 7. Probabilities relative to the mixed basis versus the
magnetic field h for the ladder Eq. (42): pI (♦), pII ((cid:3)), p3 6 Conclusions
( ), and p4 ( ). Open (full) symbols are for n.n. spins along
△ ∇
the same leg (on the same rung). The inset zooms in on the
behavior of pI,pII, and p3 near thecritical field hc. In this paper we developed a simple and effective formal-
ismthatallowstoreconstructtheprobabilityfortwospins
ofamulti-spinsystemtobeinagivenquantumstate,once
the collective state of the system is given. Remarkably,
rungs(γ =1),thefirstexcitationsintheenergyspectrum
such probabilities are found to be simple combination of
of the ladder are triplet excitations on the rungs.
standardmagneticobservables,Eqs.(22-27).Withinsuch
WhentheZeemanenergybecomeslargerthanthegap, formalism it is very natural to understand how concur-
forh>h ,thegroundstatestartstoevolvewiththefield, rence quantifies the amount ofentanglementbetween two
c
whose immediate effect is that of pushing the quantities spins by comparing the probabilities for those spins to be
relative to spins on the rungs and along the legs towards indifferentBellstates.Inparticulartheexpressionforthe
eachother:Infact,forh>1 n.n.spins alongthe legsand concurrenceclearlyseparatesthecaseofparallel[Eq.(36)]
ontherungssubstantiallysharethesamebehavior.Asfor and antiparallel [Eq. (39)] spins, leading to the introduc-
the probabilities, we see that p and p keep being equiv- tion of the concept of parallel and antiparallel entangle-
II 3
alent, no matter the value of the field, and slowly vanish ment.
as saturation is reached. On the contrary the probability The knowledge of the probability distribution for a
for n.n. spins to be in u increases at the expense of the given set of two-spin states can be a useful tool to study
| Ii
probability relative to the singlet state until, for h 1.8, quantum phases dominated by the formation of particu-
≃
the two probabilities cross each other. larlocal two-spinstates andto investigate the transitions
10 AndreaFubiniet al.: Readingentanglement in terms of spin configurations in quantummagnets
given by the alternation of such states. Within this class 23. T. J. Osborne and F. Verstraete, quant-ph/0502176.
of phenomena we can cite the occurrence of short-range 24. T.Barnes,E.Dagotto,J.Riera,andE.S.Swanson,Phys.
valence-bond states in low-dimensional quantum antifer- Rev.B 47, 3196 (1993).
romagnets [27], and the transition from a dimer-singlet 25. E. Dagotto, and T.M. Rice, Science 271, 618 (1996).
phase to long-range order in systems of weakly coupled 26. G. Chaboussant, M.-H. Julien, Y. Fagot-Revurat,
dimers under application of a field or by tuning of the M. Hanson, L.P. L´evy, C. Berthier, M. Horvati´c, and
inter-dimer coupling [28]. O.Piovesana, Eur. Phys. J. B 6, 167 (1998).
27. S. R. White, R. M. Noack, and D. J. Scalapino, Phys.
Rev.Lett. 73, 886 (1994).
28. M.Matsumoto,B.Normand,T.M.Rice,andM.Sigrist,
7 Acknowledgments
Phys.Rev.B 69, 054423 (2004).
FruitfuldiscussionswithL.Amico,G.Falci,S.Haas,D.Pa-
tan`e,J.Siewert,andR.Vaiaaregratefullyacknowledged.
We acknowledge support by SQUBIT2 project EU-IST-
2001-390083(A.F.),andbyNSFundergrantDMR-0089882
(T.R.).
References
1. M.C. Arnesen, S. Bose, and V. Vedral, Phys. Rev. Lett.
87, 017901 (2001).
2. A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature
(London) 416, 608 (2002).
3. T.J.Osborne,andM.A.Nielsen,Phys.Rev.A66,032110
(2002).
4. S.Ghosh,T.F.Rosenbaum,G.Aeppli,andS.N.Copper-
smith, Nature(London) 425, 48 (2003).
5. G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys.
Rev.Lett. 90, 227902 (2003).
6. F.Verstraete,M.Popp,andJ.I.Cirac, Phys.Rev.Lett.
92, 027901 (2004).
7. T. Roscilde, P. Verrucchi, A. Fubini, S. Haas, and
V.Tognetti,Phys.Rev.Lett.93,167203 (2004);ibid94,
147208 (2005).
8. S. Hill and W.K. Wootters, Phys. Rev. Lett. 78, 5022
(1997).
9. W.K. Wootters, Phys. Rev.Lett. 80, 2245 (1998).
10. X.Wang and P. Zanardi, Phys.Lett. A 301, 1 (2002).
11. O.F. Sylju˚asen, Phys. Rev.A 68, 060301 (2003).
12. U. Glaser, H. Bu¨ttner, and H. Fehske, Phys. Rev. A 68,
032318 (2003).
13. L. Amico, A. Osterloh, F. Plastina, R. Fazio, and
G. M. Palma, Phys.Rev.A 69, 022304 (2004).
14. C. Bruder, R. Fazio, and G. Sh¨on,Phys. Rev.B 47, 342
(1993).
15. A. Anfossi, P. Giorda, A. Montorsi, and F. Traversa,
Phys. Rev.Lett.95, 056402 (2005).
16. O. F. Sylju˚asen and A.W. Sandvik, Phys. Rev. E 66,
046701 (2002).
17. C. H. Bennett, D.P. DiVincenzo, J.A. Smolin, and
W.K. Wootters, Phys. Rev.A 54, 3824 (1996).
18. For instance, the factorized state u can be written as
| Ii
|uIi=(|e1i+|e2i)/√2, implying p1=p2 =1/2.
19. B.-Q. Jin and V.E. Korepin, Phys. Rev. A 69, 062314
(2004).
20. T. Hikihara and A. Furusaki, Phys. Rev. B 69, 064427
(2004).
21. W.Du¨r,G.Vidal,andJ.I.Cirac,Phys.Rev.A62,062314
(2000).
22. V. Coffman, J. Kundu,and W. K. Wootters, Phys. Rev.
A 61 053206 (2000).