Global-to-local incompatibility, monogamy of entanglement, and ground-state dimerization: Theory and observability of quantum frustration in systems with competing interactions

0.34 MB·English

by  S. M. Giampaolo

Global-to-localincompatibility,monogamy ofentanglement, andground-state dimerization:
Theory andobservability ofquantum frustration insystemswithcompeting interactions
S. M. Giampaolo,1,2 B. C. Hiesmayr,2 and F. Illuminati1,3,4,
∗
1Dipartimento di Ingegneria Industriale, Universita` degli Studi di Salerno,
Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy
2University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna, Austria
3CNISM Unita` di Salerno, I-84084 Fisciano (SA), Italy
5 4INFN, Sezione di Napoli, Gruppo collegato di Salerno, I-84084 Fisciano (SA), Italy
1 (Dated:September17,2015)
0
2 Frustration in quantum many body systems is quantified by the degree of incompatibility between the lo-
calandglobalordersassociated,respectively,tothegroundstatesofthelocalinteractiontermsandtheglobal
t
c groundstateofthetotalmany-bodyHamiltonian.Thisuniversalmeasureisboundedfrombelowbytheground-
O statebipartiteblockentanglement. Formany-bodyHamiltoniansthataresumsoftwo-bodyinteractionterms,a
furtherinequalityrelatesquantumfrustrationtothepairwiseentanglementbetweentheconstituentsofthelocal

7 interactionterms.Thisadditionalboundisaconsequenceofthelimitsimposedbymonogamyonentanglement
shareability.Weinvestigatethebehavioroflocalpairfrustrationinquantumspinmodelswithcompetinginter-
]
actionsondifferentlengthscalesandshowthatvalencebondsolidsassociatedtoexactground-statedimerization
h
correspondtoatransitionfromgenericfrustration,i.e. geometric,commontoclassicalandquantumsystems
c
e alike,togenuinequantumfrustration,i.e. solelyduetothenon-commutativityofthedifferentlocalinteraction
m terms.Wediscusshowsuchfrustrationtransitionsseparatinggenuinelyquantumordersfromclassical-likeones
aredetectedbyobservablequantitiessuchasthestaticstructurefactorandtheinterferometricvisibility.
-
t
a
PACSnumbers:03.65.Ud,03.67.Mn,05.30.Rt,75.10.Pq
t
s
.
t
a I. INTRODUCTION frustrationof localinteractionterms. Quantumfrustrationis
m
thensingledoutbytheexistenceofaquantumlowerboundto
- thetotalfrustrationthatisrealizedintermsoftheground-state
ManybodysystemsaretypicallymodeledbyglobalHamil-
d
entanglementbetweenthelocalinteractingsubsystemandthe
n tonians that are sums of local interaction terms. Frustration
remainderofthetotalmany-bodysystem.
o characterizesthosesituationsforwhichitisimpossibletosat-
c isfy simultaneously the minimization of all local interaction In the present work we first show how frustration relates
[ energiesinthepresenceoftheglobalconstraintthatimposes not only to the global block entanglement across a biparti-
3 minimizationofthetotalmany-bodyenergy[1–7]. tionofthetotalmany-bodysystembutalsotothelocalentan-
v In interacting many-body systems frustration may be due glementamong the constituents of the individualinteraction
2 terms. Next,weshowhowtheserelationsrulequantitatively
either to complexgeometricalconfigurationsand/orcompet-
4 the onset of ground-state dimerization and the formation of
ing interactions. These types of frustration are generic in
6 valence bond solids. For many-body Hamiltonians sums of
that they are common to both classical and quantum sys-
4
two-bodyterms,thelocalinteractingpairsarenaturallyiden-
0 tems alike. On the other hand quantum systems, due to the
tified as the elementary subsystems. We deduce a direct re-
. non-commutativityofdifferentlocalHamiltonians, mayfea-
1 lationbetweenquantumfrustrationoflocalpairsandthepair
turefurther,genuinequantum,sourcesoffrustrationwithno
0
entanglementasmeasuredbytheconcurrence[16,17]. Here-
5 classical counterpart[8–10]. In particular,quantumcounter-
withweareabletoestablisharelationbetweenquantumfrus-
1 partsofclassicallyunfrustratedsystemscanbefrustrated[6–
trationandentanglementmonogamy[18,19]whichimposesa
: 8, 10–13]. Therefore, for many-body quantum systems, a
v strongconstraint,withoutclassicalcounterpart,ontheshare-
universalmeasureoftotalfrustrationencompassingallpossi-
i
X blesources,genericandspecificallyquantum,shouldquantify abilityofquantumcorrelations.
r thedegreeofincompatibilitybetweenglobalandlocalground Specifically, we investigate the interplay between generic
a statesassociated,respectively,tothetotalmany-bodyHamil- andquantumfrustrationforanampleclassofspin–1/2mod-
tonianandtothelocalinteractionterms. els with nearest-neighbor(NN) and next-to-nearest-neighbor
(NNN)interactions.Thesemodelsfeaturebothquantumfrus-
Suchameasurehasrecentlybeenintroduced,inRefs.[14,
trationduetothenon-commutativityofthedifferentlocalin-
15],exactlyintermsoftheoverlapbetweenthegroundstates
teractiontermsandgenericfrustrationduetocompetinginter-
oflocalinteractionsandthe localreducedstatesobtainedby
actionsondifferentlengthscales. Weshowthatground-state
partial trace from the global ground state of the total many-
dimerization in these systems is realized exactly at the tran-
body Hamiltonian. It provides a unified treatment of both
sition from generic to quantum frustration, i.e. at values of
generic and specifically quantum contributions to the total
theHamiltonianparametersforwhichthetotalfrustrationbe-
comesentirelyduetotheground-stateblockentanglement.
Remarkably,such modelsare amenableto quantumsimu-
∗Correspondingauthor:[email protected] lation with systems of trappedions [20–24], ultracoldatoms
2
inopticallattices[25–28],coupledcavityarrays[29,30],su- If the global ground state is degenerate, then the measure
perconductingqubits[31,32],ornuclearmagneticresonance can in general depend on which global groundstate is actu-
simulators[33,34]. Therefore,aswewillshowinthefollow- allyconsidered. Uniquenessisguaranteedbyconsideringthe
ing, changes in the ground-state patterns caused by changes aprioriequiprobablestatisticalaverage,i.e. theconvexcom-
in the frustration propertiescan be experimentallyobserved, bination of all possible degenerateglobalgroundstates with
e.g.inoptical-latticerealizations,bymeasuringthequasimo- equalweightsor,inotherwords,themaximallymixedglobal
mentumdistribution as quantifiedby the static structurefac- groundstate(MMGGS)[15].
tor[35,36],ortheinterferometricvisibility[37]. Ground-statedegeneracyinthepresenceoforderedphases
The paperis organizedas follows. In Section II we intro- with nonvanishing local order parameters is tied to sponta-
ducetheuniversalmeasureoftotalfrustrationintermsofthe neousbreakingofsomesymmetriesofthetotalHamiltonian,
global-to-local infidelity and describe its main features. In while,bydefinition,theMMGGSpreservesallsymmetriesof
Section III we consider generic spin–1/2 models with pair- the model. Choosinga particularglobalgroundstate among
wiseinteractionsondifferentlengthscales,discusstheirfrus- thedegenerateonesortheMMGGSdependsonwhetherone
tration properties according to the degeneracy of the local isinterestedinfeaturesofthefrustrationthatare,respectively,
groundspaces,establishrigorousconditionsforthepresence either state- or model-dependent. In the following we will
ofquantumfrustration,andderiveanexactinequalityonpair- studyfrustrationin differentquantumspinsystemsfeaturing
wise frustration based on entanglement monogamy. In Sec- bothnon-degenerateanddegenerateglobalgroundstates.
tionIVweusetheseresultstoinvestigatethetransitionfrom Majorization theory and the Cauchy interlacing theorem
generic frustrationto genuinequantumfrustrationin models providean immediatelink between frustrationandentangle-
with competing nearest-neighbor (NN) and next-to-nearest- ment. Indeed,itisstraightforwardtoshowthatf isbounded
ℓ
neighbor(NNN)interactions,andidentifydifferentphasesas- from below by a quantity ǫ(r) that depends uniquely on the
ℓ
sociated, respectively, to generic frustration and to genuine reducedlocaldensitymatrixρ [14,15]:
ℓ
quantumfrustration. InSectionVwe discusstheobservable
characterizationofsuchfrustrationtransitionsintermsofthe f ǫ(r) , (2)
staticstructurefactorandthedirectobservationoftheuniver- ℓ ≥ ℓ
salmeasureoftotalfrustrationintermsoftheinterferometric where
visibility.SummaryandoutlookarediscussedinSectionVI. r
ǫℓ(r) =1− λ↓i . (3)
Xi=1
II. FRUSTRATION
The index r denotes the rank of the local groundspace pro-
Here we review the main aspects of a universal measure jectorΠℓ and λ↓i are the thefirst r largesteigenvaluesof the
reducedlocaldensitymatrixρ indescendingorder.
of total frustration expressed in terms of the incompatibility ℓ
Inthecaseofanon-degenerateglobalgroundstateρ the
betweenlocalandglobalorders,andbrieflydiscusshowthis G
universalmeasure(global-to-localincompatibility,orglobal- quantityǫ(r) coincideswiththeHilbert-Schmidtdistancebe-
ℓ
to-local infidelity) connects to quantum entanglement. The tweenρG andthesetofpurestateswithSchmidtranklessor
class of many-body Hamiltonians H under investigation equaltor[14].
G
comprises all those that can be written as sums of different This quantity is a proper entanglement monotone [38].
localterms: H = h ,whereGstandsfortheglobal(to- Moreover, for r = 1, i.e. in the case of a non-degenerate
G ℓ ℓ
tal) system and ℓ PG are the local subsystems associated groundspaceofthelocalinteractiontermhℓ,itcoincideswith
to the local interac⊂tions h . Let us consider one such local thebipartitegeometricentanglementintroducedinRef.[39].
ℓ
subsystem ℓ. Denoting the remainder of the total system by Thisistheground-stateentanglementbetweenthelocalblock
R=G ℓ,frustrationofℓoccurswhenthelocalprojectionof ℓ and the remainder R of the total many body system G as
theglob\algroundstate ofHG, obtainedbypartialtraceover measured by the Hilbert-Schmidt distance DHS of the pure
R,i.e. thereducedstateofsubsystemℓ,failstobelongtothe ground state ρG = |ΨiGhΨ|G from the set of the pure bi-
localgroundstatespaceofh . Consequently,suchfrustration separablestates Ψ definedas Ψ ψ φ ,sothat
ℓ | ibs | ibs ≡| iℓ⊗| iR
isdirectlyquantifiedintermsoftheoverlapbetweenthepro-
jteocntioarnΠhℓℓoanndtoththeerelodcuaclegdrlooucnadldsteantseitsypamcaetroifxtρhℓe=locTalrRHaρmGil- ǫ(ℓ1) ={|mΨiibns}DHS(|ΨiG,|Ψibs) . (4)
{ }
fromtheglobalgroundstateρG ofthetotalHamiltonianHG. Therefore ǫ(1) realizes a quantum lower bound to total frus-
The ensuing universalmeasure of total frustration f is thus ℓ
ℓ trationintermsofthebipartitegeometricentanglement.
definedasfollows[14]:
The fact that the above measure of frustration is bounded
from below by quantum entanglement when the many-body
f =1 Tr (Π )ρ =1 Tr Π ρ . (1)
ℓ ℓ R G ℓ ℓ
− { ⊗1 } − { } system features a pure, non-degenerate global ground state
ThemeasureisuniquelydefinedaslongasGfeaturesa non is a direct consequence of the fact that for pure states all
degenerateglobalgroundstate space, while alllocalsubsys- quantum correlations reduce to entanglement. On the other
temsℓ mayfeaturelocalgroundspacesofarbitrarydegener- hand,mixedstatesallowforquantumcorrelationsmoregen-
acy. eralthanentanglement,suchasthequantumdiscord,thatcan
3
be presentevenin separablestates. One needsto keeptrack the parity symmetry along the three independentspin direc-
ofthisimportantdifferencewhenconsideringthegeneralcase tions,sothatthetotalmany-bodyHamiltonianreads
ofdegenerateglobalgroundstates. Indeed,inordertointro-
duce a measure of frustration that should not depend on the H = Jασασα = h , (5)
ij i j ij
choice of one particularstate amongthe possible degenerate iX,j,α Xij
ones,weneedtointroduceareferenceglobalgroundstatethat
properly averages over all possible degenerate ground states where σiα are the Pauli spin operators (α = x,y,z) for the
andpreservesallsymmetriesofthemany-bodyHamiltonian. i-thspinandJα istheinteractioncouplingstrengthbetween
i,j
This can be achievedeither by takinga coherentquantum spinsi,jalongtheαdirection.ForsuchHamiltonianstheele-
superposition of all the degenerate ground states with equal mentarylocalinteractionsubsystemsℓarenaturallyidentified
amplitudes, or by introducing the maximally mixed global astheonesmadeupbypairsofdirectlyinteractingspinswith
ground state (MMGGS), i.e. the convex combination of all local Hamiltonian hij whose energy eigenstates are the four
thepossiblepuregroundstateswithequalprobabilityweights. maximallyentangledBellstates(singletandtripletstates).
Insuchamixedstate,correlationsarenotcompletelycharac- For such Hamiltonians that include, among others, the
terizedbyquantumentanglementalone,andthequantityǫ(r) Ising,XY,andHeisenbergmodels,itispossibletointroduce
ℓ aquantumgeneralizationoftheclassicalToulousecriteriafor
turnsintothesumoftheground-stateblockentanglementbe-
frustration-free systems [14, 15]. The first step of this con-
tween ℓ and R, as measuredby the convexroofof the pure-
structionconsistsin defininga prototypemodel. A quantum
statebipartitegeometricentanglement,plustheclassicalcor-
spin Hamiltonian of the type Eq. (5) is a prototype modelif
relations that are established between the local subsystem ℓ
thereexistsatleastonelocalgroundstate commontoalllo-
and an ancillary system when the latter performs a general-
calinteractiontermsh andallpairwisecouplingsareferro-
izedquantummeasurementonℓ[15,40]. ij
magnetic (Jα 0). Having defined the prototypemodel, a
Relation(2)allowsforamorerefinedclassificationoffrus- ij ≤
quantumversionoftheToulousecriteriacanbeformulatedas
trationinthequantumdomainthanmerelydistinguishingbe-
follows:
tweenfrustratedandnon-frustratedsystems,asintheclassical
i)–TheMMGGSofaprototypemodelsisINESforallpair-
domain. Namely,wecannowintroducethefollowingclassi-
wiseinteractiontermsh .
ficationofquantumsystemsaccordingtofrustration: ij
ii)–Forallmodelsobtainedfromaprototypemodelbylocal
unitary operations and partial transpositions acting on any
1. Frustration-freesystems(FF):
arbitrarysubsystem,theMMGGSisstillINESonallpairwise
fℓ =ǫ(ℓr) =0 ∀ℓ, intTerhaectqiounantteurmmsThoiujl.ouse criteria allow to make predictions
2. Inequalitysaturatingsystems(INES): on the frustrationof a modelwithoutrequiringthe complete
and exact diagonalization of the system. In fact, they allow
fℓ =ǫ(ℓr) >0 ∀ℓ, to state powerful exact relations between the degeneracy of
the local ground spaces, entanglement, and the presence of
3. Inequalitynon-saturatingsystems(non-INES):
quantumfrustrationinsystemsdescribedbyHamiltoniansof
f >ǫ(r) 0 ℓ. theformEq.(5).Indeed,akeyfeatureofaprototypemodelis
ℓ ℓ ≥ ∀ thatallinteractingpairsadmitatleastonelocalgroundstate
Accordingly,if genericfrustrationis absent, the system is incommon.Then,thefollowingholds:
alwaysinanINESgroundstateandthuseitherfrustration-free
Theorem. Ifamanybodysystemwithpairwiselocalinterac-
ifthegroundstateisclassicallycorrelatedorgenuinelyquan-
tions satisfies the quantum Toulouse criteria and the degen-
tumfrustratedifitisquantumcorrelated. Whentheinequal-
eracy of the ground space common to all local interacting
ity is not saturated (non-INES) the system’s total frustration
pairs has dimension r 2, then the system is necessarily
ingeneralcomprisesbothgenericandgenuinequantumcon- ≥
frustration-free,i.e.: f =ǫ(r) =0 ℓ.
tributions.Indeed,aswewillshowinthefollowing,theinter- ℓ ℓ ∀
playbetweengenericfrustrationandgenuinequantumfrustra-
Proof. IfamodelsatisfiesthequantumToulousecriteriaand
tionandthetransitionfromnon-INEStoINESconfigurations
thecommongroundspace hasdimensionr = 2, thenallin-
correspondtotransitionsfromphaseswithclassical-likemag-
teractingpairsadmitascommongroundstate two outof the
neticordertophaseswithgenuinequantumorder.
four maximally entangled Bell states. Obviously any linear
coherentsuperpositionofthesetwoBellstatesisagainalegit-
imategroundstateofeachlocalinteractingpair. Allpossible
III. FRUSTRATIONANDMONOGAMYOF superpositions include separable ones of the form i j .
| i⊗| i
ENTANGLEMENT Hence, the fullyfactorizedstate Ψ = N i is a le-
| facti i=1| i
gitimategroundstateofthetotalmany-bodyNHamiltonian,in
Letusconsidertranslationally-invariant(periodicboundary close analogyto the situation that occursat the factorization
conditions)many-bodysystem consisting of N spin–1/2el- points of many-body Hamiltonians in transverse fields [41–
ementaryconstituents,featuringgenerictwo-body(pairwise) 43]. By immediate inspection, the ground state Ψ has
fact
| i
interaction terms with the only constraint that they preserve vanishingfrustration,single-siteentanglement,andpairblock
4
entanglementforallinteractingpairs. Moreover,sincetheto- where is the concurrence of the spin pair i,j and we
ij
C { }
tal energy associated to Ψ is the sum of the minimum have exploited the explicit analytic form that it takes on the
fact
| i
energiesofalllocalpairinteractionterms,anypossibleother setofthefourBellstates. Equalityholdsonlywhenthefrus-
totalgroundstatemustfeaturethesamepropertyandhenceits trationisINES,i.e. whenf = ǫ . Therefore,saturationof
ij ij
projectionontoanypairofinteractingspinsmustbelongtothe theinequalityimpliesadirectcorrespondencebetweenquan-
pairlocalgroundspace,implyingthatitsfrustrationvanishes tumfrustration,aglobalfeatureofthepairasmeasuredbyits
on all pairs. Exactly the same argument applies also to the block geometric entanglementwith respect to the remainder
casesoflocalgroundspacedegeneracyr =3andr =4. oftheentiremany-bodysystem,andthelocalquantumentan-
glement within the pair. The more frustrated the subsystem
We are thus left with the only non-trivial case of a local
i,j iswithrespecttotheremainderofthemany-bodysys-
groundspacedegeneracyr = 1,correspondingtolocalpairs { }
tem, theweakeristhe internalentanglementbetweenspinsi
ofinteractingspinsadmittingoneofthefourBell statesasa
and j forming the pair. Moreover, since the quantum lower
localgroundstate.Insuchacase,thequantumlowerboundto
boundǫ isthe entanglementbetweenthe localpairandthe
ij
frustrationalways coincideswith the ground-stategeometric
remainderofthe many-bodysystemif andonlyif the global
entanglement between the local block (pair) ℓ = i,j and
ground state is pure, from Ineq. (11) it follows for all pure
{ }
therestR=G ℓofthesystem. Droppingfromnowonthe
global ground states that the stronger the entanglement be-
\
indexr =1,itreads
tweenspinsin the localpair, the weakeris the entanglement
ofthepairwiththeremainderofthemany-bodysystem.
ǫ =1 λ , (6)
ℓ max
−
These relations allow for a restatement of frustration in
where λ is the largest eigenvalue of the reduced local
max terms of entanglement shareability. Indeed, exploiting the
density matrix ρℓ. From now on, without loss of general- Coffman-Kundu-Wootters-Osborne-Verstraetemonogamyin-
ity, we will consider the case of anti-ferromagnetically in-
equality [18, 19] that sets precise constraints on the entan-
teracting pairs of spins admitting as unique, non-degenerate
glement that can be shared among many spins, we find the
ground state the antisymmetric singlet state: ψij =
following“frustrationmonogamy”:
1 ( ). Thecorrespondingprojector|ontiothe
√2 |↑i↓ji−|↓i↑ji
localgroundspacethenreads
(max(0,1 2f ))2 2
Πℓ=ij =|ψijihψij|. (7) jX∈Li − ij ≤ jX∈LiCi,j
Exploitingthesymmetryunderparityalongthethreespindi- ≤ Ci2,j ≤τi . (12)
rections,thetwo-bodyreducedlocaldensitymatrixtakesthe jXG i
∈ \
generalform
1 Intheabove,Li denotesthesubsetofspinsthatinteractwith
ρℓ=ij = 41ij +Xα giαj σiασjα . (8) is.piTnhie,uwphpielerbGou\niddτeniostethsethtaentgolteal[1sp8i,n1s9y].stWemhewnitthhoeugtlospbianl
i
Here gα denote the spin-spin correlation functions: gα = groundstate isnondegenerate,itcoincideswith the squared
ij ij
SαSα , where Sα = (1/2) σα are the spin operators and concurrencemeasuringthesingle-siteentanglementofspini
h iindjiciatestheexipectationvaluieoverthegroundstateofthe withallotherspinsinthesystem. Forspinsystemsdescribed
hs·yistem.Consequently,thefrustrationofeachinteractinglocal by Hamiltoniansof the form Eq. (5) the tangle takes the ex-
spinpairreads pressionτi = 1−m2x−m2y −m2z,wheremα = hσiαiisthe
localmagnetizationalongthe α direction. Thereforeτ = 1
i
3
f = + gα , (9) in the MMGGS, since the latter preserves all symmetries of
ij 4 ij the total many body Hamiltonian and therefore does not al-
Xα
lowforaspontaneousmagnetizationalonganyspindirection.
and the quantumlower boundǫij on the total frustrationfij Thisfactandthe frustrationmonogamyIneq.(12) implyex-
ofeachlocalinteractingpairreads actboundsontheminimumpossibleamountoffrustrationin
3 amany-bodysystem. Consideringforsimplicitytheuniform
ǫij = 4 +min gixj +giyj +gizj,gixj −giyj −gizj , case fij = f ∀{i,j}, and denoting by NL the number of
gx(cid:8)+gy gz , gx gy +gz . (10) spinsinteractingwithspini,wehave:
− ij ij − ij − ij − ij ij
(cid:9)
Sincetheeigenvectorsofρℓ=ij arethefourBellstateswecan 1 1
establishadirectconnectionbetweenthetotalfrustrationf , f 1 . (13)
ij ≥ 2(cid:18) − √N (cid:19)
L
the quantumlowerboundǫ , i.e. theground-stateentangle-
ij
mentbetweenthe localinteractingpairij andthe remainder
We thus find that the minimum frustration threshold varies
of the systems, and the entanglement of formation between
from f = 0 for N = 1 for a many-body system con-
thespinsiandj inthepair,asmeasuredbytheconcurrence min L
sistinginjustasinglepairofinteractingspins,tof =1/2
[16,17]. Wehavethatthefollowinginequalityholds: min
ij
C atthethermodynamiclimitoffullyconnectedmany-bodysys-
=max(0,1 2ǫ ) max(0,1 2f ), (11) tems,i.e. withN .
ij ij ij L
C − ≥ − →∞
5
1(cid:144)4
J2 a)
ygij
xºgij 0 J1
-1(cid:144)4
-1(cid:144)4 0 1(cid:144)4 b)
gz
ij 1 2
FIG.1:(Coloronline)BehaviorofIneq.(2)forthelocalpairfrustra-
tionasafunctionoftheadmissiblevaluesofthecorrelationfunctions
gixj = giyj andgizj inthereducedlocalpairdensitymatrixρℓ=ij of
anXXZ modelwithcompetingNNandNNNinteractions. Verti-
callystriped green area: reduced states ρℓ=ij withnon-INES frus- FIG.2: (Coloronline)a)Schematicstructureofaone-dimensional
tration (strict inequality, generic frustration). Horizontally striped quantum spin model with competing NN (J1) and NNN (J2)
orange area: reduced states withINESfrustration(inequality satu- anti-ferromagnetic interactions (periodic boundary conditions). b)
rated,genuinequantumfrustration).Redarea:disentangledreduced Ground-state structure at the integrable Majumdar-Ghosh point
stateswithINESfrustration(inequalitysaturated,vanishingpaircon- J2/J1 = 1/2. Atthispointthemodelfeaturestwoexactlydimer-
currence,enhancedgenuinequantumfrustration). Remainingwhite ized ground states that are the tensor product of NN spin singlets.
area:non-physicalstates. Thetwovalencebondsolidsbreakthetranslationalinvarianceofthe
Hamiltonianwithrespecttothelatticespacinga.
IV. FRUSTRATIONANDDIMERIZATION
NNN(J )interactions,intheclassofEq.(14).Exactground-
2
state dimerization is realized at the Majumdar-Ghosh point
Thefrustrationf ofthelocalpairinteractionh isingen-
ij ij
eral a function of the parameters entering in the total many- φMG 0.45solutionoftanφ=1/2[45].Thisistheonlyin-
≃
tegrablepointofmodelsintheclassEq.(14).Atthispointthe
bodyHamiltonian. Moreover,theinterplaybetweendifferent
groundstate becomesdoublydegenerate,breakingexplicitly
sources of frustration may lead to significant changes in the
the translationalinvarianceof the lattice, which is recovered
structureoftheglobalmany-bodygroundstate. Inparticular,
we wish to investigate the changes in the structure of quan- atdoublelatticespacing2a. Thetwogroundstatesaretensor
productsof NN spin singlets, realizingthe first and simplest
tum ground states when frustration turns from non-INES to
paradigmaticinstanceofvalencebondstates[46–48]. Except
INES,i.e. whengenuinequantumfrustrationoverrulescom-
forthe Majumdar-Ghoshpoint, theglobalgroundstate is al-
pletelyothergenericsourcesoffrustration. Wewillfocuson
the paradigmatic class of frustrated one-dimensionalmodels waysunique. At φMG frustrationisevaluatedin eachof the
two dimerized ground states as well as in the MMGGS, i.e.
with competing nearest-neighbor (NN) and next-to-nearest-
theirconvexcombinationwithequalweights.
neighbor(NNN)anti-ferromagneticinteractions,theso-called
J J models[44]: InFigs.3and4wereportthebehaviorofthepairfrustration
1 2
− f , the quantum lower bound ǫ , and the pair concurrence
ij ij
H=cosφ cosδ SixSix+1+SiySiy+1 +sinδSizSiz+1 Cij asfunctionsofφ,respectivelyfortheNN(i,i+1)andthe
Xi (cid:0) (cid:1) NNN(i,i+2)localpairinteractionterms.Theplotsaretaken
atanisotropiesδ =0(XX symmetry), π (XXZ symmetry),
+sinφ cosδ SxSx +SySy +sinδSzSz .(14) 6
i i+2 i i+2 i i+2 and δ = π (classical Ising symmetry). The fully isotropic
Xi (cid:0) (cid:1) Heisenberg2case (XXX) realized by δ = π is reported in
4
These systems are characterized by the ratio between NNN Fig.5forwhichweprovidealsoinsetscomparingthedetailed
and NN interaction strengths J /J = tanφ, with φ behavior of the pair frustrations in the MMGGS and in the
2 1
[0,π]), and the ratio between anisotropies tanδ, with δ ∈ twodegeneratedimerizedpuregroundstatesattheMajumdar-
[0,π2]. The translationally invariant case (periodic boundar∈y GhoshpointφMG.
2
conditions)isstudiedbyexactdiagonalizationwithN = 24 Remarkably,fromFigs.3and4,weseethatforallquantum
spins. InFig.1wereportthebehaviorofthepairfrustration models (XX, XXZ, and Heisenberg)it is always fi,i+1 =
fij, accordingto the classification inducedby Ineq.(2), asa ǫi,i+1, i.e. frustration of the NN interacting local pair is al-
functionofthespin-spincorrelationfunctions,wherewehave waysINES(genuinequantum),notwithstandingthepresence
exploitedEqs.(8),(9),and(10),andthesymmetrygx =gy. ofa sourceoffrustration,due to geometry,in termsof com-
ij ij
Bothnon-INES(generic)andINES(genuinequantum)frus- petinginteractionsondifferentlengthscales.
trations are realized in a significant range in the parameter The NN pair frustration increases for increasing relative
space. strengthoftheNNNinteractions,reachingasymptoticallythe
InFig.2wereportschematicallythestructureofthequan- maximumvalue3/4. Thiscoincideswiththemaximumpos-
tum ground states for models with competing NN (J ) and sible valueof ǫ , i.e. themaximumpossiblevalueof the
1 i,i+1
6
ground-stateblockentanglementbetweentheNNpairandthe
1
restofthesystem. Suchamaximumisachievedforareduced
pairdensitymatrixρ withallfoureigenvaluesdegenerate
i,i+1
andequalto1/4. 0.5
In other words, maximizing the relative strength of the δ = 0
NNNinteractionmaximizestheground-stateblockentangle- 1
+
ment ǫi,i+1 between NN pairs and the rest of the system fi,i 01
andthecorresponding(genuinequantum)NNpairfrustration =
fi,i+1.Hence,bymonogamy,itminimizesthepairwiseentan- +1
glementCi,i+1 betweenthespinsformingtheNNinteracting εi,i 0.5
δ = π/6
local pair. Indeed, for a sufficiently large value of the ratio
1
J /J , theNN pairconcurrenceC vanishesexactlyand +
re2mai1nszerothereafter. i,i+1 Ci,i 0
1
Even more important, Figs. 3 and 4 show that frustration
1 δ = π/2
f oftheNNNinteractinglocalpairundergoesafulltran- +
siit,iio+n2fromnon-INES(genericfrustration)toINES(genuine fi,i
0.5
quantum frustration) exactly at the Majumdar-Ghosh point
tanφ = 1/2. Therefore, transition to exact ground-state
dimerization corresponds to exact transition from non-INES 0
toperfectINESfrustrationoftheNNNlocalinteractionterms. 0 φMG π/4 π/2
For increasing relative strength of the NNN interactions the φ
NNN pair frustration decreases, reaching asymptotically its
minimum(whoseactualvaluedependsonδ).Suchtransitions FIG.3: (Coloronline)Frustrationfi,i+1oftheNNinteractinglocal
infrustrationexhibituniversalfeaturesintheXX,XXZ,and pair,quantumlowerboundǫi,i+1,andNNpairconcurrenceCi,i+1,
Heisenbergcases, irrespectiveof the correspondingdifferent asfunctionsof theparameter φrulingtheratiotanφ betweenNN
symmetryclasses. andNNNinteractions. Upperpanel: J1−J2modelwithXX sym-
As shown in Figs. 3 and 4, the frustration pattern differs metry,δ=0.Solidgreenline:fi,i+1 =ǫi,i+1,asthetwoquantities
radicallyonlyinthe Isingcase, as shouldbe expected,since inthiscasecoincide∀φ. Dashed blueline: Ci,i+1. Central panel:
alllocalinteractionscommuteinIsingsystems,entanglement J1−J2 modelwithXXZ symmetry, δ = π/6. Alsointhiscase
fi,i+1 =ǫi,i+1∀φ.Botharedenotedbythesolidgreenline.Dashed
vanishes, and, as discussed in Section III, there are no gen-
blueline:Ci,i+1.Lowerpanel:J1−J2modelwithIsingsymmetry,
uine quantum sources of frustration. In this latter case we δ=π/2.Dot-dashedredline:fi,i+1betweentheMajumdar-Ghosh
havesimplytwowelldistinguishedregimesseparatedbythe point φMG and φ = π/2. Solidgreen line: fi,i+1 = ǫi,i+1 = 0
Majumdar-Ghosh point φMG. Indeed, at this point the NN betweentheMajumdar-GhoshpointφMG andφ = 0. Dashedblue
frustration jumps from zero to half the maximum value 1/2 line: Ci,i+1. NN frustration isalways INES, fi,i+1 = ǫi,i+1, for
and remains constant thereafter, while the NNN frustration thequantumXX andXXZ models. Intheregimeofdominating
jumps from the maximum value 1 to a vanishing value and NNN interactions, fi,i+1 → 3/4, which isthe maximum possible
remainszerothereafter. blockentanglementǫi,i+1betweentheNNinteractinglocalpairand
therestofthesystem. Theleftmostverticallinecorrespondstothe
On the other hand, in the nontrivial quantum cases with
Majumdar-Ghosh point φMG. The rightmost vertical lines corre-
non-commutinglocalinteractions(XX,XXZ,Heisenberg)
spondtotheboundaryvalueφ oftheintermediatequantumregime:
b
we observe a rich structure with three different patterns of φ ≃ 0.99andφ ≃ 1.08,respectivelyfortheXX andtheXXZ
b b
frustrationastheratioofNNNtoNNinteractionsvaries. At symmetries. ModelswithIsingsymmetryfeatureonlyafrustration
small φ well below the Majumdar-GhoshpointφMG (domi- duetogeometry. Itundergoesasharptransitionfromfi,i+1 = 0to
nating NN interactions) NN spin pairs are weakly frustrated fi,i+1 = 1/2exactlyatφMG andremainsconstantthereafter. The
inoppositiontoNNNpairsthattendtobestronglyfrustrated. quantitiesǫi,i+1andCi,i+1arealwaysvanishingintheIsingcase.
Indeed, for φ 0 the NNN pair frustrationachievesvalues
→
veryclosetounity,i.e. themaximumpossiblevalue,whichis
achievedexactlyintheclassicalIsingcase. Forφ’sfarabove 2aandNNanti-ferromagneticinteractions. Twosimpleanti-
the Majumdar-Ghosh point φ (dominating NNN interac- ferromagneticordersarethenestablishedoneachchain.
MG
tions)thesituationreverses: NNNpairsbecomeweaklyfrus- However, in the quantum models a further intermediate
tratedwhileNNpairsbecomestronglyfrustrated. regime is established in the central interval starting at the
Thesetwolimitingregimesarecharacterizedbytwosimple Majumdar-Ghoshpoint and extending to the right, i.e. with
antiferromagneticorders that essentially coincide with those increasingvaluesofφ. Inthisinterval,weobserveasmooth
of the classical Ising case. Fig. 6 summarizes this behav- crossover in terms of a succession of small but sizable step-
ior pictorially in terms of the lattice structure. In the limit jumpsin the behaviorof the NN and NNN pair frustrations.
φ 0thelatticereducestoasinglechainwithlatticespac- Thisintermediate,centralregimecorrespondstotheonsetof
→
ing a andNN anti-ferromagneticinteractions,allowingfora aquantumorderthatiswelldistinguishedfromthetwoclas-
simpleanti-ferromagneticorder.Inthelimitφ π/2thelat- sical(Ising-like)anti-ferromagneticregimesdiscussedabove.
→
ticeeffectivelysplitsintotwochains,eachwithlatticespacing Thelower anduppervaluesofthis intervalare, respectively,
7
1 1
r=2
δ = 0
+2 0.5 +r 1
fi,i Ci,i 0.5 +1 àà
=2 0 +r fi,i
ε+i,i 1 fi,i 0φ 0.4 φææMG 0.6
δ = π/6 +r 0
C+2i,i 0.5 εi,i 1 r=1 +21 ææ
+2 0 f+i,ir fi,i0
fi,i 1 = 0.5 φ 0.4 φMG 0.6
r
+
ε+2i,i 0.5 δ = π/2 εi,i
0
0 φMG π/4 π/2
0 φ
0 φMG π/4 π/2
FIG.5:(Coloronline)SameasinFigs.3andFig.4butthistimefor
φ
J1−J2 modelswithHeisenberg(XXX)symmetry,δ = π/4. In
thiscasetheboundary value of theintermediatequantum regionis
FIG.4:(Coloronline)SameasinFig.3butthistimeforthefrustra-
φ ≃ 1.18. Insets: zoomofthebehavioroftheNNandNNNpair
tionfi,i+2,quantumlowerboundǫi,i+2,andconcurrenceCi,i+2of frbustrations at and around the Majumdar-Ghosh point. Away from
theNNNinteractinglocalpairs.Transitiontogenuinequantumfrus-
Mtraatjiuomnd(naor-nG-IhNoEshSptooinINt.EFSru),stfria,tii+o2nr=emǫai,iin+s2I,NoEcScutrhsereexaafctetlry. Ianttthhee φciMdeGswthiethptuhreeMglMobGaGl gSr.oEuxnadctsltyataetiφsMuGni,qbueesiadnedstthhuesavaelwraagyesdcooniens-
intheMMGGS,wealsoreportthenon-averagedNNandNNNpair
regimeofdominatingNNNinteractions,fi,i+2 →1/4,whichisthe frustrationsinthetwoexactlydimerizedpureglobalgroundstates.
tmeriancimtinugmlopcoaslsipbaleirbaloncdktheentarensgtleomfethnetǫsiy,is+te2mb.etMweoedneltshewNithNNIsiinng- TheNNpairfrustrationfi,i+1vanishesinthegroundstateforwhich
{i,i+1}belongtothesamedimerandarethusmaximallyentan-
symmetry feature only frustration due to geometry. It undergoes
gledbetweenthemselves(purplefullcircle).Viceversa,itcoincides
aMsahjuamrpdatrra-nGshitoisohnpforoinmtafnid,i+re2ma=ins1vtaonifshi,iin+g2th=ere0afteexra.ctly at the withtheblockentanglementǫi,i+1atitsmaximumvalue3/4inthe
groundstateforwhich{i,i+1}belongtodifferentdimers(green
fullsquare). TheNNNpairfrustrationfi,i+2 = 3/4(greenfullcir-
cle)inbothgroundstates,since{i,i+2}alwaysbelongtodifferent
the Majumdar-Ghosh point and the boundary point φb cor- dimers.
responding the last step-jump in the increasing behavior of
f anddecreasingbehavioroff .
i,i+1 i,i+2
Thefactthatasharpchangeinthebehaviorinthefrustra- the numerical data obtained from exact diagonalization that
tionpropertiesappearsexactlyattheMajumdar-Ghoshpoint at these pointsexact crossoversare realized between ground
mayseematfirstsurprising,sincethetransitiontothedimer- states of different parity. These crossovers are then respon-
izedphaseintheJ J model,regardlessoftheanisotropy, sible forthe suddenincreasingofthe NN frustrationand the
1 2
−
occursatacriticalvalueφ = φ muchlowerthanφ . For correspondingsuddendecreasingoftheNNNfrustrationthat
c MG
example,inthecaseofthefullyisotropicHeisenbergmodel, areobservedintheinterval[φ ,φ ].
MG b
δ = π/4, the critical value φ tan 1(0.2411) 0.237, Thisstructureofground-statecrossoversappearssimilarto
c −
≃ ≃
asdeterminedbyextensivenumericalinvestigationsbasedon theoneoccurringinXY chainsintransversefieldh,forval-
thedensitymatrixrenormalizationgroup(DMRG)andother uesof h belowthe factorizingfieldh [41–43]. Indeed,one
f
numericalalgorithms[44]. However,thefrustrationdoesnot candrawaverycloseparallelbetweenthetwo-sitefactoriza-
featureanysharpchangewhencrossingφc. tionofthegroundstateoftheJ1 J2modelattheMajumdar-
−
Thesmoothbehaviorofthefrustrationatφ andtheappear- Ghoshexactdimerizationpointφ ,andthesingle-sitefac-
c MG
anceofstep-jumpsinthefrustrationwhenthesystemisinthe torization of the ground state of the XY model at the exact
intermediatequantumregimedeservefurthercomments. Let factorizingfieldh . Firstly,inbothcases,thesharpchangein
f
us first consider the physical origin of the step-jumps. One thebehaviorofvariousphysicalquantities,includingfrustra-
mightconjecturethattheyareduetofinite-sizeeffectsandthat tionandground-stateentanglement,thatoccursexactlyatthe
they should disappearwhen consideringmuch largerchains. respectivefactorizationpoints(single-sitefactorizationinthe
In fact, this is not the case, as can be already understoodby XY modelath ,two-sitefactorizationintheJ J model
f 1 2
−
consideringchainsofrelativelysmallsize. Indeed,bylooking at φ ), is always present, independentlyof the size of the
MG
at the structure of the ground state in correspondenceto the system [43, 49, 50]. Moreover, in the finite-size XY chain,
occurrenceof the step-jumps, we have verified by analyzing the crossovers occur between ground states of different par-
8
ity andappearfor0 < h < h . Such crossoversarealways
f
present, buttheir numberand position dependon the size of
the system [43]. The crossoversterminate exactly at h = 0
and h = hf. At each crossover, several physical quantities, φ → 0 φ→π/2
including frustration and ground-state entanglement, feature
step-jumpscompletelyanalogoustotheonesthatarefeatured
inthecentralquantumregimeofthefinite-sizeJ J model,
1 2
−
i.e.intheregionφ <φ<φ .Theanalogyiscompleteby
MG b
observingthatthecrossoversterminateexactlyatφ = φ
MG
and φ = φ , and that both exact ground state factorization
b
FIG. 6: (Color online) Latticeseffectively realized in the different
andexactgroundstatedimerizationarenotassociatedtoany
limitsofφ.Forφ→0NNNinteractionsarestronglysuppressedand
quantumphasetransition.Indeed,asalreadymentioned,both
thelatticereducestoaNNinteractingchainwithlatticespacinga.
theMajumdar-GhoshpointintheJ J modelandthefac-
1 2 Forφ → π/2thelatticesplitsintotwoindependentNNinteracting
−
torizationpointintheXY modeloccurindependentlyofthe chains,eachwithlatticespacing2a.
size ofthesystem. Moreover,bothpointsarealwaysassoci-
atedtoadoubledegeneracyoftheglobalgroundstate.
In the XY model, where all quantities can be evaluated ical order [51]. However, in the case of the J J model
1 2
−
analyticallyandcontrolledpreciselybothatfinite-sizeandin this behavior can become evident only once more powerful
the thermodynamiclimit, it is possible to verify exactly that numerical techniques are developed in order to evaluate the
the number of the step-jumps increases while the height of frustration in chains of sizes much larger than the ones that
thejumpsdecreasesasthe size ofthechainincreases; in the areallowedbyresortingtoexactdiagonalization.
thermodynamiclimittheheightofthejumpsvanishesexactly
and a fully analytic behavior is restored. Unfortunately, the
non-solvabilityof the J1 J2 modelpreventsus, at least at V. OBSERVABILITY:STRUCTUREFACTORAND
−
themoment,toestablishwhetherthecentralquantumregion INTERFEROMETRICVISIBILITY
extendsindefinitely,i.e.theboundarypointφ increasesasthe
b
sizeofthechaindiverges,andwhetheranalyticityisrestored
The nature of the intermediate, centralquantumregime in
inthethermodynamiclimit.
comparison to the two standard anti-ferromagnetic orders is
Concerningthesmoothbehaviorofthefrustrationobserved bestcapturedbystudyingthe behaviorofthe static structure
atφc insmallchains, weremarkthatthe dramaticchangeto factor, as defined below and reported in Fig. 7. Indeed, as
exact ground state dimerization occurs at and only at φMG. changes in the patterns of quantum frustration imply a tran-
Furthermore,ourinvestigationreliesonexactdiagonalization, sition between different lattice geometries, we can connect
whichisfeasibleonlyforchainsofratherlimitedsize. Toad- themtothequantummomentumdistributionobservedinex-
dress larger sizes, one would need to resort to DMRG and periments, which is quantified by the static structure factor
relatedalgorithmssuchasmatrixproductstates(MPS).How- S (k):
f
ever, the existing codes are not suitable, as they stand, for
theevaluationofthefrustrationmeasure. Thefirststumbling N
2
blockisthattheDMRGalgorithmneedsprofoundmodifica- S (k)= cos(kai j ) −→S −S→ , (15)
f i j
N | − | h · i
tionsandimprovementsinordertocomputereduceddensity iX,j=1
matrices and projectors onto the local ground spaces, their
eigenvectors, and their spectra, in the presence of periodic whereaisthelatticespacing,k isthewavevector,and−→S is
i
boundary conditions. The second stumbling block is that, thespinoperatorvectoroftheithwhosecomponentsarethe
whileDMRGallowstoevaluatewithoutmucheffortthecor- threePaulispin-1/2matrices. Inmostexperimentalsituations
relation functions between neighboring sites, it works much directaccesstothecorrelationfunctionsforeachpaircanbe
less efficientlywhen one needs to computecorrelationfunc- extremelychallenging.Insuchcasesan“averaged”and“col-
tions between non-neighboringspins in the chain. Unfortu- lective”informationsuchasthestaticstructurefactor,thatcan
nately, exactly such correlation functions are those needed beobtainedfromtheanalysisofthevisibilitywhichquantifies
in order to evaluate, for instance, the static structure factor. thecontrastofthetime-of-flightimages[35,36],mayprovide
These difficulties have motivated us, whenever direct exact usefulcoarse-grainedinformationonthechangesinthequan-
analyticalevaluationprovedtobeimpossible,toresorttoex- tumordersduetochangesinthefrustrationpatterns.
act diagonalization algorithms based on augmented Lanczos InFig.7wereportthebehaviorofthestaticstructurefac-
methods. tor for three different values of φ, respectively well below,
In case of sufficiently large chains, the frustration mea- around, and well above the Majumdar-Ghoshpoint φ
MG
≃
sure is indeed expected to detect the transition to approxi- 0.45, comparing the quantum XX and Heisenberg models
matedimerizationbyshowinganinflectionatφ ,inthesame with the classical Ising case. For φ 0.14, well below
c
≃
way as it does in exactly solvable models that we are cur- φ quantumandIsing modelsallfeaturesa single peakat
MG
rently verifying, such as the 1-D XY model, or the 1-D Ki- k = π/a, corresponding to the same anti-ferromagnetic or-
taevandcluster-Isingmodelsofsymmetry-protectedtopolog- der. Atφ 1.4,i.e. wellaboveφ allmodelsfeaturetwo
MG
≃
9
6 both quantities on sums of correlation functions, a fact that
canbeimmediatelyappreciatedseencomparingEq.(9)with
4
φ≃0.14 Eq. (15), a direct connection between correlation functions
andǫ ishinderedbythehighlynonlinearnatureoftheirmu-
2 ij
tual relationship, where the nonlinearity originates from the
0 nontrivialminimizationthatoccursinEq.(10).
4 Asalreadymentioned,thegenuinequantumregimeestab-
lishedattheMajumdar-Ghoshpointφ extendsuptoacrit-
MG
) ical boundary value φ correspondingto the last step-jumps
(kf 2 φ≃0.55 inthepatternofincreasbingNNpairfrustrationanddecreasing
S
NNN pair frustration, as reported in Figs. 3, 4, and 5. The
0 extensionofthequantumregionisanon-universalfeature,as
4 itincreases,asshouldbeintuitivelyexpected,withincreasing
symmetryofthedifferentquantummodels.Indeed,forδ =0
(XX),φ 0.99. Forδ = π/6(XXZ),φ 1.08. Finally,
2 φ≃1.4 b ≃ b ≃
forδ =π/4(Heisenberg),φ 1.18.
b
≃
Concerning observability, besides measuring the static
0 structure factor in time-of-flightexperiments, a complemen-
tary strategy would be the measurement of the visibility in
0 1/2 1 3/2 2
interferometricexperimentswithatoms[37]. Inquantumop-
ka/π
tics,theinterferometricvisibilityV byfeedingtwosuccessive
photonsfromasourceintoa50:50beamsplitteris
FIG.7:(Coloronline)Behaviorofthequasi-momentumdistribution
Sf(k) as a function of the quasi-momentum k for different values V =Tr(ρ ρ ) , (16)
a b
of φ. Blackdot-dashed curve: classicalJ1 −J2 model withIsing
symmetry (δπ/2). Redsolidcurve: quantum J1 −J2 model with whereρa andρbarethestatesofthetwoincomingphotons.
XX symmetry (δ = 0). Blue dashed curve: quantum J1 − J2 Infact, itispossibleto envisageverygeneralschemesfor
model withHeisenberg symmetry(δ = π/4). Upper panel: dom-
theestimationandthedirectmeasurementoflinearandnon-
inating NN interactions (φ ≃ 0.14). The static structure factor
linear functionals of quantum states, based on simple quan-
features a single peak for all models, corresponding to a simple
tum networks [52]. Such schemes have been recently in-
Ising-likeanti-ferromagneticorderinasimplechainwithNNanti-
vestigated and specialized to quantum simulators realized
ferromagneticinteractionsandlatticespacinga.Lowerpanel: dom-
inatingNNNinteractions(φ ≃ 1.4). Thelatticesplitsintwosub- with atomic ensembles in optical lattices, either in terms of
chains,eachwithlatticespacing2a. Correspondingly,twosymmet- quantum switches [53] or multiparticle atomic interferome-
ricpeaks arefeatured by allmodels, corresponding tosimpleanti- ters[54],andareclosetoexperimentalrealizability[37].
ferromagneticorderoneachsub-chain. Centralpanel: intermediate
regime (φ ≃ 0.55). In this regime, the classical model still fea-
turestwosimpleanti-ferromagneticorders,whilethequantummod- VI. CONCLUSIONSANDOUTLOOK
elsfeatureanontrivialquantumorderandnopeaks(themanysmall
secondarypeaksareduetofinite-sizeeffectsthatvanishinthether-
In summary, we have shown that a measure of frustration
modynamiclimit).
based on the global-to-localincompatibility overlap(global-
to-localinfidelity)quantifiesefficientlybothgeneric(geomet-
ric, commonto andequalforclassicalandquantumsystems
symmetric peaksat k = π/2a and k = 3π/2a, correspond-
alike)andgenuinelyquantumsourcesoffrustration.
ingtotwoanti-ferromagneticorders,oneforeachofthetwo
For a large class of Hamiltonianswe have proventhat the
effective chains of Fig. 6. These are the geometric regimes
presenceofgenuinequantumfrustrationispossibleonlyifthe
for which classical Ising and quantum XX and Heisenberg
groundstatesofthelocalinteractiontermsarenondegenerate,
ordersareindistinguishable.
that is, the rank of the projection operator onto the ground
Finally, for φ 0.55, a value above but comparable to spaceofthelocalinteractingsubsystemsisr =1.
≃
φMG,weobservethatthequantummodelsdonotfeatureany Insuchcasesthetotalfrustrationcanberelated,viadiffer-
peak,correspondingtoaquantumordertotallydistinguished entupperandlowerbounds,withtwotypesofentanglement:
from the two classical anti-ferromagneticorders featured by the internal(local)entanglementbetween the constituentsof
theIsingmodelonthetwoeffectivechains. the local interaction terms, and the block (global) entangle-
Ontheotherhand,althoughtheanalysisofthestaticstruc- mentbetweenthelocalinteractiontermsandtheremainderof
turefactorprovidesusefulinformationaboutthebehaviorof thetotalmany-bodysystem. Moreover,wehaveestablisheda
the frustration measure, in general it cannot yield complete “monogamyoffrustration”relationquantifyingthe trade-off
informationabouttherelationbetweenthefrustrationandits on frustration due to the interplay between local and global
quantum lower bound ǫ . Indeed, while the existence of entanglement.
ij
a direct relationship between the static structure factor and WehavethenappliedthesegeneralresultstospinHamilto-
thefrustrationisduetoaqualitativelysimilardependenceof nians with competing anti-ferromagneticinteractionson dif-
10
ferentlengthscales,thusfeaturingsimultaneouslygenericand anonvanishingsub-leadingconstantcontributiontothearea-
genuine quantum contributions to the total frustration. We lawleadingscalingoftheground-stateblockentanglement,as
have then shown that the quantum phase transition to ex- measuredbythevonNeumannentropyoftheblockreduced
act ground-statedimerizationand valencebondsolids corre- densitymatrix,knownastopologicalentropy.
spondstoanexacttransitiontogenuinequantumfrustration. On intuitivegrounds,the universalmeasure of frustration,
We have investigated the behavior of the total frustration Eq.(1), shouldbeperfectlysuitedforthecharacterizationof
asa functionofthephysicalparametersofagenericclassof topologicalorder and other maximally quantumorders, as it
many-bodyHamiltonians. We have identified three different measurespreciselythedegreeoftheirincompatibilitywiththe
regimes,twogenericonesinwhichfrustrationduetogeome- localorderinducedbythelocalinteractionterms.
trydominatesandonlyoneofthecompetinginteractionterms The global-to-local incompatibility overlap is defined in
is strongly frustrated while the remaining one is essentially any spatial dimension (at variance with topological entropy,
frustration-free, giving rise to phases with a well defined, which is not defined for one-dimensional systems), it is in
classical-like, anti-ferromagnetic order, and an intermediate principlecomputableinaclearlyandunambiguouslydefined
one in which genuine quantum frustration is non-negligible, way(topologicalentropyhastobecomputedbycomplexand
the total frustrations of the competing interaction terms are extremely delicate subtraction schemes), and is amenable to
comparable,andaquantumvalencebondsolidphaseisestab- direct observation by strategies relying on time-of-flight or
lishedwithnodefinitemagneticorder. interferometric experiments (Von Neumann topological en-
We have further shown how these different many-body tropy is not directly accessible to experimental observation,
regimes can be clearly identified in an experimentby inves- and one should rely instead on the measurementof the two-
tigatingthebehaviorofthestaticstructurefactor,acollective Re´nyi topological entropy). On this basis, we are currently
observableamenabletodirectobservationintime-of-flightex- investigating the scaling behavior of frustration as a charac-
periments for quantum spin chains realized by atom-optical terizationoftopologicalorder[51].
quantum simulators. Finally, we have also briefly discussed In general terms, quantum ground states can be classified
thepossibilityofthedirectobservationoffrustration,asquan- according to their distinct patterns of long-range entangle-
tifiedbyourglobal-to-localincompatibility,bymeasuringthe ment. This provides the most basic categorization of quan-
visibilityinexperimentswithatomicinterferometers. tumphasesofmatter,morefundamentalthanLandau’ssym-
Theinvestigationwascarriedoutforone-dimensionalsys- metry breaking paradigm. It would then be worth pursuing
tems with competing nearest neighbor (NN) and next-to- theinvestigationoftherelationbetweenfrustration,topolog-
nearestneighbor(NNN)interactions,butthemethodological icalorder,andpatternsofgenuinemultipartiteentanglement,
frameworkis generaland in principle can be appliedalso to alongthelinessuggestedbysomerecentstudies[57–59]. In-
the study of systems with long-range interactions and/or to deed,theobservedanalogybetweentwo-sitedimerizationand
higherdimensionsanddifferentlatticegeometries. single-sitefactorizationsuggeststhatk-siteground-statefac-
Theconceptsintroducedinthepresentworkmightbepar- torizationmightbeausefultooltounderstandhierarchiesof
ticularly relevant to the investigation of systems that feature quantum many-body orders in terms of complex patterns of
maximally quantum (nonlocal) and global orders, such as multipartiteground-stateentanglement.
quantum spin liquids [55] featuring topological order [56]. Acknowledgments:-SMGandFIaregratefultoMarcello
Topologicallyorderedphases do notbreak any symmetry of Dalmonteforvaluablediscussions.SMGandFIacknowledge
themany-bodyHamiltonian,donotadmitlocalorderparam- the FP7 CooperationSTREP Project EQuaM - Emulatorsof
eters, and feature a robust ground-state topologicaldegener- QuantumFrustrated Magnetism (GA n. 323714). SMG and
acy anda complexpatternof long-rangeground-stateentan- BCHacknowledgetheAustrianScienceFund(FWF-P23627-
glement. A typical global signature of topological order is N16).
[1] G.Toulouse,Commun.Phys.2,115(1977);J.Vannimenusand (2004).
G.Toulouse,J.Phys.C10,L537(1977). [9] A. Ferraro, A. Garc´ıa-Saez, and A. Acin, Phys. Rev. A 76,
[2] J.Villain,J.Phys.C10,1717(1977); 052321(2007).
[3] S.Kirkpatrick,Phys.Rev.B16,4630(1977). [10] S.M.Giampaolo,G.Adesso,andF.Illuminati,Phys.Rev.Lett.
[4] K. Binder and A. P.Young, Rev. Mod. Phys. 58, 801 (1986), 104,207202(2010).
andreferencestherein. [11] M.M.Wolf,F.Verstraete,andJ.I.Cirac,Int.J.Quant.Inf.01,
[5] M. Mezard, G. Parisi, and M. A. Virasoro (eds.), Spin Glass 465(2003).
Theory and Beyond (World Scientific, Singapore, 1987), and [12] N.deBeaudrap,M.Ohliger,T.J.Osborne,andJ.Eisert,Phys.
referencestherein. Rev.Lett.105,060504(2010).
[6] C.Lacroix,P.Mendels,andF.Mila(eds.),IntroductiontoFrus- [13] J.Schnack,DaltonTrans.39,4677(2010).
tratedMagnetism (Springer-Verlag, Heidelberg-Berlin, 2011), [14] S. M. Giampaolo, G. Gualdi, A. Monras, and F. Illuminati,
andreferencestherein. Phys.Rev.Lett.107,260602(2011).
[7] H.T.Diep(ed.),Frustratedspinsystems(WorldScientific,Sin- [15] U.Marzolino, S.M.Giampaolo, andF.Illuminati,Phys.Rev.
gapore,2013),andreferencestherein. A88,020301(R)(2013).
[8] C. M. Dawson and M. A. Nielsen, Phys. Rev. A 69, 052316 [16] S.HillandW.K.Wootters,Phys.Rev.Lett.78,5022(1997).