Haldane Topological Orders in Motzkin Spin Chains

0.38 MB·

by  L. Barbiero

HaldaneTopologicalOrdersinMotzkinSpinChains
L. Barbiero,1 L. Dell’Anna,2 A. Trombettoni,1,3 and V. E. Korepin4
1CNR-IOM DEMOCRITOS Simulation Center, Via Bonomea 265, I-34136 Trieste, Italy
2DipartimentodiFisicaeAstronomia“G.Galilei”,Universita` diPadova,viaF.Marzolo8,I-35131,Padova,Italy
3SISSA and INFN, Sezione di Trieste, Via Bonomea 265, I-34136 Trieste, Italy
4C. N. Yang Institute for Theoretical Physics, Stony Brook University, NY 11794, USA
Motzkinspinchainsarefrustration-freemodelswhoseground-stateisacombinationofMotzkinpaths. The
weightofsuchpathcontributionscanbecontrolledbyadeformationparametert. Asafunctionofthelatter
thesemodels,besidetheformationofdomainwallstructures,exhibitaBerezinskii-Kosterlitz-Thoulessphase
7 transition for t=1 and gapped Haldane topological orders with constant decay of the string order parameters
1
fort < 1. BymeansofnumericalcalculationsweshowthatthetopologicalpropertiesoftheHaldanephases
0
dependonthespinvalue. ThisallowstoclassifydifferentkindsofhiddenantiferromagneticHaldanegapped
2
regimesassociatedtonontrivialfeatureslikesymmetry-protectedtopologicalorder. Ourresultsfromoneside

n allowtoclarifythephysicalpropertiesofMotzkinfrustration-freechainsandfromtheothersuggestthemasa
a newinterestingandparadigmaticclassoflocalspinHamiltonians.
J

1 PACSnumbers:
3

] Spinchainsplayacrucialroleinmanyfundamentalphys-
h
ical phenomena like magnetism [1], quantum phase transi-
c
tions [2], topological orders [3] and quantum computation
e
m [4]. A fundamental contribute to the understanding of spin
chains is provided by the the seminal papers by Haldane
-
t [5] where a new topological phase, the Haldane phase (HP),
a
t uniquelydetectableviaanon-localstringorderparameter[6]
s
. hasbeendiscoveredforspin-1XXZHeisenbergchains. This
t
a hasdrivensignificanteffortstolookfornewkindsofmodels
m whosetopologicalordercanbedescribedintermsofastring
- orderparameter[7]motivatingthediscoveryofthecelebrated
d
Affleck-Kennedy-Lieb-Tasaky (AKLT) model [8]. Although
n
theargumentofHaldaneisgivenforintegerspinchains,only
o
c integerspinXXZ-likeandAKLT-likechainsowntopological
[ HP and is therefore non-trivial to find and study new classes

ofHamiltonianswhereHPemerges.
2
v Thankstothestrongestquantum”resource”,namelytheen-
8 tanglement, spin models have also a fundamental role in the
7 simulationofquantumlogicalgatesforquantumcomputation
8
[4]. For this reason finding and studying Hamiltonians with
5
highlyentangledspinsiscurrentlyoneofthemostchalleng-
0
. ingandintriguingfieldsinquantumphysics. Figure 1: (Color online) Upper panels: Cartoons of a possible
1
In this direction local integer frustration-free spin Hamilto- Motzkin path and its representation in terms of spins for the two
0
nianswhoseground-statecanbeexpressedasacombination cases a) uncolored s = 1 and b) colored s = 2. Central panels:
7
DMRGlocalmagnetizationforasystemoflength2n = 60atdif-
1 ofMotzkinpaths[9]havebeenrecentlyintroduced[10,11].
ferenttdeformationvalues(cid:104)Sz(cid:105)forc)s = 1andd)s = 2. Lower
: Amongothersinterestingaspects,theirimportanceisgivenby i
v panels:Thermodynamiclimitofthegap∆=E −E asafunction
1 0
thefactthattheyownalevelofentanglemententropywhich
i oftfore)s=1(redcircles)andf)s=2(bluesquares).Thecontin-
X stronglyexceedstheoneexhibitedbyotherpreviouslyknown uoslinesarefittedwiththeform∼(exp−b/(cid:112)|t−t |)witht =1
c c
r local models. Relevantly, also for half-integer spins, a simi- andbafittingparameter. Thethermodynamiclimitisextrapolated
a
larclassofHamiltonians,theFredkinspinchains,exhibiting byusingchainsoflengthupto2n=60. AlltheDMRGsimulation
the same features [12, 13] has been introduced. In addition are performed by keeping at most 1024 DMRG states and 5 finite
sizesweepswithanerrorenergy<10−9(10−7)fors=1(s=2).
to their entanglement properties Motzkin chains own further
verypeculiarproperties. Indeed,eveniftheyarepurelylocal
models, for high spin values s (i.e., s ≥ 2) they behave as
defactolongrangeHamiltoniansbeingabletoviolatecluster versionofMotzkin[14]andofFredkin[15]chainshavebeen
decomposition properties (CDP) and area law (AL) scaling introduced,andtheirgapstudied[16],withthecontributionof
of the entanglement entropy [12]. Very recently, a deformed MotzkinorFredkinpathstotheground-statebeingweighted
throughtheintroductionofaparametert.
2
Duetotheaforementionedargumentsitappearsclearasthese mations and |φ(t)k(cid:105) = (1 + t2)−1/2(|k,0(cid:105) − t|0,k(cid:105)),
new models are both very interesting by themself and they |ψ(t)k(cid:105) = (1 + t2)−1/2(|0,−k(cid:105) − t|−k,0(cid:105)) and
could open the path towards fundamental applications. This |Θ(t)k(cid:105) = (1 + t2)−1/2(|k,−k(cid:105) − t|0,0(cid:105)). The defor-
motivatesustoinvestigateaMotzkinchainfordifferentspin mation induced by t (cid:54)= 1 keeps the model frustration-free
valuesandinpresenceofpathdeformations. Hereafteranin- [14],and,whilefort = 1werecovertheundeformedmodel
troductionofthemodelintermsofdeformedMotzkinpaths, [10–12],fort>1(t<1)thepathshavinglarger(smaller)h
we present density matrix renormalization group (DMRG) arefavoredinthegroundstate.
[17]calculationswhichallowtorevealtheappearanceofdif- t≥1Regime. Thislatterpointexplainsthet>1behav-
ferent phases as a function of the deformation parameter t. iorofthelocalmagnetization(cid:104)Sz(cid:105)observedinFig.1c)andd)
j
Inparticularweshowthatlocalmagnetizationisabletocap- fors = 1ands = 2respectively. Indeed,sincet > 1makes
ture the t > 1 regime where a clear domain wall structure moreprobablehigherpaths,intermsofspinsthiscorresponds
takesplaceindependentlybythespinvalues. Fromtheother toadomainwall(DW)structurewheretheupanddownspins
side once t < 1 the system undergoes to a phase transition are separated in two different regions of equal length n [23]
of a Berezinskii-Kosterlitz-Thouless (BKT) type [18] as sig- andthezerospinsarebasicallyabsent. Relevantly,asshown
naled by an exponential opening of the gap. Moreover our inFig. 1e)andf),thislatterregimeisgapless(∆=0)mean-
numericalcalculationsconfirmthatforthiskindofdeforma- ing that the difference between the ground-state E and the
0
tion the entanglement entropy is bounded and size indepen- first excited state E energy goes to zero in thermodynamic
1
dent [14]. Crucially we find that this gapped regime can be limit (TDL). The aforementioned features allow to find the
describedsolelybyanon-vanishingvalueofstringorderpa- analogybetweenEq. 1andtheγ <1regimesofXXZchains
rameters thus showing the topological nature of the t < 1 for both spin 1 and 2 (being γ the z-anisotropy parameter
deformedMotzkinchains. Fors=1onlyonestringisfinite, [24]). Further similarities can be also noticed for the t = 1
similarly to what happens in the SU(2)-Haldane phase for casewhereagaplessregimeisassociatedtoapower-lawde-
XXZorAKLTspin-1models,thusrevealingthepresenceof cayofthecorrelationfunction(cid:104)S+S−(cid:105)and,asexactlyshown
i j
a symmetry-protected topological (SPT) order. On the other in[12],anexponentialdecayof(cid:104)SzSz(cid:105)−(cid:104)Sz(cid:105)(cid:104)Sz(cid:105)[25]thus
i j i j
hand,fors = 2differentkindsofHaldanephaseshavebeen resembling the XY phase of XXZ models but, with the key
obtained[19,20]. Inparticular,forthespin2Motzkinchain, featurethatbothALdecayandCDPareviolatedfors=2.
weshowthattwostringsdisplayaconstantdecayexhibiting t<1Regime. Fromtheotherside,asalreadymentioned,
a phase similar to the SO(5)-topological Haldane order oc- a t < 1 deformation minimizes the height of the possible
curing in s = 2 AKLT model [21]. Interestingly, unlike the paths. This is clearly visible in the (cid:104)Sz(cid:105) behavior shown in
j
undeformedcaset=1,fort<1theCDP[22]isvalid. Fig. 1 c) and d) where an almost totally flat local magneti-
Model. The spin model we consider has the peculiarity zation with h = 0 is observed. Crucially (cid:104)Sz(cid:105) shows also
j
of having a ground state which can be expressed in terms antiparallel peaks at the edges of the chain thus supporting
of Motzkin paths describing all the possible 2n moves that thepossiblepresenceofedgestates. Thiseffect,asexplained
one can make to go from a point of height h = 0 to an before, is produced by the Π (s) term in Eq. (1) which has
∂
otherpointofthesamehwithoutcrossingthe0line[10,11]. alsotheroleofbreakingtheground-statedegeneracy. Theal-
As shown in Fig. 1 a) and b), spins can be seen as moves mostflatmagnetizationcanexplainanotherbehavior,namely
by imposing that up/zero/down spin corresponds to increas- that the entanglement entropy, S(A) = −Trρ log ρ of a
A 2 A
ing/conserving/decreasing the height of the path. Of course, subsystemA,isboundedanddoesnotdependonneitherthe
for spin s = 1 only uncolored steps (uncolored Motzkin chain nor on the partition length [14], meaning that the area
chain)areallowedandthesystemcanbeseenasaspins=1 lawscalingisfulfilled. Indeed,asitispossibletoseeinFig.
chain,whilelargervaluesofscanbeachievedwhencolored 2a),wefindthatS(A)isconstantatfixedtforany2nwhile
stepsarepossible(coloredMotzkinchain). TheHamiltonian it grows almost linearly with the deformation strength. The
reads: latteriseasilyexplainedbythefactthatfort<1thestrength
of t actually affects only the first and the last move with flat
2n−1 2n−1
H = (cid:88) Πj,j+1(s,t)+Π∂(s)+ (cid:88) Πcj,rjo+ss1(s) (1) (cid:104)Sjz(cid:105) = 0 in the bulk. Consequently a larger/smaller t will
produce an higher/lower value of |(cid:104)Sz(cid:105)| in the first and last
j=1 j=1 i
sitethusgeneratingmore/lessentropy. Noticethat,asshown
where Πj,j+1(s,t) = (cid:80)sk=1(cid:0)|φ(t)k(cid:105)(cid:104)φ(t)k|j,j+1 + in Fig. 2 a), this behavior holds for any considered s value.
|ψ(t)k(cid:105)(cid:104)ψ(t)k| | + |Θ(t)k(cid:105)(cid:104)Θ(t)k| (cid:1) is the bulk Alesstrivialaspect,conjecturedin[14],emergesbylooking
j,j+1 j,j+1
term,Π (s) = (cid:80)s |−k(cid:105)(cid:104)−k| +|k(cid:105)(cid:104)k| istheboundary Fig. 1e)andf),namelyt < 1deformationssupportthepres-
∂ k=1 1 2n
term which makes more favorable for the first spin to point enceoffinitegapinthethermodynamiclimit.Asvisibleinthe
upward, |k(cid:105), and the last downward, |−k(cid:105). The latter samefigures,forboths=1ands=2thegapopenscompat-
term in Eq. (1) Πcross(s) = (cid:80) = |k,−k(cid:48)(cid:105)(cid:104)k,−k(cid:48)|, iblywithanexponentialdecay∆∼exp−b/(cid:112)|t−t |,being
j,j+1 k(cid:54)=k(cid:48) c
present only for s > 1, ensures the color matching of t = 1 and b a fitting parameter, thus signaling a BKT-like
c
up and down spins with the same height. The param- phasetransition. Inintegerspinchainsgappedregimecanbe
eter t appearing in Π (s,t) describes path defor- usuallyassociatedtoeitherantiferromagnetic(AF)orderde-
j,j+1
3
Figure2:(Coloronline)a):EntanglemententropyS(A)forasubsystemhavinglengthnwith1≤i≤nfors=1(redsymbols)ands=2
(bluesymbols). TheinsetshowstheconstantbehaviorofS(A)asafunctionofthesize2n. b): C(|i−j|)fordifferentt < 1values. c):
O1,−1(|i−j|)fordifferentt<1values. Thecorrelationsinthepanelsb)andc)areevaluatedinasystemofsize2n=60withipinnedin
thefirstchainsite.Wecheckedthatdifferenti-valuesdonotalterthephysicalbehaviorofthecorrelations.
scribedbythetwopointscorrelationfunctions by the geometrical meaning of deformations. Indeed, as ex-
plained before, a small t favors the paths with low h. Intu-
C(|i−j|)=(cid:104)SzSz(cid:105)−(cid:104)Sz(cid:105)(cid:104)Sz(cid:105) (2)
i j i j itively, one can argue that the path with smaller h is the one
ortoHaldaneordersdescribedbyastringorderparameter where the first and last move corresponds to respectively the
rising and the lowering steps with in the middle a series of
Ok,k¯(|i−j|)=(cid:104)Lk,k¯eıπ(cid:80)i≤(cid:96)<jLk(cid:96),k¯Lk,k¯(cid:105) (3) flatmoves. Thismeansthatthenumberof+1,−1spinspro-
i j
ducingthehiddenAForderisminimizedbyreducingt,thus
where Lk,k¯ = |k(cid:105)(cid:104)k|−|−k(cid:105)(cid:104)−k|. Notice that, for s = 1, producingalowersaturationvalueofthestringorder. Never-
k(k¯) can be solely equal to 1(−1) thus L1,−1 = Sz while thelesswecheckedthatevenverysmalldeformationssupport
i i
for s = 2, k(k¯) can take the values 1(−1) and 2(−2) and thepresenceofaconstantO1,−1(|i−j|),suddenlydisappear-
Sz = 2L2,−2+L1,−1. Theimportantinformationsencoded ingfort = 1, thusallowingtounambiguouslyconcludethat
i i i
in such non-local order parameters is that, once it is finite, theuncoloredt<1Motzkinchainhastopologicalorderwith
Eq.(3)describesatopologicalphase,usuallycalledHP,with hiddenAF.ThisphaseisusuallycalledSU(2)-Haldanephase
hiddenantiferromagnetism. ThehiddenAForderisgivenby and it has been observed both in spin-1 XXZ [6, 26] chains
thefactthatitcannotbedescribedbyusualtwopointcorrela- andintheAKLTmodel[27]. Wewillkeepthisnomenclature
tionfunctionsEq. (2)thusdescribingaphasewherespinsup even if for our model only the operator (cid:80) Sz = (cid:80) L1,−1
i i i i
anddownarerigorouslyalternatedandseparatedbyarandom commuteswiththeHamiltonian. Thisissimilartowhathap-
numberofzerospins. Ofcourse,whilefors = 1theHPcan pens in the spin-1 XXZ model when a single ion anisotropy
be given only by alternating +1 and −1 spins thus signaled termisincluded,breakingtheSU(2)invariancebutpreserv-
byafiniteO1,−1(|i−j|),fors = 2thehiddenordercanbe ingtheHaldanephase.
signaled, as it will be clear in the following, by two or even For systems with conformal invariance like the spin-1 XXZ
solelyonefiniteOk,k¯(|i−j|). model, the topological order is also captured by an even de-
generacy of the entanglement spectrum (ES) [28]. From the
s = 1 case. Here we start our analysis with the s = 1 other side, when Hamiltonians cannot be described by con-
casebyevaluatingbothO1,−1(|i−j|)andC(|i−j|)fordif- formal field theory (CFT), like for instance the AKLT [29]
ferentt<1values. Fig. 2clearlyshowsthatwhileC(|i−j|) orexoticbosonicHamiltonians[30],theESdoesnotpresent
rapidlydecaystozerothestringorderparameterremainscon- even degeneracy but the topological order is assured by the
stantasafunctionofthedistancethussignalingthepresence presence of edge modes and finite strings. We checked that
of a HP. This aspect, in analogy with XXZ chains, supports duetothelackofapossibledescriptionintermsofCFT[11]
our prediction regarding the BKT nature of the phase transi- the ES of t < 1 deformed Motzkin chains does not display
tion. Wealsocheckedthatthestringsalongtransversedirec- anydegeneracy. Neverthelessedgemodes,visibleinFig. 1c)
tions decay. Furthermore, as visible in Fig. 2 c), the string andd),andthefinitestringinFig2assurethetopologicalor-
O1,−1(|i−j|) saturates to a constant value which becomes der. Thelatterhasafurtherfundamentalpropertyduetofact
bigger the larger is t. At a first look this aspect could seem thatitappearsforanoddvaluesofthespin. Indeed, oncea
counterintuitive since one expects that the larger is the gap, Haldane phase takes place for odd spins s, SPT order [7] is
thestrongeristhestringorder. Neverthelessaneasyinterpre- generated. Thisisgivenbythefactthattheedgemodesfrac-
tation of the O1,−1(|i−j|) behavior as function of t comes tionalize in two half-integer spins which cannot be removed
4
Figure3:(Coloronline)a):C(|i−j|)fordifferentt<1values.b):O1,−1(|i−j|)fordifferentt<1values.c):O2,−2(|i−j|)fordifferent
t<1values.Allthecorrelationareevaluatedinasystemofsize2n=60withipinnedinthefirstchainsite.Wecheckedthatusingdifferent
valuesofidonotalterthephysicalbehaviorofthecorrelations.
unlessinpresenceofaphasetransitionoranexplicitlysym- withthes = 1case, thestringsbecomestrongerbyincreas-
metriesbreaking. Thisconsiderationallowstoconcludethat ing t. Fig. 3 b) also shows a further information encoded in
thes=1versionofEq.(1)witht<1deformationssupports theC(|i−j|)behavior. Indeed,onthecontrarytothet = 1
thepresenceofSPTtopologicalorderwithboundedandsize regime[12],itsexponentialdecayisassociatedtoa0edge-to-
independententanglementthusstronglycharacterizingthethe edge value thus holding the CDP. The opening of a gap in a
Motzkinchains. coloredMotzkinchain,therefore,restoresthepurelocalityof
the model in Eq. (1), in agreement with the general findings
s = 2 case. As shown in [11, 12], the s > 1 unde- forgappedlocalHamiltonians[35].
formed t = 1 chains have much more intriguing properties Conclusions and Perspectives. In conclusion, our results
with respect to lower spin case. These are induced by the demonstratetheexistenceoftopologicalHaldaneordersina
presence of colors which increase the symmetry of the sys- new class of spin Hamiltonians. Furthermore we shown the
tem. As for the s = 1, s = 2 XXZ Heisenberg models and behavior of Motzkin chains as a function of the deformation
AKLTmodelscansupportthepresenceofgappedphasesfor strength t. While the undeformed t = 1 case has XY-like
positive z-anisotropies. The gap can again be associated to features,fort > 1thesystempresentsagaplessdomainwall
AForderdetectedbyC(|i−j|)ortodifferentkindsofHal- structure. From the other side at t = 1 a BKT-like phase
daneorder,seeforinstance[19,31]andreferencetherein. In transition,characterizedbyaexponentialopeningofthegap,
particular in such systems SPT topological order is signaled occursforanyt<1values. Thegappedregimeisassociated
by finite values of both O1,−1(|i − j|) and O2,−2(|i − j|). toSPThiddenHaldaneantiferromagneticorderssignaledby
Moreover, asconjecturedin[32]andshownin[31,33], sin- finite values of string non-local order parameters. The two
gle ion anisotropy terms can support the formation of a SPT possibleHaldaneordershavethepeculiarityofhavinganen-
SU(2)-Haldane order even for s = 2. Our calculations in tanglement entropy independent from both block and chain
Fig. 1f)showthatagainthecoloredMotzkinchainisgapped size. Moreoverourresultssuggestthatitwouldbeveryinter-
for t < 1 and the gap is associated to a hidden AF order estingtohaveaphysicalimplementationoftheMotzkinspin
since C(|i − j|) has a clear exponential decay rapidly sat- chains. Inthisrespectcoldatomicsystems,whichhavebeen
urating to 0 as shown in Fig. 3 a). From the other side both already proposed to simulate several kinds of spin Hamilto-
O1,−1(|i−j|)andO2,−2(|i−j|)haveaconstantandbasically nians with topological orders [36], could provide a possible
equalbehaviorthusclarifyingthatthes = 2Motzkinchains physicalplatformtoimplementMotzkinchains.Theirexperi-
with t < 1 deformations support the presence of an SO(5)- mentalrealizationcouldberelevantfortechnologicalachieve-
Haldane phase. It is worth stressing that SO(5) is not the ments since, from one side, symmetry-protected topological
symmetryofourmodel,ratherU(1)×U(1)×Z ,sinceonly orders have been proposed as ideal candidates towards the
2
(cid:80) L2,−2 and(cid:80) L1,−1 commutewiththeHamiltonian,like realization quantum devices like quantum repeaters [37] and
i i i i
inthes=2AKLTmodelwhentheterm(cid:80) (Sz)2 isswitched substrate for measurement-based quantum computation [38]
i i
on. AlsointhatcaseSO(5)-Haldanephasesurvivesoncethe while, from the other, side Motzkin paths may have appli-
symmetry is lowered from SO(5) to U(1)×U(1) [34]. In cations in the field of polymer absorption [39]. Finally we
our case the symmetry is supplemented by the invariace un- underline that, in future works, it would be very interesting
derinterchangingthetwocolors(Z ). Thisisthereasonwhy to study the gapped regimes in the fermionic version of the
2
O1,−1(|i−j|)andO2,−2(|i−j|)arethesame, asshownin s = 3/2 Fredkin model where exotic Haldane regimes can
Fig. 3. Moreover it is important to notice that, in analogy takeplace[40].
5
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forkindhospitality. LDacknowledgesfinancialsupportfrom [23] Thefactthespinsupanddownareseparatedintworegionsof
MIUR through FIRB Project No. RBFR12NLNA 002. LB equallengthisduetothefactthatweareimposingavanishing
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