Chiral spin liquids in triangular lattice SU(N) fermionic Mott insulators with artificial gauge fields

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by  Pierre Nataf

ChiralspinliquidsintriangularlatticeSU(N)fermionicMottinsulatorswithartificialgaugefields
PierreNataf,1 Miklo´sLajko´,2 AlexanderWietek,3 KarloPenc,4,5 Fre´de´ricMila,1 andAndreasM.La¨uchli3
1InstituteofTheoreticalPhysics,EcolePolytechniqueFe´de´raledeLausanne(EPFL),CH-1015Lausanne,Switzerland
2Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
3Institut fu¨r Theoretische Physik, Universita¨t Innsbruck, A-6020 Innsbruck, Austria
4Institute for Solid State Physics and Optics, Wigner Research Centre for Physics,
Hungarian Academy of Sciences, H-1525 Budapest, P.O.B. 49, Hungary
5DepartmentofPhysics,BudapestUniversityofTechnologyandEconomics,1111Budapest,Hungary
(Dated:January6,2016)
6 Weshowthat,inthepresenceofaπ/2artificialgaugefieldperplaquette,Mottinsulatingphasesofultra-
1 coldfermionswithSU(N)symmetryandoneparticlepersitegenericallypossessanextendedchiralphasewith
0 intrinsictopologicalordercharacterizedbyamultipletofN low-lyingsingletexcitationsforperiodicboundary
2 conditions,andbychiraledgestatesdescribedbytheSU(N)1Wess-Zumino-Novikov-Wittenconformalfield
theoryforopenboundaryconditions.ThishasbeenachievedbyextensiveexactdiagonalizationsforNbetween
n
3and9,andbyapartonconstructionbasedonasetofN Gutzwillerprojectedfermionicwave-functionswith
a
fluxπ/N pertriangularplaquette.Experimentalimplicationsarebrieflydiscussed.
J

5
PACSnumbers:67.85.-d,71.10.Fd,75.10.Jm,02.70.-c

]
s
a The search for unconventional quantum states of matter istheSU(N)HubbardHamiltonian
g
in realistic models of strongly correlated systems has been
- N
t an extremely active field of research over the last 25 years. (cid:88)(cid:88) (cid:88)
n H =−t (eφijc† c +H.c.)+U n n (1)
Mott insulating phases in which charge degrees of freedom i,α j,α i,α iβ
a
u aregappedhavebeenarguedtopotentiallyhostseveralfami- (cid:104)i,j(cid:105)α=1 i,α<β
q liesofquantumspinliquidsrangingfromResonatingValence
where the phases φ are chosen in a such a way that the
t. Bond Z2 quantum spin liquids [1–3] to U(1) algebraic spin (gauge-invariant) fluixj through each triangular plaquette is
a
liquids [4–6] and chiral spin liquids [7–14]. The topological
m equal to π/2. Then, at a filling of one particle per site,
propertiesofthesephaseshaveattractedalotofattentiondue
and for large enough U/t, the effective model is an SU(N)
-
to their potential impact on the implementation of quantum
d Heisenbergmodelwithlocalspinsinthefundamentalrepre-
n computers[15]. sentationofSU(N)endowedwithrealpairwisepermutations
o
and purely imaginary three-site permutations defined by the
c
[ Coldatomsopennewperspectivesinthatrespect.Inpartic- Hamiltonian[24,25]

ular,alkalinerareearthsallowtorealizeSU(N)Mottphases (cid:88) (cid:88)
1 H =J P +K (iP +h.c.) (2)
with N as large as 10 [16–19], and if a chiral phase can be ij 3 ijk
v
8 stabilized,itslow-energytheoryisexpectedtobetheSU(N) (cid:104)i,j(cid:105) (i,j,k)
5 levelk =1Chern-Simonstheory. Thefirstproposalofachi-
wherethesumover(i,j,k)runsoveralltriangularplaquettes,
9 ralphaseinthiscontextgoesbacktotheworkofHermeleetal
andP andP arecircularpermutationoperators. Tosec-
0 ij ijk
[20,21],whoshowedthatamean-fieldapproachleadstothe
0 ondorder,theamplitudeofthepairwisepermutationissimply
stabilizationofchiralphasesonthesquarelatticeinthelimit
. givenbyJ = 2t2/U,whilethe3-sitepermutationappearsat
1 oflargeN andlargenumberofparticlespersitemwithN/m
0 third order in perturbation theory with K3 = 6t3/U2 [26].
integer and ≥ 5. The same mean-field applied to SU(6) on
6 In the following, we will discuss the properties of the model
the honeycomb lattice with one particle per site has also led
1 (2) as a function of J and K using the parametrization
3
: to the prediction of a chiral state, with a competing plaque-
v J = cosθ and K3 = sinθ. We will discuss the experimen-
tte state very close in energy [22]. More recently Ref. [23]
i talprospectsofrealizingthisHamiltoniantowardstheendof
X suggested the stabilization of SU(N) chiral spin liquids on
the manuscript. It is interesting to note that parent Hamilto-
r thesquarelatticeusingstaticsyntheticgaugefields,basedon
a nians for SU(N) chiral spin liquids have been proposed re-
aslave-rotormean-fieldapproach. Inallthesescases,there-
cently[27,28]. Theyinclude both thetwo-sitepermutations
sultscallforfurtherinvestigationwithmethodsthatgobeyond
and the imaginary part of the cyclic three-site permutations,
mean-fieldtheory.
butalsoinadditiontherealpartofthecyclicthree-sitepermu-
tations, whichweomit. Therangeofthetermsintheparent
In this Letter, we show that the ground state of the Mott Hamiltonians are however not restricted to nearest neighbor
phase of N-color fermions on the triangular lattice with one ortheelementarytriangularplaquetteonly,buttheamplitudes
particle per site is a SU(N) chiral spin liquid in a large pa- dependinapower-lawfashiononthedistancesamongthetwo
rameterrangeifthesystemissubjecttoastaticartificialgauge orthreespins. Whiletherearesomestructuralsimilarities,it
fieldwithfluxπ/2pertriangularplaquette. Thestartingpoint isnotobviousthatthespatiallycompactHamiltonian(2)fea-
2
-0.5 1 4 4
SU(3), Ns=21(a) (b) (c) (d)
SU(4), Ns=20
SU(5), Ns=25
SU(6), Ns=24 0.9 SU(N)
SU(7), Ns=21
-1 SU(8), Ns=24 3 CSL 3
SU(9), Ns=27
E/Ns-1.5 E / EVMCED00..78 SU(N) ∆GS2 Ns=21, SU(3) Ns=24, SUNN(4ss==)2270,, SSUU((34)) 2∆singlet
CSL
1 Ns=25, SU(5) SU(N) 1
0.6 Ns=24, SU(8)
CSL
-2 Ns=24, SU(6)
0 0.1 0.2 0.3 0.4 0.5 0.50 0.1 0.2 0.3 0.4 0.5 00Ns=21,0 S.1U(7) 0.N2s=27, S0U.(39) 0.4 0.5 0 0.1 0.2 0.3 0.4 0.50
θ/π θ/π θ / π θ / π
FIG.1. Panel(a):GroundstateenergypersiteasafunctionofθforvariousN andN . Opensymbols(fulllines)denoteED(VMC)results.
s
(b): QualityoftheVMCwavefunctionasmeasuredbytheratioE /E . (c): EnergysplittingamongtheexpectedN singletstates
VMC ED
formingthegroundspacemanifoldofaSU(N)chiralspinliquid.(d)Energygapfromthegroundstatetothefirstexcitedsingletstatewhich
isnotpartoftheexpectedgroundspacemanifold.
turesCSLphases. ItisthegoalofthisLettertoprovidecom- that the small and large θ regimes for all considered N are
pellingnumericalevidence,basedonlarge-scaleExactDiag- mostlikelyotherphases,whiletheintermediateregioncould
onalizations(ED)andGutzwillerprojectedpartonwavefunc- harbourchiralspinliquids.
tions,thattheaboveHeisenbergHamiltonianindeedfeatures
SU(N)chiralspinliquidsareintrinsicallytopologicallyor-
extendedregionsofSU(N)CSLsforallvaluesofN = 3to
dered: Theyexhibitanon-trivialgroundstatedegeneracyon
9consideredhere.
thetorus[21]andfractionalexcitations. Thegroundstatede-
Exact diagonalizations – We start by investigating finite generacy on the torus is expected to be N for these partic-
periodictriangularlatticeclustersasafunctionofθ forvari- ular states with N different abelian anyons [20, 21]. In our
ous values of N. We focus on the range θ ∈ [0,π/2] in the numerical simulations, we can detect this degeneracy by in-
following. θ > π/2 is likely to be dominated by ferromag- vestigating the low-energy spectrum on samples with a total
netism, while θ < 0 yields the time-reversed, but otherwise numberoflatticesitesNs thatisanintegermultipleofN. In
identical physics as −θ. For small values of N = 3,4 we Fig. 1(c) we display the energy spread ∆GS of these N ex-
usedthestandardEDapproachemployingallthespacegroup pected ground states for different N as a function of θ. As
symmetries,whileonlyconsideringtheindividualcolorcon- a general trend we observe that the splitting reduces signifi-
servation, corresponding to an abelian subgroup of SU(N). cantly as we increase N. On the other hand several samples
ForallotherN arecentlydevelopedEDapproachbytwoof still show a substantial splitting. Naively one would expect
the authors [29], exploiting the SU(N) symmetry at the ex- asimpleexponentialsuppressionofthesplittingwithsystem
pense ofspatial symmetries, iscurrently the only wayto ad- size,howeverintherelatedcontextoffractionalCherninsula-
dress these systems within ED. Depending on N, the largest torsamoresubtledependenceofthegroundspacesplittingon
systemsizesN rangefrom21to27latticesites. the actual shape of the clusters has been observed and ratio-
s
nalized[30]. Wethinkthatsimilarconsiderationsapplyhere
InFig.1(a)weplottheEDresultsfortheenergypersiteof
aswell.
thegroundstateasafunctionofθforallconsideredN (open
symbols). WhilethecurvesforN (cid:46) 5lookrathersmoothat Finallywealsomeasurethegap∆singlet fromtheabsolute
first sight, it is visible that the energy per site displays kinks groundstatetothefirstsingletlevelthatisnotpartoftheex-
aroundθ/π ∼0.05−0.1andatθ/π ∼0.35−0.4forN =6 pectedgroundstatemanifold. Thisisameasurefortheexci-
to9. Forcomparisonweplottheenergyexpectationvalueof tationgapinthegappedchiralspinliquidstates. InFig.1(d),
parameter-free Gutzwiller projected chiral spin liquid model one observes an approximate dome-shaped behaviour of this
wavefunctionsforallvaluesofN (fulllines).Wewilldiscuss gapforallN,andfurthermorethisgapseemstodependonly
thepropertiesofthesewavefunctionsinamoment. Interest- weakly on N. The approximate region in θ where the N-
ingly, these model wave functions have very competitive en- fold ground state degeneracy splitting is small compared to
ergies,especiallyintheθ regionslightlyabovethefirstkink. the excitation gap (for large N) is indicated as a shaded re-
ForaquantitativecomparisonweshowinFig.1(b)theratioof gioninallthepanels,andindicatesaroughstabilityregionfor
thevariationalenergydividedbytheEDgroundstateenergy. the SU(N) chiral spin liquids on the triangular lattice. One
It is impressive that for N beyond 3 the best ratio exceeds shouldnotehoweverthatthepreciseextentofthechiralspin
0.98forthesystemsizesconsidered. Sothepicturesofaris liquidsforsmallN isanopenquestionatthispoint.
3
10
x 3.5
atri
m 1 3
p
a
erl 0.1 S2.5
v G
o
Eigenvalues of 0.00.000.000111 SSSSSSSUUUUUUU(((((((3456789))))))),,,,,,, NNNNNNNsssssss=======8818891440446008 E - E 01..5521 three-sublattice flavor order S CUS(3L)
1e-05
1 2 3 4 5 6 7 8 9 10 0
index 0 0.1 0.2 0.3 0.4 0.5
θ / π
FIG.2. VMCgroundspacedegeneracy: orderedsequenceofeigen-
valuesoftheoverlapmatrixofGutzwillerprojectedwavefunctions FIG. 3. Summed squared overlaps of the VMC model wave func-
with 30 different values of threaded flux. The overlap matrix has tionswithEDeigenstatesforN =3andN =12.Thebluecrossed
s
preciselyN largeeigenvaluesforanSU(N)chiralspinliquid. denoteEDeigenstates,whilethediameterofthefilledredcirclesde-
notesthetotalsquaredoverlaponthoseeigenstates. Inthebestcase
thesummedoverlapsonthelowestthreeEDeigenstates(degeneracy
1+2)accountforover90%ofthetotalweight.
Variational parton approach – An appealing way to de-
scribetheSU(N)chiralspinliquidsistouseaparton-based
mean field approach [20, 21, 31–36], complemented with a at the mean-field level leads to a robust rank-N overlap ma-
Gutzwiller projection. The idea is to fractionalize the ele- trix,thereforecorroboratingtheexpectationofanN-foldde-
mentary spin degree of freedom into fermionic spinons (par- generate ground state manifold in the thermodynamic limit
tons) with N flavors. For an exact description a dynamical alsoattheVMClevel.
gauge field needs to enforce the physical constraint of one Since the variational energies for SU(3) turned out not to
fermion per site. At the mean-field level however it is suffi- be very competitive, as shown in Fig. 1(a)/(b), we explicitly
cienttospecifythebandstructureandfillingofthefermionic calculated the overlaps of individual ED eigenstates of the
spinons. In the SU(N) chiral spin liquids of interest here, Hamiltonian (2) with the three orthogonal Gutzwiller wave
thespinonbandstructureconsistsofN bands,wherethelow- functionsobtainedonthesamesystemsize. InFig.3weplot
est band is completely filled for all N flavors and separated the summed squared overlap of all three wave functions (di-
by a gap from the other bands. In addition this band is re- ameteroffilledcircles)withtheEDeigenstates(crosses)asa
quiredtohaveChernnumber±1.Forthetriangularlatticewe functionofθ.HereweconsideraN =12sitesystem,where
s
use a Hofstadter-type tight-binding Hamiltonian with a uni- themomentaofthethreeEDgroundstatesinthechiralspin
form flux of π/N per triangular plaquette [37], fulfilling the liquid phase are at the zone center (one) and at the corners
requirementsonthebandstructure. Thismean-fieldstatecan of the Brillouin zone (twofold degenerate). Around θ = 0
nowbeturnedintoavalidspinwavefunctionbytheapplica- the SU(3) triangular lattice Heisenberg model is in a three-
tionofanexactGutzwillerprojection,enforcingthepresence sublatticeflavororderedstate[41,42],howeverintheregion
of exactly one fermionic spinon per site. Such a wave func- around θ/π ∼ 0.25, the three lowest ED eigenstates indeed
tioncanbehandledbyVariationalMonteCarlo(VMC)tech- have sizeable overlap with the VMC model wave functions,
niques,andinparticularonecaneasilycalculatetheenergyof therebyunderliningthepresenceofanSU(3)chiralspinliq-
the Hamiltonian (2) on rather large lattices. The VMC ener- uidforsufficientlylargevaluesofθalsoforN =3.
giesdisplayedinFig.1(a),(b)havebeenobtainedthisway. Edge states – Another hallmark of chiral topological
ThenextquestionishowtheVMCapproachisabletoac- phasesisthepresenceofchiraledgemodesintheenergyspec-
countforthenon-trivialgroundstatedegeneracyonthetorus. trumofsystemswithaboundary. Ithasbeenunderstoodthat
Itturnsoutthatbythreadingfluxthroughthenon-contractible thecharacteristicenergylevelstructureoftheedgeexcitations
loopsaroundthetorus,oneisabletospananN-dimensional asafunctionofthemomentumalongtheboundaryservesas
subspace of Gutzwiller projected wave-functions, with al- a fingerprint of the type of topological order realised in the
most identical local properties on finite lattices. From the bulk [43]. The SU(N) CSLs considered here are expected
viewpoint of topological order this corresponds to a charge to exhibit a chiral edge energy spectrum described by the
pumping procedure, where one cycles through the N differ- SU(N) Wess-Zumino-Novikov-Witten(WZNW)conformal
1
entgroundstatesbythreadingdifferentanyonicfluxthrough fieldtheory(CFT)[21].ThisisthesameCFTthatgovernsthe
the interior of the torus. These concepts have recently been low-energyspectrumofwell-studiedone-dimensionalcritical
exploredinthecontextofSU(2)CSLonseverallattices[38– SU(N)spinchains[27,28,44,45].
40]. We have checked in Fig. 2 that the subspace of wave Inordertotestthishypothesisnumerically, onehastode-
functionsspannedbyusing30differentboundaryconditions sign a setup where one can detect the edge modes in a clean
4
SU(3) SU(4) SU(5) SU(6) SU(7) SU(8)
Ns=7 Ns=13 Ns=15 NN=s=1199 168
C6 C6 C7 sC6 x2 224 x2 56
15
3 20 21
168
4 490
45 6
3 20 224 56
3 6 5 120 735 2800
6 36 50 21 168 56
20 4 45 47505 6 218604 224 2281 56 550044
4 5 84 84 2281 504 1008
3 4 5 6 21 56
0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2
l-l l-l l-l l-l l-l l-l
0 0 0 0 0 0
FIG. 4. Edge states in SU(N) chiral spin liquids: the leftmost panel displays the N = 19 sites triangular cluster with open boundary
s
conditionsused.Inthevariousotherpanelsweexhibitthelowenergyspectrumasafunctionoftheangularmomentumaroundthecentralsite
(l denotesthegroundstateangularmomentum). Thechiraledgestatesareclearlyvisible,withacharacteristicSU(N)multipletstructure,
0
whichcorrespondstoaparticularsectorofachiralSU(N) Wess-Zumino-Novikov-Wittenconformalfieldtheory.Theanalyticalpredictions
1
areindicatedbythedimensionsoftheSU(N)multipletsandcanbefoundinTab.Iofthesupplementarymaterial.
way. Startingfromatorusanaturalwaywouldbetocutthe summarytheanalysisofthestructureoftheedgeexcitations
torus open into a cylinder. This geometry however has two performed here confirms the SU(N) WZNW CFT predic-
1
independent,counter-propagatingedges,makingacleananal- tionsandthusstrengthensthecaseforabelianSU(N)chiral
ysisdifficult,giventhesystemsizesN accessibletoED.We spinliquidsinthemodelHamiltonian(2).
s
therefore choose to emulate a disk geometry by considering
Experimental considerations – With the recent realiza-
thespecificN =19sitetriangularlatticewithopenboundary
s tion of the Mott-crossover regime in 3D optical lattices with
conditions depicted in the left panel of Fig. 4. Such a lattice
fermionicYtterbiumatoms[46,47]theprospectforthereal-
might actually be built in future ultracold atom experiments
izationofstronglycorrelatedSU(N)quantummagnetismis
withopticallatticesandatightconfiningpotential. Thissam-
becoming bright. Our proposal for triangular lattices builds
plestillhasasixfoldrotationaxisaboutthecentralsite,yield-
oningredientswhichhavebeendemonstratedseparately: the
inganangularmomentumquantumnumberwhichweuseto
possibility to realize Mott insulators in optical lattices, and
plottheenergyspectrum.Theenergyspectrumofthedischas
to create static artificial gauge fields in an optical lattice (for
no topological ground state degeneracy, but features gapless
alkaline atoms) [48, 49]. Beside, working with the triangu-
edge modes whichtypically propagate only inone direction.
lar lattice is a big advantage because the 3-site permutation
Theprecisemultipletstructureoftheedgemodesdependson
term is the first and only term to appear to third order per-
theanyonicsector. Inoursetupthissectorcanbesimplyla-
turbation theory starting from the Hubbard model with one
beled as a = (N mod N). In Tab. I of the supplementary
s particlepersite,bycontrasttoe.g. thesquareandhoneycomb
materialwehavecompiledtheSU(N) WZNWCFTpredic-
1 lattice, where they appear at order 4 and 6 respectively, and
tions for the different irreducible representations of SU(N)
are not the first corrections. The chiral phase typically ap-
which appear at a given excitation energy, here qualitatively
pears for θ (cid:39) 0.3, which, using the perturbation expressions
labeledbytheexcessangularmomentuml−l0.Intheremain- ofJ =2t2/U andK =6t3/U2,correspondstot/U (cid:39)0.1.
3
ingpanelsofFig.4wedisplaytheactualEDenergyspectrum
This might besmall enough to be still inthe Mott insulating
of the Hamiltonian (2) for a fixed value of θ/π = 0.25 for
phase, and to ensure that higher order corrections are negli-
N =3upto8asafunctionoftheangularmomentuml−l .
0 gible. Infuturestudiesonemightalsorelaxtheπ/2fluxper
For all N one can clearly identify a branch of chiral excita-
plaquette condition, and explore the extent of the expected
tions propagating to the right. The analytical predictions are
stabilityregionoftheSU(N)CSLphases.
indicated by the dimensions of the SU(N) irreducible rep-
Several interesting questions need to be addressed in fu-
resentations. For all N the numerical data for the first three
turework. Forexample,isitpossibletodirectlyengineerthe
sectors(l−l =0,1,2)isinfullagreementwiththeanalyti-
0
required three site exchange terms in Hamiltonian (2) using
calpredictions.Thesplittingbetweenthemultipletsatagiven
sophisticatedquantumopticsschemes? Thereishopethatthe
valueoflisexpectedtovanishasN grows,andthespectrum
s
current activity on lattice gauge-theory implementations will
should become linear with a certain edge state velocity. In
bringtechniquestoaddressthisquestion. Anotherintriguing
5
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6
NinSU(N) N mod N l=0 l=1 l=2
s
2 1 (2) ⊕ (4)
3 1 (3) ⊕ (¯6) 2× ⊕ ⊕ (15)
4 3 (¯4) ⊕ (20) 2× ⊕2× ⊕ (3¯6)
5 4 (¯5) ⊕ (45) 2× ⊕2× ⊕ (50)⊕ (7¯0)
6 1 (6) ⊕ (8¯4) 2× ⊕2× ⊕ (120)⊕ (21¯0)
7 5 (2¯1) ⊕ (2¯8)⊕ (224) 3× ⊕ ⊕2× ⊕ (490)⊕ (73¯5)
8 3 (56) ⊕ (168)⊕ (5¯04) 3× ⊕2× ⊕2× ⊕ (10¯08)⊕ (2800)
9 1 (9) ⊕ (3¯15) 2× ⊕2× ⊕ (396)⊕ (27¯00)
TABLEI. ThethreefirstangularmomentumsectorsofthechiraledgemodeoftheN =19dropletforSU(N) .Thenumber(N mod N)
s 1 s
selectstheanyonicsector(primaryfieldoftheCFT)whichconsequentlydeterminestheedgespectrum.
7
SUPPLEMENTARYMATERIAL variantcombinationofN −1LuttingerliquidCFTs(thusthe
SU(N) WZNWCFTcentralchargec=N −1).
1
SU(N) WZWNpredictionsforthechiraledgestates
1
InTab.IweexplicitlylisttheexpectedSU(N)irreducible SU(N) countingrule OEISidentifier
representationswiththeirmultiplicityforthefirstthreeexci- SU(2) 1,1,2,3,... A000041
tationlevelsl = 0(primaryfield)andl = 1,2(firsttwode- SU(3) 1,2,5,10,... A000712
scendantlevels)ofachiralSU(N) WZNWconformalfield
1 SU(4) 1,3,9,22,... A000716
theory. TheprimaryfieldforeachN isdictatedbytheopen
SU(5) 1,4,14,40,... A023003
boundaryclusterssizeN =19viathelength(N mod N)
s s SU(6) 1,5,20,65,... A023004
of the single-column young diagram at l = 0. We have de-
SU(7) 1,6,27,98,... A023005
rived these results using a successive SU(N) coupling se-
SU(8) 1,7,35,140,... A023006
quence with the adjoint representation starting from the irre-
duciblerepresentationatl=0andsubsequentthenull-vector SU(9) 1,8,44,192,... A023007
elimination based on the SU(N) counting rule restrictions SU(10) 1,9,54,255,... A023008
listed in Tab. II. This simplified procedure uses the fact that
TABLEII.SU(N)countingrulesusedtoderivetheresultsinTab.I.
the SU(N) CFT can also be seen as particular SU(N) in-
1