Realizing and Detecting the Haldane's Quantum Hall effect with Ultracold Atoms

0.34 MB·English

by  L. B. Shao

Realizingand Detecting theHaldane’s Quantum Halleffect withUltracoldAtoms
L. B. Shao1,2,3, Shi-Liang Zhu1, L. Sheng2, D. Y. Xing2, and Z. D. Wang3
∗
1InstituteforCondensedMatterPhysicsandDepartmentofPhysics,SouthChinaNormalUniversity,Guangzhou,China
2NationalLaboratoryofSolidStateMicrostructureandDepartmentofPhysics,NanjingUniversity,Nanjing,China
3Department of Physics and Center of Theoretical and Computational Physics,
The University of Hong Kong, Pokfulam Road, Hong Kong, China
WedesignaningeniousschemetorealizetheHaldane’squantumHallmodelwithoutLandaulevelsbyusing
9 ultracoldatomstrappedinanopticallattice. Threestanding-wavelaserbeamsareusedtoconstructawanted
0 honeycomblattice,wheredifferenton-siteenergiesintwosublatticesrequiredinthemodelcanbeimplemented
0 throughtuningthephaseofonelaserbeam. Thestaggeredmagneticfieldisgeneratedfromthelight-induced
2 Berry phase. Moreover, we establish a relationship between the Hall conductivity and the atomic density,
enablingustodetecttheChernnumberwiththetypicaldensity-profile-measurementtechnique.
n
a
PACSnumbers:73.43.-f,05.30.Fk
J

8
2
The quantum Hall effect (QHE) [1] in two-dimensional Peierls phase from site j to its second neighbor l, which is

electron systems is one of the most peculiar quantum- assumed to take the form ϕ = ϕ. The hopping direc-
] jl ±
l mechanicalphenomenaobservedinnature. TheQHEisusu- tions for which ϕ = +ϕ are shown in Fig. 1a. The most
l jl
a allyassociatedwithauniformexternalmagneticfield,which interesting and unique feature of the model lies in that the
h
splits the electronenergyspectruminto discrete Landaulev- phase of the system can be changed from a normal insula-
-
s els(LLs).WhentheFermienergyliesinthegapbetweentwo tor to a Chern insulator by the simulation of parity anomaly
e LLs, the Hall conductivity in units e2/h is accurately quan- [5, 6, 7, 8, 9, 10, 11]. However, it is extremely hard to
m
tizedtoaninteger. TheprecisequantizationoftheHallcon- realize the Haldane’s model experimentallyin ordinary con-
.
t ductivitywas explainedbyLaughlin[2] basedupona gauge densedmattersystemsbecauseoftheunusualstaggeredmag-
a
invariance argument, which is fundamentalto the picture of neticfluxassumedinthemodel.
m
edge states proposed by Halperin [3]. On the other hand,
- Ontheotherhand,thetechnologyofultracoldatomsinan
d Thouless, Kohmoto,NightingaleandNijs (TKNN)[4] inter- opticallatticeprovidesaperspectiveapproachtoexplorerich
n pretedtheHallconductivityasthetopologicalChernnumber
fundamentalphenomenaofcondensedmatterphysics[12,13,
o of the U(1) bundle over the magnetic Brillouin zone of the
14].Inparticular,howtorealizetheQHEwithcoldatomshas
c
bulkstates.
[ attractedconsiderableinterest[15, 16, 17, 18, 19]. Neverthe-
Twentyyearsago,HaldaneshowedinprinciplethataQHE
less,theatomicQHEhasnotbeenobservedyet,mainlydueto
2 may also result from breaking of time-reversal symmetry
challengesinboththerealizationanddetectionoftheatomic
v
withoutanynetmagneticfluxthroughaunitcellofaperiodic
0 Halleffects. Althoughaneffectivemagneticfieldforneutral
two-dimensional(2D)system,wheretheelectronstatesretain
5 atomscanbesimulatedeitherbyrotatingtheatoms[20]orby
8 their usual Bloch state character [5]. In his work, Haldane laser-induced Berry phases[21, 22, 23, 24], the strong mag-
1 constructedatight-bindingmodelonahoneycomblatticein-
neticfieldregionrequiredforQHEhasnotbeenreachedyet
4. cludingacomplexsecondnearest-neighborhoppingintegral. inexperiments.Fortherotatingmethod,thesystemiscloseto
0 ThehoneycomblatticeconsistsoftwotriangularsublatticesA¯
thepointatwhichthecentrifugalpotentialcancelstheexter-
8 andB¯ with differenton-siteenergiesM and M, as shown
nalharmonictrap,andtheatomsmayflyapartattherotation
0 −
in Fig. 1a. For M = 0, the inversion symmetry is broken
: 6 speedrequiredbyQHE[16,19]. Forthelaser-inducedBerry
v and the lattice possesses the point group C3υ symmetry. A phaseapproach,thecoldatomsmovinginaspatiallyvarying
i periodicvectorpotentialA(r) is appliedto thelattice, given
X laser field feel an effective gauge potential[21, 22, 23, 24],
that the total magnetic flux through each unit cell vanishes,
r but the region of the strong uniform field is rather small for
a i.e., the first-neighbor hopping integral t is unaffected. The
twotypicalcounterpropagatingGaussianlaserbeams. Inad-
′
second-neighborhoppingintegralt acquiresa Peierlsphase
dition, the detection method for cold atoms is very differ-
factorexp(ie A dr/¯h), wheretheintegrationisalongthe
ent from that for condensed matter systems; especially the
·
hoppingpath.RTheHamiltonianofthemodeliswrittenas
widely used technique for QHE based on the transportmea-
surementsisnotworkableforatomicQHE.InthisLetter,we
H = (ta†lbj +H.c.)+ M(a†jaj −b†jbj) design an ingenious scheme to realize the Haldane’s quan-
Xl,j Xj tumHallmodelwithoutLLsbyusingultracoldatomstrapped
h i
+ t′eiϕjl(a†laj +b†lbj), (1) in an optical lattice. We work out a distinct method to con-
Xl,j structthehoneycomblatticesthathavedifferenton-siteener-
hh ii gies by three standing-wavelaser beams. Althoughit is still
where a and b are the annihilation operators on site R in hardtoachieveastronghomogenousmagneticfieldrequired
i i i
sublattices A¯ and B¯, respectively. ϕ is the accumulated by the conventional QHE in atomic system, we may evade
jl
2
(a) (b) are trappedat the minimaof the potential, forminga honey-
2
comblattice.Anamazingfeaturehereisthatthedifferentsite-
1 energiesofsublatticesA¯andB¯iscontrollablebythephaseof
laser beam χ. For instance, we get exactly the honeycomb
0 lattice with the same on-siteenergies(M = 0) forχ = 2π,
3
−1 as shown in Fig.1b, while the two sublattices have different
on-siteenergies(M =0)forχ= 2π.
−2 6 6 3
−2 −1 0 1 2 Now we elaborate how to simulate the staggered mag-
netic field in the Haldane’s model. Since the net flux per
(d) (c)
2 unit cell vanishes, the vector potential applied to the lat-
tice must be periodic. Such magnetic fields can be cre-
1
ated by Berry phase induced from two opposite-travelling
0 standing-wave laser beams [22]. For the two laser beams
−1 with Rabbi frequencies Ω1 = Ω0sin(yk2L + π4)eixk1L and
Ω2 = Ω0cos(yk2L + π4)e−ixk1L, the effective gauge poten-
−−22 −1 0 1 2 tial is generated as A1(r) = h¯k1Lsin(2yk2L)ex [22]. Here,
kL =kLcosθandkL =kLsinθwithkL thewavevectorof
1 2
FIG.1:(Coloronline)(a)IllustrationoftheHoneycomblatticestruc- thelaserandθ theanglebetweenthewavevectorandtheex
tureofgraphene,whereopenandsolidcirclesrepresentsitesinsub- axis. We emphasizethatthechoiceofwavevectork2L ofthe
lattices A¯ and B¯. a1 = (12a,−√23a), a2 = (12a,√23a) are the laser beamsmust be a multiple of 2√3a3π in order to be com-
unit vectors of the underlying triangular sublattice. s1, s2 and s3 mensuratewiththeopticallattice. We takekL = 2√3π. The
arethreevectorspointingfromaB¯ sitetoitsthreenearest-neighbor 2 3a
Peierlsphasesforthenearest-neighborhoppinginFig.1aare
sites. (b)and(c)showthecontoursofthepotentialVforχ=2π/3
andχ = 39π/60,respectively. Thevertical(horizontal)axisrepre- ϕ12 = ϕ61 = ϕ34 = ϕ45 = ϕ0 andϕ23 = ϕ56 = 0. For
sentsykL/π(xkL/π).(d)Contoursofthemagneticfielddefinedby the next-neares−t-neighb−orhopping integrals, which are inte-
0 0
Eq.(2). grated on a period of the vectorpotential, the corresponding
accumulatedphasesare ϕ13 = ϕ24 = ϕ46 = ϕ15 = 0, and
thisbottlenecksinceLLsarenotnecessaryinthisunconven- ϕ35 = ϕ62 = ϕ, where ϕ = k1Lasina√k32L. Since the lattice
hasthesymmetryofpointgroupC3v,thevectorpotentialA1
tional QHE. We elaborate that the staggered magnetic field, is rotated by 2π to obtain the other two vector potentials.
which is hardto generatein condensedmatter systems, may ±3
Thenthetotalaccumulatedphasesalongthenearest-neighbor
be rather easy to set up by other three standing-wave laser
directions are found to cancel out because of the symmetry
beams. Inthisscenario,differenton-siteenergiesintwosub-
ofhoneycomblattice. However,thetotalaccumulatedphases
latticescanbeeasilyadjustedthroughtuningthephaseofone
for the next-nearest-neighborhopping along the arrowed di-
of the laser beams, and thus the whole phase diagram[5] in-
rectionsof the dashed lines in Fig. 1a are just ϕ. Therefore,
cluding the exotic topological phase transition predicted by
thetotalvectorpotentialandmagneticfieldcanbewrittenas
Haldanemayberevealedexperimentally.Furthermore,based
on (2 + 1)-dimensional relativistic quantum mechanics cal- A = h¯k1L[sin(2yk2L)+cos(√3xk2L)sin(yk2L)]ex
culations,weestablishadirectrelationshipbetweentheHall √3h¯kLsin(√3xkL)cos(ykL)e (2)
1 2 2 y
conductivityandtheequilibriumatomicdensity,suchthatthe −
B = 2h¯kLkL[cos(2ykL)+2cos(√3xkL)cos(ykL)]e .
famoustopologicalChernnumbermaybeexperimentallyde- 1 2 2 2 2 z
−
tectedwiththestandarddensityprofilemeasurementusedin The contours of the magnetic field are plotted in Fig. 1d, in
atomicsystems[12]. which the red lines indicate where the magnetic field van-
Let us first consider single component fermionic atoms ishes.Thetotalmagneticfluxthrougheachhexagonvanishes,
(e.g.,40K,6Li,etc.)ina2Dhoneycomblattice[14,25],which as the Peierls phase accumulatedalong its edgesis zero. As
can be realized by three detuned laser beams. A detuned aconsequence,thetotalHamiltonianofthiscoldatomicsys-
standing-wavelaser beam will create a potential in the form tem can be described by Eq. (1). With Fourier transforma-
V0sin2(kL0 ·r), where V0 is the potential amplitude and kL0 tionaj = √1N keik·Rjak andbj = √1N keik·Rjbk, the
is the wave vector of the laser. To generate the honeycomb HamiltonianofPthesystemcanbewrittenbyPusing“spinors”
lattice withdifferenton-siteenergiesinsublatticesA¯andB¯, (a ,b )tas
k k
thethreelaserbeamswiththesamewavelengthbutdifferent
polarizationsare applied along three differentdirections: e Hk =h0(k)+h1(k)σ1+h2(k)σ2+h3(k)σ3, (3)
y
aVwnhd=er√e2V3α0e[xsi±n=2(12√αe3+yx,+kreLsπ2/p2)ec+tivyseiknlyL2.(/Ty2kh.e0LTph+oetepπ3no)ttie+anltsiisianlt2hc(uoαsn−tgoi−uvresnχ2ab)r]ye, 2swith)′,esrienhh2ϕ(0k(k))s=i=n(2kt′tcPoasi)ϕs.iPns(ki(cio·s=s(ki)1·,a,ia2)n,,dh31)h(ak3r)(ek=t)hetP=thrieceMosv(ekc+-·
plottedin±Fig. 1ban0dFi±g. 1c0. Undertheseconditions,atoms torspointPingi froma· B¯isiteitoitsthreenearestneighbors.The
3
vectorh(k) = [h1(k),h2(k),h3(k)]isaneffectivemagnetic (a)6
field of the “spinors”. The energyspectraare E = h0(k) 4 C=0
bh|ha3(n(kKd)|−toa)un)cdwhetihsthethKeen±evrag=lyen±gcae43pπabi(as1n,|dh0)3a.|t,Itwwf hhoe3Dre=irha03c,p=thoeinhct3so(nKKd+u+c)t(ainoo±dnr ’--0242M/t C=-1 C=0 C=1
-6 -3 -2 -1 0 1 2 3
K . ThefamousTKNNindexorChernnumberforthissys-
twemi−thishˆ(gkiv)eansbtyheCun=it8v1πecRt1oBrZofd2hk(kǫµ)ν.hˆIt·i(s∂dkµemhˆo×ns∂tkraνthˆe)d[t2h6a]t (b)-0121 (c)-0121
-2 -2
the gauge invariance C can only take integer values and the -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
quantumHallconductivityisproportionaltotheChernnum- (d)2 m+ (e)2 m+
ber[4]. FortheHaldane’smodel,differentphasesofthesys- 1 1
0 0
temcanbecharacterizedbydifferentvaluesofC. Thephase -1 -1
-2 -2
diagramisdepictedinFig. (2a)[5], in whichthesolidline is -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
thecriticalboundarybetweenthenormalinsulatorwithC =0 m+ m+
and the Chern insulator with C = 1. It is notable that the FIG.2: (a)Phasediagramofthesystem,wherethephaseboundary
phase ϕ = πtanθ can be controll±able by simply choosing (solidline)correspondstoh3 = 0. Thesystembehaveslikeanor-
mal insulator when C = 0 and aChern insulator when C = ±1.
thelaserangleθ,whiletheenergydifferenceM betweensub-
(b)-(e) Charge density in units B/φ0 for |m | = 0.5|m+| and
lattice can be tuned by the phase of laser beams. With such eB¯hvF2 = 4|m+| as a function of normalize−d chemical potential
controllability,itispromisingtorealizetheexotictopological µ/|m+|(correspondingtoarescaledatomicdensityprofileinatrap)
phasetransitionbetweendifferentChernnumbersinFig. 2a. forfourdifferentcases: (b)m+ < 0 < m ,(c)m+ > 0 > m ,
We now turn to establish a direct connection between the (d)m+ <m <0,and(e)m+ >m >0−.In(b)and(c)thesh−ift
topologicalChermnumberandtheatomicdensity,notingthat ofchargeden−sityatµ=0ispresents−othattheHallconductivityis
nonzero. In(d)and(e),theshiftoftheatomicdensityisabsentand
the latter can be detected with density profile measurements
sotheHallconductivityvanishes.
typicallyusedinatomicsystems. We firstdevelopaGreen’s
functionmethodtocalculatetheatomicdensity.Thesystemis
actuallydescribedbyaDirac-likeHamiltonianwhichcanbe Euclideanspace.FromthestandardGreenfunctionapproach,
obtainedbyexpandingEq.(3)aroundtwoDiracpointsK = theatomicdensityisexplicitlyobtainedas
4π(1,0). By the substitution of k K + p, we h±ave
H±3a= υFp1σ1 υFp2σ2 + m σ3→at K± , respectively,
w3√h±e3rte′sυi∓nFϕ=. U√n32daetr−uisnitthaeryFterramnsifover±mloactiitoynaσn2d±Hm±σ2=, wMe ca±n ρ± = |φB0|sgn(µ)(cid:26)int(cid:20)2µh¯2v−F2|meB2±|(cid:21)+ 12(cid:27)Θ(|µ|−|m±|)
writetheHamiltonianinamoresymmetricform− m
B ± Θ(m µ), (7)
H = vFp1σ1 vFp2σ2+mσ3, (4) ±2φ0|m±| | ±|−| |
± − − whereΘstandsfortheunitstepfunctionandint[x]meansthe
where the notation m = m is introduced for simplicity
± ± largestintegerlessthanx. ThesecondtermofEq. (7)isthe
(i.e.,wetemporallyomitsubscripts below).
± atomicdensityinducedintovacuumasµ 0bytheuniform
Itisobservedthattheconductivityσ (Chernnumber)of →
xy magneticfield. Itisofparityanomalysinceitscorresponding
thesystemisrelatedtotheatomicdensityρaccordingtothe
Hallcurrentisindependentofthe magneticfield aftermulti-
Stredaformulaσ =∂ρ/∂ onceanadditionaluniform
xy B|µ,T plyingthedriftvelocity / ,where istheelectricfield.
magneticfield isapplied.Suchamagneticfieldcanbesim- E B E
B The total atomic density is given by the sum of the den-
ulated by rotating the optical lattice at a constant frequency
sities of the two components ρ = ρ+ + ρ . At µ = 0,
ω =eB/2m.WechoosethevectorpotentialasA0 =A1 =0 the Hall conductivity at = 0 can be obt−ained from the
ean>dA0,2E=q.(B4)xc.aBnybeussionlgvetdheinsuthbestriteuatliospnapce→. Thpe+eigeeAnewneitrh- density by using the StreBda formula as σxy = Ceh2, where
giesoftheHamiltoniancanbeobtainedas[5] C = 12[sgn(m+)−sgn(m−)]istheChernnumber. Toshow
how to detect the Chern number of the system, we consider
msgn(e ) n=0 a finite magneticfield . The calculateddensityρ in unitof
En =(cid:26)− m2+B2n¯hv2 e n=1,2,3... , (5) e /φ0 is plotted as aBfunction of the normalized chemical
and the dege±nepracy of each LFL| Bis| /φ0 per unit area with p|inoBtFe|ingt.ia2l.µ/It|mis+e|s(sfeonrt|imal−th|a=t t0h.e5|smpa+ti|aalnddeen¯hsivtFy2Bpr=ofi4le|mρ+(r|))
|B|
φ0 the flux quantum. The density ρ in terms of the Green’s is uniquelydeterminedby the functionρ(µ/m+ ) in the lo-
functionfortheDiracHamiltonianisgivenby | |
caldensityapproximation,whichistypicallywellsatisfiedfor
(D/ +m)G=1, ρ=−Tr[γ0G(x,x′)]|x→x′ (6) tTrhapeppeldatfeearumsiionntsh.eFaitgoumreisc2dbe-n2seitystahnadvefoornfeo-tuor-doinffeecroenrrtecsapsoens-.
whereD/ =γτDτ,withD0 =h¯∂0 µ,D1 =h¯vF∂1,D2 = dencetotheplateausintheHallconductivityduetothefinite
¯hvF∂2 + ievF x, γ0 = σ3, γ1 =−σ2, and γ2 = σ1 in magneticfield > 0. We herefocusonµ = 0, whichisof
B − B
4
ourmaininterest. Form+ < 0 < m ,whichcorrespondsto aware that a relation between the density profile and the
−
C = 1, the atomic density ρ = /φ0 < 0, as shown in HallconductivityinconventionalQHEwasalsoaddressedin
− −B
Fig. 2b. Form+ > 0 > m , whichcorrespondsto C = 1, Ref.[28]. We thankDr. H.Zhaiforbringingourattentionto
−
thedensityρ= /φ0 >0,asshowninFig.2c. Fortheother thatwork.
B
twocasesm+ < m < 0andm+ > m > 0,m+ andm
− − −
havethesamesign,correspondingtoC = 0,andthedensity
ρ = 0,asseenfromFigs.2dand2e. Therefore,asimpledi-
rect relation between the Chern number and the equilibrium
atomicdensityisestablishedas ∗ Electronicaddress:[email protected]
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494(1980).
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[3] B.I.Halperin,Phys.Rev.B25,2185(1982).
The important relation (8) actually provides us a feasible
[4] D.J.Thoulessetal.,Phys.Rev.Lett.49,405(1982).
waytoexperimentallydetecttheChernnumberC indifferent
[5] F.D.M.Haldane,Phys.Rev.Lett.61,2015(1988).
phases. In the absence of , the density of the cold atoms
[6] G.W.Semenoff,Phys.Rev.Lett.53,2449(1984).
B
at µ = 0 is first measured, which is denoted as ρ0. Then [7] S.Deser,R.JackiwandS.Templeton,Phys.Rev.Lett.48,975
the opticallattice isrotatedto generatethe effectiveuniform (1982).
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[11] Y.Zhangetal.,Nature438,201(2005).
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B
Finally, webrieflyaddressanalternativeapproachtoreal- [15] N.R.CooperandN.K.Wilkin,Phys.Rev.B60R16279(1999).
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s-band of the honeycomb lattice. As for fermions in the p-
ibid,85,4648(2000);B.Paredesetal,ibid.87,010402(2001).
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[17] A.S.Sorensen,E.Demler,andM.D.Lukin,Phys.Rev.Lett.94,
can also be implementedby rotating each optical lattice site
086803(2005).
aroundits own center[27]. Nevertheless, how to detectsuch [18] R.N.PalmerandD.Jaksch,Phys.Rev.Lett.96,180407(2006).
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We haveestablishedarelationshipbetweentheHallconduc-
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tivity and the equilibrium atomic density, which provides a
Y.Li,C.Bruder,andC.P.Sun,ibid.99,130403(2007).
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differentphases. [23] K.Osterlohetal.,Phys.Rev.Lett.95,010403(2005).
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for Basic Researches of China (Nos. 2006CB921800, [25] S. L. Zhu, B. G. Wang and L. M. Duan, Phys. Rev. Lett 98,
2004CB619004, 2007CB925104, 2007CB925204, 260402(2007).
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and 2009CB929504), the RGC of Hong Kong (Nos.
196805(2007).
HKU7045/05P, HKU7049/07P and HKU7044/08P), the
[27] C.Wu,arXiv:0805.3525(May,2008).
URC fund of HKU, NCET and the NSFC under Grant Nos. [28] R.O.Umucalilar,H.Zhai,andM.O¨.Oktel,Phys.Rev.Lett.100,
10429401,10674049,and10874066. 070402(2008).
Note added – After this work was completed we became