Chern numbers hiding in time-of-flight images

0.83 MB·English

by  Erhai Zhao

Chernnumbershidingintime-of-flightimages
Erhai Zhao1,2, Noah Bray-Ali2, Carl J. Williams2, I. B. Spielman2 and Indubala I. Satija1,2
1SchoolofPhysics,Astronomy,andComputationalSciences,GeorgeMasonUniversity,Fairfax,VA22030and
2JointQuantumInstitute,NationalInstituteofStandardsandTechnologyandUniversityofMaryland,Gaithersburg,MD20899
(Dated:January21,2013)
Wepresentatechniquefordetectingtopologicalinvariants–Chernnumbers–fromtime-of-flightimagesof
ultra-coldatoms.WeshowthattheChernnumbersofintegerquantumHallstatesoflatticefermionsleavetheir
fingerprintsintheatoms’momentumdistribution.Weanalyticallydemonstratethatthenumberoflocalmaxima
inthemomentumdistributionisequaltotheChernnumberintwolimitingcases,forlargehoppinganisotropy
andinthecontinuumlimit.Inaddition,ournumericalsimulationsbeyondthesetwolimitsshowthattheselocal
2 maximapersistforarangeofparameters. Thus,aneverydayobservableincoldatomexperimentscanserveas
1 ausefultooltocharacterizeandvisualizequantumstateswithnon-trivialtopology.
0
2
I. INTRODUCTION servable of cold-atom experiments, into an unexpected, use-
n
ful tool for characterizing and visualizing topological states
a
J ofmatter.
Topologicalinsulators(TI)andsuperfluidsaremany-body
6 Figure 1a summarizes the main results of this paper ob-
quantum systems with energy gaps in the bulk but topolog-
tained for the Hofstadter model [14]. We highlight the pa-
ically protected, gapless excitations on the boundary [1, 2].
] rameterregimeswherethestructureinn(k)ismostevident:
s Eachclassoftopologicalphasesischaracterizedbyatopolog-
a inthecontinuumlimit(Fig.1b)orforhighlyanisotropiclat-
icalinvariantwhichcanbeeitheranintegerorabinaryquan-
g tices(Fig.1c). Inparticular,weprovideatransparentunder-
tity. Forexample,theintegerquantumHallinsulators[3]are
- standing of n(k) for large hopping anisotropy. In this limit,
t distinguishedfromothertwo-dimensionalinsulatorswiththe
n states with energy at the band edges dominate, and they are
a samesymmetrybytheChernnumber,anintegerthatcapture dimerized(theirlocalizedwavefunctionsarepeakedatpairs
u the global properties of all occupied bands [4]. The Chern
ofspatiallyseparatedlatticesites). Thesizeofthese“Chern-
q numbercoincideswiththenumberofchiraledgemodes,and
dimers,”inunitsofthelatticespacingd,isequaltotheChern
t. theHallconductanceinunitsoftheconductancequantum. A
a numberofthecorrespondinggap. Asaresult,n(k)variessi-
centralprobleminexploringnewformsoftopologicalmatter
m nusoidally (Fig. 1c) with a period determined by the Chern
ishowtoextractthetopologicalinvariantsfromexperiments,
number. Furthermore, our detailed numerical studies ver-
- andunambiguouslydeterminethetopologicalphase.
d ify that such distinctive features in the momentum distribu-
n Experimentswithultracoldatomshasbroughtusopportu- tionpersistbeyondtheChern-dimerandthecontinuumlimit
o nities to creating topological phases of matter in parameter (Fig. 2c and 2d). However, the correspondence between the
c
regimes unreachable in solids. In addition to band-structure numberoflocalmaximainn(k)andtheChernnumberisnot
[
engineering using optical lattices and tuning the atom-atom alwayspreciseforarbitraryparameters.

3 interaction by Feshbach resonance, artificial magnetic fields
v forneutralatomshavebeenproducedbyeitherrotation[5]or
0
atom-lasercoupling[6,7],makingwaytorealizingquantum
0 II. ULTRACOLDFERMIONSINSYNTHETICMAGNETIC
Hall states of cold atoms [8]. Moreover, non-Abelian syn-
1 FIELD
thetic gauge fields [7, 9] and spin-orbit coupling [10] have
3
. been demonstrated. These developments motivated theories
5 We consider the Hofstadter model [14] of spinless
on properties of ultracold topological matter, e.g., the frac-
0 fermionicatomsonasquareopticallatticeinthee -e plane
tional quantum Hall effect of interacting bosons [5, 11, 12], x y
1
:1 tahneomquaalnotuusmHsapllinstHataelsleofffepc-otrobfiftaelrmfeiromniso[n9s],[1an3d].thequantum H =− (cid:88) ts,s(cid:48)(cid:0)eiθs,s(cid:48)fs†fs(cid:48) +h.c.(cid:1)−(cid:88)µsfs†fs. (1)
v
i While cold atom systems offer new detection techniques (cid:104)s,s(cid:48)(cid:105) s
X
andgreattunability,itischallengingtomeasurethetopologi-
Here, f† is the creation operator for a fermion at site
r calinvariantsdirectly.Measurementsoflocaldensityofstates s
a s = {s ,s }; the nearest-neighbor hopping strengths t =
(of the edge states) or mass transport are marred with com- x y x
t , and t = t are real and positive; θ =
plexityincontrasttosolidstatesystems. Thus,creativemeth- s,s+ex y s,s+ey s,s(cid:48)
(cid:82)s(cid:48)
ods have to be invented. For example, Ref. [8] proposed to e A·dldependsonthevectorpotentialA[21]forparti-
s
extract the quantized Hall conductance from the derivatives cleswithchargee;andµ isalocalchemicalpotential. With
s
ofthein-situatomdensitydistributionbytheStˇredaformula. cold atoms, the hopping anisotropy λ = t /t can be tuned
y x
Here,weshowthattheChernnumberc ofalattice-quantum byvaryingtheintensityofthelasersgivingrisetotheoptical
r
Hall system of fermions can be simply determined from the latticepotential[6]. ForanappliedmagneticfieldB = Be
z
structures (ripples) in the momentum distribution n(k) as normal to the 2D plane, α = d2eB/h is the magnetic flux
measured in time-of-flight (TOF) images for a range of pa- per plaquette in units of the flux quantum. We study homo-
rameters. Ourresultstransformn(k), asimpleeverydayob- geneous gases with µ = µ, and also finite systems in the
s
2
presenceofanisotropicharmonictrappotential,capturedby numerical results are complemented by analytic analysis for
µ =µ −mω2d2s2/2. low flux and large anisotropy. In what follows, we discuss
s 0
Artificial magnetic fields have been implemented for inturnthebehaviorsofn(k )indifferentparameterregimes,
x
bosons[15],anditisstraightforwardtoapplythesetechniques startingwithanisotropiclatticeandlowflux.
to fermions. With their smaller fine-structure splittings, the
alkali fermions (6Li and 40K) are more susceptible to heat-
ing from spontaneous emission than some bosons (87Rb and III. OSCILLATIONSINMOMENTUMDISTRIBUTION
133Cs),butforthe(cid:126)/t ∼240µstimescales[22]associated
x,y
withthemajorenergygapsinthespectrum(seeFig.2a),the Figures 1b and 2b show n(k ) in this isotropic low flux
x
expected≈ 200mslifetimein40Kshouldbeacceptable;see limit,atα=0.05andλ=1. Asthechemicalpotentialisin-
Ref.[9]foranalternatesolution. creaseduptoµ=0,successivelyhigherHofstadterbandsare
We first summarize the well known properties of the ho- filled,withChernnumbersc = r wherer = 1,2,3... labels
r
mogeneous Hofstadter model [Eq. (1)] with commensurate thehighestfilledband.Correspondingly,n(k )hasexactlyc
x r
fluxα = p/q,wheretheintegersp,q arerelativelyprime. In localpeaksontopofasmoothenvelope,i.e.,countingpeaks
the Landau gauge [23] suitable for current experiments [6], inTOFimagesunambiguouslydeterminestheChernnumber,
the vector potential A = αxey scales the unit cell along ex and therefore which quantum Hall state the system is in. To
by a factor of q to a magnetic unit cell; this also creates a understandthis,werecallthatforα (cid:28) 1,thecyclotronorbit
magnetic Brillouin zone (MBZ) with kx ∈ (−π/q,π/q] and islargecomparedtothelatticespacing,sothesystemresem-
ky ∈(−π,π]. TheHofstadtermodelisdiagonalinthesingle- blesafreeFermigasinmagneticfield,andthenarrowbands
particle basis Ψsx,sy = ei(kxsx+kysy)ψsx(ky), provided ψsx intheHofstadterspectrummaptoLandaulevels. Forrfilled
satisfies[14] Landau levels, n(k ) is the summed square modulus of the
x
momentum-spacewavefunctions,
eikxψr +e−ikxψr +2λcos(2πs α+k )ψr =−E ψr
with esnxe+r1gy in unitssx−of1tx. The indexx r =y 1s,x2,...q larb(e2lss)x n(kx)∝(cid:88)r−1(2νν!)−1Hν2(kx(cid:96)B)e−(kx(cid:96)B)2, (5)
linearlyindependentsolutions,andψr = ψr . Thesingle ν=0
sx+q sx
energybandatα = 0splitsintoq sub-bands,andtheenergy whereH aretheHermitepolynomials,and(cid:96) =(cid:112)(cid:126)/eB is
spectrumhasq−1gaps[14]. Whenthechemicalpotentialµ ν B
themagneticlength. DuetothenodestructureofH ,n(k )
isinsidether-thenergygap,thesystemisanintegerquantum ν x
hasexactlyrlocalmaximalocatedatthezerosofH . This
Hall state with Chern number c , defined by the Thouless- r+1
r is reflected in our numerical results as plotted in Fig. 1b for
Kohmoto-Nightingale-Nijs (TKNN) formula [4] and related
the lattice system. As is depicted by the shaded regions in
tothequantizedHallresistance.
Fig.1a, thisassociationisvalidintheentirelowfluxregime
UsuallyweconsiderTOFimagesasapproximaterepresen-
andisinsensitivetoλ.
tationsofthemomentumdistributionn(k)obtainedbyimag-
At larger α the Landau level picture breaks down, but
ingthe2Datomiccolumndensityafterat periodoffree
TOF the correspondence between the number of local peaks and
expansion. In most cases, the final spatial coordinate of an
the Chern number appears to persist for a range of parame-
atom is x = vt where v is the velocity [16], however,
TOF ters. Fig. 2c and 2d show examples of n(k ) for large flux
vectorpotentialscomplicatetheinterpretationofTOFimages x
α = 0.45 and λ = 1 where well resolved peaks do remain.
becausethevelocityisv=(cid:126)k/m−eA/m. InRef.[17],we
Moregenerally,asαincreases(forµresidingwithinthesame
showedthatsuddenlyturningthegaugefieldtozeroisequiv-
energygap),thedistinctivefeaturesofn(k )aregenerallyre-
alent to a brief electric field that takes the initial momentum x
tained, but compared to the low flux regime, the local peaks
(cid:126)k and maps it to a new final velocity v(cid:48) = (cid:126)k/m. As a
(exceptatk = 0)movetohighermomentaandcanbesub-
result,detectedTOFimagesdoapproximatemomentumdis- x
sumedintothebackgroundstructure.Insomeinstances,these
tributions[24]. Thecrystalmomentumdistribution
peaksturnintoshoulder-likestructures.
n˜(k)=(cid:88)(cid:104)f†f (cid:105)eik·(s−s(cid:48)), (3) Intriguingly, as the hopping anisotropy becomes large,
s s(cid:48) small wiggles on a background turn into regular and pro-
s,s(cid:48) nouncedoscillationsregardlessofα. Fig.2cand 2dillustrate
the progression of n(k ) from λ = 1 to λ = 10, showing
can be reconstructed from TOF images using the recipe in x
pronouncedoscillationsdevelopwithincreasingλ. Mostim-
Ref.[18]. Here,theaverage(cid:104)···(cid:105)isoverthegroundstate. We
portantly,thesimplerulereappearsinthelimitofλ(cid:29)1: the
alsodefinea1Dcrystalmomentumdistributionbyintegrating
numberofoscillationsinn(k ),i.e.,thenumberofpeaksfor
n˜(k)alonge , x
y
k ∈ [−π,π),isagainexactlyequaltotheChernnumberc .
x r
(cid:90) π AsisemphasizedinFig.1c,n(k )asymptoticallyapproaches
x
n(k )= dk n˜(k). (4)
x y
−π 1
n(k )∝r± cos(c k ) (6)
x 2 r x
Forhomogeneoussystems,wecomputedn(k )bynumer-
x
icallydiagonalizingEq.(2)usingasufficientlylargeunitcell, forchemicalpotentialinsidether-thgap.Theone-to-onecor-
whileforfinitetrappedsystemswediagonalizedEq.(1). Our respondence,ormapping,betweenthenumberofoscillations
3
andtheChernnumberforanisotropiclattices(Fig.1a:Chern- V. EFFECTOFTEMPERATUREANDTRAP
dimer regime) is one of our key observations which we now
explainanalytically.
Our numerical simulation based on Eq. (1) confirms that
thestructuresinn(k )persist,andremainvisible,fortrapped
x
systems at finite temperatures. Due to the finite band gap of
thequantumHallinsulator,thermalfluctuationsonlyslightly
IV. CHERN-DIMERS smear the peak (or oscillation) structure in n(k ) as long as
x
T issmallcomparedtothegapofinterest, whichisoforder
t/2 for the major gaps in the Hofstadter spectrum. Fig. 4
The λ (cid:29) 1 limit is tractable in perturbation theory [19]
shows examples of n(k ) for realistic trap and temperature
wherethesolutionstoEq.(2)aresimple. Whent = 0(i.e., x
x
parameters.Thus,ourproposalofdetectingtheChernnumber
λ → ∞), there are q bands each with dispersion E (k ) =
r y
isexperimentallyfeasible.
−2t cos(2παr+k ),r ∈{1,...,q}(seeFig.3,wherep=1
y y
andq = 5). Theseeigenstatesareextendedalonge andlo-
y
calizedtosinglesitesasafunctionofx.AsdefinedinEq.(2),
the r-th band describes states ψr = δ localized at site VI. Q-MATRIXANDMOMENTUMDISTRIBUTION
sx sx,r
s =randdisplacedink ,asidentifiedinFig.3. Thebands
x y
intersectatnumerousdegeneracypointsk∗ forstatesr = s Finally, we point out a general connection between n˜(k)
y x
and r(cid:48) = s(cid:48). The spatial separation along e of these lo- and the Q-matrix [20], which plays a fundamental role in
x x
calized states is (cid:96) = s(cid:48) − s . These band-crossings occur the classification of topological insulators and in turn is re-
x x
atq−1discreteenergies(rightaxesofFig.3)withexactlyq latedtotheChernnumberofaquantumHallinsulator. Fora
crossingshavingthesame(cid:96)ateachenergy.Forfiniteλ,t hy- band insulator with band structure E (k) (with energy mea-
x r
bridizesthestatesateachcrossing,leadingtoq(q−1)avoided suredfromthechemicalpotential),thespectralprojectionop-
crossings, and(q−1)gapsinthebandstructure. Neareach eratorsP (k)andP (k)projectontostatesaboveorbelow
+ −
degeneracypointthesystemiswelldescribedbytheeffective thechemicalpotential,respectively. TheQ-operatorQ(k) =
Hamiltonian[19], P (k) − P (k) describes a “flattened” Hamiltonian that is
+ −
smoothly deformed from the original [20]. The momentum
Hˇ (k)=∆ [cos(k (cid:96))σˇ +sin(k (cid:96))σˇ ]+v (k −k∗)σˇ . (7) distributionisrelatedtothespectralprojectionoperatorofthe
2 (cid:96) x x x y (cid:96) y y z filledbandsρ(k) = P (k). Inmatrixform,ρisrelatedtoQ
−
by
σˇ are the Pauli matrices in the basis of localized
x,y,z
states |r(cid:105) and |r(cid:48)(cid:105); and the k-independent constants are Q (k)=δ −2ρ (k). (9)
ab ab ab
∆ = (−1)(cid:96)Πr(cid:48)−1 (cid:8)1(cid:2)E (k∗)+E (k∗)(cid:3)−E (k∗)(cid:9)−1
(cid:96) R=r+1 2 r y r(cid:48) y R y
and v = 2λ(cid:2)sin(2παr+k∗)−sin(2παr(cid:48)+k∗)(cid:3). The As an example, for the Hofstadter model at flux α = p/q,
(cid:96) y y
eigenstatesare“Chern-dimers”,superpositionsof|r(cid:105)and|r(cid:48)(cid:105) ρab(k) = (cid:104)fa†(k)fb(k)(cid:105),wherea,b ∈ {1,...,q}arethesite
withwavefunction indiceswithinthemagneticunitcell,andkisthecrystalmo-
mentum within the MBZ. n˜(k) as defined in Eq. (3) is sim-
ψs±x = √12(cid:0)δsx,r+eiβ±δsx,r(cid:48)(cid:1), (8) pthlye bn˜e(hka)vi=or (cid:80)ofan˜b(ρkaxb,(kky∗)). aBsyfuEnqct(i9o)n, ofofrkgxiviesndkeyter=minkey∗d,
by the Q-matrix along cut k = k∗ in the BZ. So is the
y y
1D momentum distribution n(k ). On the other hand, the
x
whereβ = −(cid:96)(k +π)+πandβ = −(cid:96)(k +π)arerel-
+ x − x Chern number is nothing but the winding number of the off-
ativephasesfortheupperandlowerbandedges,respectively.
diagonal block of the Q-matrix in chiral basis, also known
Starting from Eq. (7) it is straightforward to show [19] that
asthetransitionfunction, alongacutthatsplitstheBZ[20].
theChernnumberofeachbandgapisc =(cid:96). Thus,forsuffi-
r Thus, on general grounds, phase winding responsible for
cientlyanisotropichopping,dimerizedstatesformattheband
nonzero Chern number implies nontrivial structures in the
edgeswithspatialextentalonge equaltotheChernnumber
x momentum distribution function. Explicitly, for the asymp-
ofthecorrespondinggap. totically exact effective Hamiltonian Hˇ (k), the Q-matrix
2
Theformationofdimersofsize(cid:96) = cr attheedgesofthe is Qˇ(k) = (v(cid:96)2ky2 + ∆2(cid:96))−1/2Hˇ2(k). The Chern number
r-th gap implies a “hidden spatial correlation” described by c = (cid:96), for the off-diagonal block of Q-matrix is simply
r
correlationfunctionC(l) = (cid:80)s(cid:104)fs†fs+lex(cid:105), whichispeaked ∼ eikx(cid:96). Meanwhile,fromtheQ-matrix,weobtainn(kx) =
atl = (cid:96). Forchemicalpotentialinther-thgap,C(l)asymp- q[1−sign(∆ )cos(k (cid:96))]. Thisprovesanalyticallyourprevi-
(cid:96) x
totes to a delta function C(l) → δ(l −(cid:96)), as λ → ∞. Be- ous result Eq. (6). The oscillations adiabatically evolve into
causeβ+−β− =πforeachcrossing,thenetcontributionto local peaks for the isotropic lattice, as explicitly shown in
C(l) from dimerized states associated with gaps fully below Fig.2cand2d.
theFermienergyiszero. SinceC(l)andn(k√x)areaFourier In contrast with Chern numbers, n(kx) is not a topologi-
(cid:80)
transform pair, n(k ) = C(l)cos(k l)/ N; this yields cally invariant quantity. Still, the clear signatures of Chern
x l x
the cos(c k ) term in Eq. (6). The r offset in Eq. (6) comes numbersinTOFimagesthatwefindheremakesmomentum
r x
fromtheC(l=0)contribution. distributions into an unexpected tool for exploring topologi-
4
cal states of matter. In addition to edge states and Hall con- ploreothertopologicalstatesofmatterincludingtopological
ductance,ripplesinthemomentumdistributionanddimerized superfluids.
statesofanisotropiclatticeswhosespatialextentencodesthe We acknowledge the support of the ONR and NIST (E.Z.
topological quantum numbers provide a new way to charac- andI.I.S),theNRC(N.B.-A.),theNSFthroughthePFCatJQI
terize and visualize non-trivial topological states. Our study (I.B.S.) and the ARO with funds from both the Atomtronics
opensthepossibilityofusingmomentumdistributionstoex- MURIandtheDARPAOLEProgram(I.B.S).
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494(1980). (2007).
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(2010). disonWesley,1991),pp.287–292.
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r
[10] Y.-J.Lin,K.Jimenez-Garcia,andI.B.Spielman,Nature471, λ =2d=800nmlasers,givingt =0.085E ,wherethe
L x,y r
83 (2011), ISSN 0028-0836, URL http://dx.doi.org/ singlephotonrecoilenergyisE =h2/2mλ2 =h×7.8kHz
r L
10.1038/nature09887. andmistheatomicmass.
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023615(2011). notthemechanicalmomentumdistribution.
[14] D.R.Hofstadter,Phys.Rev.B14,2239(1976).
5
(a) Schematic map of parameter space
10
(c) Chern
dimer
nisotropy, Continuum Continuum
A
Numerics
only
(b)
1
0.0 0.5 1.0
Flux,
(b) Isotropic, nearly-continuumm: = 1, = 0.05
c = 2
r
) 3.0 cr = 3
nk(x cr = 4
d
e 2.0
z
ali
m
r
o 1.0
N
0.0
- - /2 0 /2
Crystal momentum, k
x
(c) Chern dimer: = 1000, = 0.45
1.2
1.0
)
kx
n( 0.8
d
ze 0.6 cr = 2
mali cr = 3
or 0.4 cr = 4
N
0.2
0.0
- - /2 0 /2
Crystal momentum, k
x
FIG.1:(Color)RelationbetweentheChernnumberc andstructure
r
in TOF data as a function of magnetic flux per unit cell α in units
ofthefluxquantumh/e,andhoppinganisotropyλ. (a)Schematic
mapoftheα-λplaneshowingtheChern-dimerandthecontinuum
regimes.Numericalresultsfortheremainderofparameterspacewill
beshowninFig. 2and3. (b,c)Normalized1Dcrystalmomentum
distributionsn(k ),seeEq.(4),computedatthreedifferentchemical
x
potentialscorrespondingtoc =2,3,4;ineachcase,thenumberof
r
peaksoroscillationsisequaltoc .Thedatain(b)isforanisotropic
r
λ = 1latticeatα = 0.05,andthedatain(c)isforananisotropic
λ=1000latticeatα=0.45.k ismeasuredinunitsof1/d,where
x
disthelatticespacing.
6
(a) Energies for = 2 (b) n(kx) for λ = 1, α = 0.05
tn units of x 024 µtpotential, /x --210...000
Energy i --42 hemical -3.0
C
0.00 0.25 0.50 0.75 1.00 - - /2 0 /2
Crystal momentum, k
x
(c) n(kx) for α = 0.45, cr = 2 (d) n(kx) for α = 0.45, cr = 3
6 3
λ = 1
k)x k)x λ = 1
n( 4 n( 2
d d
e e
aliz aliz λ = 10
m 2 m 1
or λ = 10 or
N N
0 0
- - /2 0 /2 - - /2 0 /2
Crystal momentum, kx Crystal momentum, kx
FIG.2: (Color)1Dcrystalmomentumdistributionsn(k ). (a)En-
x
ergyspectrumofEqs.(1)and(2)forλ=2. (b)Evolutionofn(k )
x
withincreasingchemicalpotentialµatλ = 1andα = 0.05show-
ingc increasingintandemwiththenumberofoscillationsinn(k ).
r x
(c,d)Evolutionofn(k )withincreasinganisotropyλatα = 0.45
x
for(c)c =2and(d)c =3. Thesecurvesdepictthedistributions
r r
approachingthecosinefunctionwithincreasingλ. Theunitfork
x
is1/d.
Crystal momentum, k
y
-π -π/2 0 π/2 π
2
ty E1

of 1

s
t E
ni 2
u 0

n
y i -E
g 2
r -1 δs = 1
e x
n
E -E
1
-2
r=s=2 1 0 4 3
x
FIG. 3: (Color) Energies in the anisotropic limit for p = 1,
q = 5 showing the perfect sinusoidal energies when λ → ∞
(solidblackcurves),andtheavoidedcrossingswhenλ = 10(solid
blue curves). The hybridized states at these crossings correspond
to “Chern-dimers,” and the red-circled example has spatial extent
δs =1.Theunitfork is1/d.
x y
7
)x 2.0
k
(
n

d
e
z
ali
m
1.0
r
o
N
α = 0.2
α = 0.3
0.0
-π -π/2 0 π/2 π
Crystal momentum, k
x
FIG.4: (Color)Themomentumdistributionfunctionn(k )forthe
x
isotropic lattice at temperature T = 0.2t and a trap potential with
mω2d2 = 10−3t. µ atthetrapcenterissetto−0.75t,−0.5tfor
0
α = 0.2,0.3respectively. Thetwinpeaks,indicatedbythearrows,
allow to determine the Chern number near the trap center to be 2.
Theunitfork is1/d.
x