Tomography of band insulators from quench dynamics

3.6 MB·

by  Philipp Hauke

Tomography of band insulators from quench dynamics
Philipp Hauke,1,2,∗ Maciej Lewenstein,3,4 and Andr´e Eckardt5
1Institut fu¨r Quantenoptik und Quanteninformation,
O¨sterreichische Akademie der Wissenschaften, Technikerstr. 21A, A-6020 Innsbruck, Austria
2Institut fu¨r Theoretische Physik, Universita¨t Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria
3ICFO – Institut de Ci`encies Fot`oniques, Parc Mediterrani de la Tecnologia, E-08860 Castelldefels, Spain
4ICREA – Instituci´o Catalana de Recerca i Estudis Avan¸cats, Lluis Companys 23, E-08010 Barcelona, Spain
5Max-Planck-Institut fu¨r Physik komplexer Systeme, No¨thnitzer Str. 38, D-01187 Dresden, Germany
(Dated: July 18, 2014)
Weproposeasimpleschemefortomographyofband-insulatingstatesinone-andtwo-dimensional
optical lattices with two sublattice states. In particular, the scheme maps out the Berry curvature
in the entire Brillouin zone and extracts topological invariants such as the Chern number. The
measurementreliesonobserving—viatime-of-flightimaging—thetimeevolutionofthemomentum
distribution following a sudden quench in the band structure. We consider two examples of exper-
imental relevance: the Harper model with π-flux and the Haldane model on a honeycomb lattice.
4
Moreover, we illustrate the performance of the scheme in the presence of a parabolic trap, noise,
1
and finite measurement resolution.
0
2

l Band insulators are a fascinating form of quantum (1D)andtwo-dimensional(2D)opticallatticesthatisnot
u
matter. Their very existence relies on band structure restricted to a specific system. In particular, the scheme
J
andquantumstatisticsandthegeometricphaseinherent allowsformapping-outtheBerrycurvatureasafunction
7
in their wave function gives rise to intriguing physics: of quasimomentum and for measuring the Chern num-
1
Topological quantum numbers, such as the Chern index ber. Our scheme is based on the momentum-resolved

] or the Z invariant, separate them into classes with fun- monitoring—via TOF imaging—of the dynamics follow-
2
s
damentally different behavior [1, 2], some of which sup- ing an abrupt quench in the band structure. In the fol-
a
g portchiralorhelicaledgemodes[3,4].Onlyrecently,first lowing, we first introduce the basic protocol underlying
- evidence of such topological insulators has been found our method, and then discuss two relevant applications:
t
n in solid-state materials [5–7] and photonic systems [8]. theπ-fluxHarpermodel[37,38],andtheHaldanemodel
a The recent creation of artificial gauge fields in optical [39].
u
lattices [9–16] makes it seem likely that topological in-
q Scheme for the tomography of band insulators.— Con-
sulators will be realized in the near future also in highly
. sider spin-polarized (i.e., non-interacting) fermions in a
t tunablesystemsofultracoldatoms. However,theexperi-
a 2D optical lattice. In each elementary cell (cid:96), the lat-
m mental characterization of apparently structureless band
tice shall have two sublattice states s=A,B, located at
insulators poses a challenge. So far, there are schemes
- r [see Fig. 1(a), left]. The corresponding tight-binding
d that are designed to measure specific topological prop- (cid:96)s
Hamiltonian is characterized by matrix elements h
n erties of the system. For example, Zak’s phase, i.e., the (cid:96)(cid:48)s(cid:48),(cid:96)s
that obey the translational symmetry of the lattice. The
o Berry phase acquired during the adiabatic motion along
c a path through the Brillouin zone (BZ), was measured diagonal terms refer to on-site potentials, h(cid:96)s,(cid:96)s ≡ vs,
[ and the off-diagonal matrix elements describe tunneling
recently from Bloch oscillations [12]. Also, the loca-
between near neighbors. Thanks to the translational
3 tion of Dirac cones was mapped out in a honeycomb
symmetry, the Hamiltonian is diagonal with respect to
v lattice using Landau-Zener transitions [17–19]. Other
0 quasimomentum k. With respect to the basis states
4 proposed schemes are designed to directly measure ei- |ks(cid:105)∝(cid:80) e−ikr(cid:96)s|(cid:96)s(cid:105) it is represented by a k-dependent
ther the Chern number (from density profiles [20], wave- (cid:96)
82 packet dynamics [12, 21–24], time-of-flight (TOF) imag- 2×2 matrix hs(cid:48)s(k)=(cid:80)(cid:96),(cid:96)(cid:48)h(cid:96)(cid:48)s(cid:48),(cid:96)se−i(r(cid:96)(cid:48)s(cid:48)−r(cid:96)s)k, which
we decompose as h (k) ≡ h (k)δ +h(k)σ . Here,
. ing [24–26], or unidirectional TOF imaging with single- s(cid:48)s 0 s(cid:48)s s(cid:48)s
1 σ denotes the vector of Pauli matrices in sublat-
site resolution [27]), or to probe the presence of chiral s(cid:48)s
0 tice space. For every quasimomentum k, the 2D
edgemodes(viatransportmeasurements[28–31],inpar-
4
sublattice space defines a Bloch sphere, with north
1 ticular using quench-based schemes, or Bragg-scattering
and south pole given by |kA(cid:105) and |kB(cid:105), respec-
: [32–35]). A method allowing for a full tomography of a
v bandinsulatorhassofaronlybeenproposedforaspecific tively. The two eigenstates |k±(cid:105) lie at ±hˆ(k) on
i
X experimental realization of a topological insulator based this Bloch sphere, where hˆ(k) ≡ h(k)/|h(k)| ≡
(cid:0) (cid:1)
r on spin-dependent hexagonal lattices [36]. sin(ϑk)cos(ϕk),sin(ϑk)sin(ϕk),cos(ϑk) . Their ener-
a Here, we propose a simple scheme for the complete gies ε (k)=h (k)±|h(k)| define the band structure of
± 0
tomography of band-insulting states in one-dimensional the lattice.
We consider the system to be in a band-insulating
state with complete occupation of the single-particle
∗ [email protected] states of the lower band, |k−(cid:105) = sin(ϑ /2)|kA(cid:105) −
k
2
a) A
  B A
  A
  B A
  dynamics in the momentum distribution
quench

A
  B A
  A
  B A

J J’

v v n(k,t)=f(k)[1−sin(ϑk)cos(ϕk+ω(t−tm))], (1)
A B
A
  B A
  A
  B A

whose oscillatory time dependence directly reveals both
b)
ϕ and sin(ϑ ) = 1−|hˆ (k)|2. The time-dependence of
k k z
quench

|ψB >

|nhˆ(k(,kt))|afrlloomwsthues atompr|aeluixc>to

undsetrauncdt hˆthxe(kp)h,ahˆsye(kof),thaes owseclilllaas-
E z
F tions. It is suffitracnisefenr

t to consider data for k from the first
rotate

BZ; the Wannier envelope f(k) does not spoil the mea-
|ψ >

k A k surement as it just g|ψi B v>

es an irrelevant overall prefactor
for each value of k.|ψ AF >

or a full tomography, it remains
to reveal the sign of hˆ (k). Since the overall sign is
FIG. 1. General procedure. (a) A lattice with two sublat- z
not important, one has to determine those lines where
tice states is quenched abruptly such that both sublattices
are energetically separated by (cid:126)ω = vA−vB and tunneling hˆz(k) changes sign. These lines can be clearly iden-
is suppressed (drawn example: a square lattice with alter- tified by a characteristic cusp-like behavior of |hˆ (k)|,
z
nating tunneling matrix elements J and J(cid:48)). (b) Initially, |hˆ (k)|∝|k−k |, which sharply contrasts with
z sign-change
the system is a band insulator with the lower band occupied
thesmoothvariationofhˆ (k)asitresultsfromtunneling
completely. Ateveryquasimomentumk,thetwo-dimensional z
between near neighbors.
state space is represented by a Bloch sphere, with the state
of the lower band lying at −hˆ(k). With the quench, two flat Moreover, hz(k), including its sign, can also be mea-
bandsare created, ontowhichthestate isprojected. There- sured via band mapping: After abruptly switching on a
sultingdynamicscorrespondstoarotationaroundthez-axis strong potential off-set lifting B with respect to A sites
of the Bloch sphere with the same frequency ω for every k. asbefore,thelatticeisswitchedoffwithoutwaitingtime
Monitoring this dynamics in momentum space allows for re- ataslowratesuchthatquasimomentumismappedonto
constructionoftheinitialpositionontheBlochsphere,giving momentum. Absorption images after TOF reveal then a
a complete tomography of the initial band-insulating state.
momentumdistributionwheretheA(B)population,cor-
responding to the lowest (first excited) band, is mapped
onto the first (second) BZ.
cos(ϑk/2)eiϕk|kB(cid:105) [41]. Here, we assume that the gap Edge states in the Harper model with π-flux.— Once
is much larger than the temperature, allowing to neglect hˆ(k) is reconstructed, one can infer whether the sys-
thermal excitations, which would decrease the observed
tem supports edge modes or not, by invoking the bulk-
contrast. Thisband-insulatingstateisrepresentedbythe
boundary correspondence. According to the procedure
map hˆ(k) from the first BZ onto the Bloch sphere. The derived in Ref. [40], for that purpose one has to identify
topological properties of this map determine the proper- a closed path k(λ) in k-space such that hˆ(k(λ)) lies in
ties of the system. Our aim is to design a feasible mea-
a plane E that contains the origin (i.e., it lies on a great
surementschemethatallowsforareconstructionofhˆ(k). circleoftheBlochsphere). Iftheunitvectorhˆ describes
The momentum (not quasimomentum) distribution of a closed circle around the origin when moving along the
the band insulator, which is obtained from TOF im- path, thesystemdoespossesszero-energyedgemodes; if
ages taken after suddenly switching off the lattice po- not, it does not.
tential, is given by n(k)=f(k)|(cid:104)k−|kA(cid:105)+(cid:104)k−|kB(cid:105)|2 = Motivated by recent experiments [15, 16], let us con-
f(k)[1−sin(ϑk)cos(ϕk)]. Here, f(k) is a broad enve- sider the example of a square lattice with nearest-
lopefunctiongivenbythemomentumdistributionofthe neighbortunnelingandwithafluxofπ (halfafluxquan-
Wannier function; the expression in square brackets pos- tum) per plaquette, which can be generated via laser-
sesses the periodicity of the reciprocal lattice. Unfortu- assisted tunneling or lattice shaking [42]. We consider a
nately, n(k) does not provide sufficient information to gaugewherethetunnelingmatrixelementsiny-direction
reconstruct hˆ(k) or, equivalently, both ϑk and ϕk. In alternate between −J and +J when moving through the
order to obtain the missing information, at the measure- lattice in x-direction, giving two inequivalent sublattices
menttimetm thesystemshallbesubjectedtoanabrupt s=A,B. Additionally, weassumedifferenton-siteener-
quench h(cid:96)(cid:48)s(cid:48),(cid:96)s → h(cid:48)(cid:96)(cid:48)s(cid:48),(cid:96)s, such that a potential off-set gies vA = ∆/2 and vB = −∆/2, and that the tunneling
v(cid:48) − v(cid:48) ≡ (cid:126)ω between A and B sites is created and matrixelementinx-directionalternatesbetween−J(cid:48)and
A B
tunneling suppressed [see Fig. 1(a), right]. In quasimo- −J [Fig. 1(a)]; both can be achieved by a superlattice in
mentum representation, the Hamiltonian is now charac- x-direction. TheextentofthefirstBZisgivenbyπ (2π)
terized by a constant vector h(cid:48)(k) ≈ ((cid:126)ω/2)e gener- in k (k ) direction. The quasimomentum-space Hamil-
z x y
ating a rotation around the z-axis of the Bloch sphere tonian of this model is characterized by
with frequency ω. Starting from the band-insulating
state, this dynamics is, thus, captured simply by replac- h(k)=(−J−J(cid:48)cos(2k ),J(cid:48)sin(2k ),−2Jcos(k )+∆).
x x y
ing ϕ → ϕ +ω(t−t ). This leads to an observable (2)
k k m
3
states: Edge states, expected for the two plots on the
left where J(cid:48) > J, are clearly indicated by data points
describing a circle around the origin. Qualitatively, one
can see this information already in the time evolution of
n(k ,k0;t) in the upper row: If the band insulator sup-
x y
ports edge states, the maximum winds around the time
period.
Our scheme also permits to monitor the topological
transition of the model happening when ∆ exceeds 2J,
whereforJ(cid:48) =J bothDiracconesmerge(see[43],where
edge modes for open boundary conditions are discussed
also).
FIG. 2. (a-d) Time-resolved TOF images for the π-flux
Harper model (∆ = 0 and k0 = π/2). To mimic a realistic Measuring Berry curvature and Chern number in a
y
experiment, we added normal-distributed noise with a stan- Haldane-like system.— Edge currents are topologically
dard deviation of 0.1 of the average signal and assumed a protected only if the associated integer Chern number,
limited resolution of 21 points along k and 41 time points. given by the integral of the Berry curvature over the
x
(e-f) If edge states are supported, the temporal maximum as whole BZ, is finite. In the above example, this is not
a function of quasimomentum winds around the time period thecase, sincetheedgemodesalwaysappearincounter-
once (panels a,b), and hˆky0(kx) = (hˆx,hˆy,0) describes a unit propagating pairs located at the two Dirac cones. We
circlearoundtheorigin(e,f), contrarytowhennoedgestate now turn to a lattice model where a finite Chern num-
issupported(c,dandg,h,respectively). Redline: exactcase. bercanbefoundanddemonstratehowourschemecanbe
Bullets: values extracted from the noisy, resolution-limited
usedtomapouttheBerrycurvatureinquasimomentum.
data of the upper row, using Eq. (1) [k from 0 (light) to π
x The Berry curvature of the lower band is given by
(dark)].
1(cid:16) (cid:17)
Ω (k)= ∂ hˆ ×∂ hˆ ·hˆ , (3)
− 2 kx ky
The two parameters ∆/J and J(cid:48)/J allow us to ex-
plore various situations with qualitatively different band and is readily obtained from hˆ(k). It describes the po-
structures. Most notably, for J = J(cid:48) and ∆ < 2J, larizability[44]andtheanomalousHallconductivity[45]
one finds two Dirac cones lying at kx = ±π/2 and also at lower filling. The Chern number reads
k = arccos(∆/2J). For imbalanced tunneling matrix
y
elements J(cid:48) (cid:54)= J, a band gap opens at the Dirac points 1 (cid:90)
and edge states appear if J(cid:48) > J. We focus here on the w− = 2π d2kΩ−(k). (4)
case∆=0(for∆(cid:54)=0see[43]). Inthiscase,bychoosing
ky0 = π/2 (hˆz = 0), we can confine hˆky0(kx) to a plane It counts how often hˆ(k) wraps around the Bloch sphere
that contains the origin of hˆ’s Bloch sphere, a necessary when k covers the full first BZ. It is proportional to the
condition for observing edge modes at zero energy [40]. Hall conductivity of the completely filled band and indi-
Edge states (in the equivalent system with open bound- cates the presence of robust chiral edge modes [3].
ary conditions in x-direction) do appear if the origin is Let us consider the Haldane-like model sketched in
encircled by hˆ (k ) [40]. Fig. 3(a) with the lower band completely filled. The
k0 x
y atoms live on a honeycomb lattice [17, 46] with real
Inanexperiment,wewishtoreconstructhˆ (k )from
k0 x tunneling matrix elements J between nearest neighbors
y
the dynamics following a sublattice quench in order to (NNs)andcomplextunnelingmatrixelementsJ(cid:48)eiθ with
conclude whether the system possesses edge states or Peierlsphaseθbetweennext-nearestneighbours(NNNs).
not. A typical result will look like the upper row of The model can be realized, e.g., in a shaken optical lat-
Fig. 2, where we plot n(k,t) at ky0 = π/2, ∆ = 0, and tice [47]. For J(cid:48) =0, the vector h(k) lies in the xy-plane
four values of J(cid:48)/J. In order to illustrate the robust- andthebandstructurepossessestwoDiracpointswhere
ness of our method, we have contaminated n(k,t) with h(k) = 0. For finite NNN tunneling, J(cid:48) > 0, hˆ (k) ac-
z
normal-distributed uncorrelated noise, with a standard quires a finite value and a direct band gap opens at the
deviation of ten percent of the average signal. Further- Diraccones(thoughonestillhasanindirectbandtouch-
more, we assumed a mediocre experimental resolution ing). While for J(cid:48) = 0 the unit vector hˆ(k) was con-
of 21 points in the first BZ along k and 41 points in
x fined to the equator, when approaching the Dirac points
time. We can reconstruct hˆ (k ) from n(k ,k0;t) sim-
ky0 x x y it will now visit either the north or the south pole, de-
plybyidentifyingϕ(k)withthepositionofthefirstmax- pending on the sign of hˆ (k). The unit vector hˆ(k) can
z
imum in time and sin(ϑk) = 1−|hˆz(k)|2 with the dif- only wrap around the Bloch sphere, as required for a fi-
ference between maximum and minimum. The resulting niteChernnumber, ifhˆ (k)hasoppositesignatthetwo
z
graphs hˆk0(kx) = (hˆx,hˆy,0), plotted in the lower row Dirac points such that it visits both poles. We find that,
y
of Fig. 2, clearly reveal the presence or absence of edge while for |θ| < θ the system is a trivial band insulator,
c
4
sum over differences is unreliable close to the topological
transition (see [43]). Much better results are obtained
by the gauge-invariant description in terms of effective
field strengths developed by Fukui et al. [48]. In the
Supplemental Material [43], we show how their formula
can be expressed in terms of hˆ. Since this method en-
forcesanintegerresult, itgivestheexactansweralready
for very small numbers of reciprocal lattice points. We
demonstratethisinFig.3(f), whereweuseforn(k)only
4×4 coarse-grained pixels in the first BZ and take only
10 time steps. We again obtain sinϑ from the maximal
k
amplitudeofthedatapointsandϕ fromthepositionof
k
the maximum (for such a low resolution, the sign of h
z
can be obtained from band mapping). Remarkably, even
forthisextremelyresolution-limitedsituation,theChern
number can be reproduced accurately.
A natural question concerns the role of the trapping
FIG. 3. (a) Haldane-like lattice model with real tunnel-
ingparameterJ andcomplextunnelingparameterJ(cid:48)eiθ. (b) potential. Asshownin[43],aharmonictrapmodifiesthe
n(k,t) changes strongly with time (J(cid:48) = 0.3J, θ = 0.4π). measured momentum distribution (1) roughly by a pref-
(c) The winding of the phase ϕk is opposite around the two actor(cid:16)µ0−(cid:15)−(k)(cid:17)2sinc2(cid:16)(µ0−(cid:15)−(k))t(cid:17),withFermienergy
Diracpoints(whitecirclesinb),independentlyofθ. (d,e)In µ0−(cid:15)min 2
the topologically trivial phase (d, J(cid:48) = 0.3J and θ = 0), the µ0 and band minimum (cid:15)min. The first term describes
plottedquantityhˆ hasthesamesignatthetwoDiraccones, the reduced contrast of modes with high energy, since
z
while in the topological phase (d, J(cid:48) =0.3J and θ=0.4π) it theseareonlypopulatedinthecentralregionofthetrap,
hasoppositesign;thissignchangeisclearlyvisibleasakinkin andthesecondtermcapturesdephasingduringthepost-
themeasuredquantity|hˆz|(lowerplotsshowing|hˆz|alongthe quench time-evolution, caused by the spatially varying
dashed lines; lines: ideal case, bullets: data for trapped sys- potential energy. For realistic parameters, both effects
tem with normal-distributed noise of variance 0.05, using re- are small, and the proposed scheme works reliable even
alisticparameters[13,15],J/(cid:126)=2π0.26kHz,ω=2π10kHz,
in the presence of a trap [43] (cf. the lower panels of
trappingfrequency2π50Hz,latticespacing380nm). (f)The
Fig. 3). A newer generation of experiments may enable
coarse-grainedChernnumbercomputedfollowingRef.[48]re-
avoidingaspatiallyvaryingtrappingpotentialaltogether
produces the exact result already for a limited resolution of
[49].
4×4 reciprocal lattice points.
Discussion, conclusion, andoutlook.—Therobustand
simplemethodforthetomographyofbandinsulatorsde-
it becomes a topological Chern insulator, characterized scribed here is not restricted to the two discussed exam-
by a Chern number |w |=1, once |θ|>θ ≈0.18π. ples. It can be applied to any 1D or 2D band-insulator
− c
As exemplified in Fig. 3(b) for an initial state in the with two states per elementary lattice cell—interesting
topological phase (θ = π/2), n(k,t) changes its pattern examples include the Su–Schrieffer–Heeger [50] or Rice–
strongly as a function of time. From this dynamics we Mele model [51], which was recently realized in an op-
can extract the position of hˆ(k) on the Bloch sphere tical lattice [12]. Here, an interesting application would
[Fig. 3(c-e)]. The sign change of hˆ for θ/π > 0.18 be to measure a topological charge pump [52] to extract
z the Chern number quantizing the transport of matter.
[Fig. 3(e)] between the two Dirac cones identifies a finite
Moreover, the method can also be employed to measure
Chern number |w | = 1. Although within our scheme
− systemswithonlypartially-filledlowestband,anditpro-
one can directly measure only the absolute value |hˆ |,
z vides a means to validate Hamiltonians synthesized for
the sign change in hˆz(k) can clearly be identified from the purpose of quantum simulation. As an outlook, it
the pronounced kink where |hˆz| touches zero, either be- will be interesting to generalize the scheme to lattices
tween the two Dirac cones, indicating the topological with more than two sublattice states and to include in-
phase [Fig. 3(e)], or elsewhere, as in the trivial phase ternal atomic states.
[Fig. 3(d)]. The fact that the sign change of h (k) can
z Acknowledgements.— We acknowledge discussions
occur between the Dirac cones, where the band gap is
with Alexander Szameit and Leticia Tarruell. This work
largest, allows to indirectly identify a topological band
wassupportedbytheEUIPSIQS,EUSTREPEQuaM,
structure even if the system is not in a perfect band
ERC AdG OSYRIS, ERC synergy grant UQUAM, Fun-
insulating state. Namely, thermal excitations or small
daci´oCellex,andSFBFoQuS(FWFProjectNo. F4006-
deviations from unit filling are relevant mainly near the
N16).
Dirac cones, and not where the sign change occurs.
In a realistic situation, the resolution of n(k,t) will be
restricted. Approximating the Chern number (4) by a
5
Supplemental material circles hˆ(k) = 0. In the considered model, the second
condition can only be fulfilled if hˆ (k ,k0) = 0, i.e., if
z x y
In this Supplemental Material, (i) we provide addi- ∆ < 2J and k0 = arccos(∆/2J). Otherwise, the system
y
tional information on how the topological phase transi- may still support edge modes, but these then do not lie
tionsintheHarperandHaldanemodelsmaybeobserved, at zero energy. The fulfilment of the first condition, the
(ii) we adapt the method of Ref. [48] to compute Chern closing of the curve, can be identified by a small value of
numbers directly from the Hamiltonian in quasimomen- min|ϕ | [see Figs. 2(e-h) of the main text for exam-
tum representation, using only a small number of grid kx,ky0
ples].
points, and (iii) we present analytical and numerical cal-
Indeed,intheentiretopologicalphaseatJ(cid:48)/J >1,we
culations including a possible harmonic trapping poten-
find min|ϕ | ≈ 0 [shown in Fig. S2(a), where ∆ = 0
tial, showing that its presence is not detrimental to the kx,ky0
and k0 = π/2]. For these parameters, we moreover find
proposed scheme. y
|hˆ |≈0,indicatingthattheedgestateslieatzeroenergy
z
[Fig. S2(b)].
A. Possible observation of topological phase As Fig. S2(c) exemplifies for the point ∆ = 0 and
transitions in the π-flux Harper model J(cid:48)/J = 2, edge modes actually exist for all k0 as long
y
as J(cid:48) > J and ∆ < 2J (see Fig. S1). However, these
Here, we discuss the possibility of observing the topo- modes lie at zero energy only at the Dirac point, which
logical phase transitions supported by the Harper model can be seen by a sharp kink of |hˆz| as a function of ky0
at half a flux quantum [as given by Eq. (2) of the main [Fig. S2(d)]. As illustrated in Fig. S2(e,f) for J(cid:48)/J = 2,
text]. We first describe under which circumstances zero- when increasing ∆ to values larger than 2J, we find
energy edge states are to be expected. Afterwards, we |hˆ | > 0: the Dirac cones have merged and, although
z
turntothequestionhowthesecanbedetectedusingthe edge states still exist, they are no longer at zero energy.
quench-based band tomography.
The spectrum of the Harper model with π flux is ex-
emplifiedinFig.S1,upperrow,forseveralvaluesofJ(cid:48)/J B. Chern numbers in a Haldane-like model,
and ∆/J (we consider an infinite lattice in y-direction, extracted from discrete k-space grids
and L = 80 sites in x-direction). In these spectra, edge
modescanclearlybedistinguishedaslinesthataresepa-
In a realistic TOF image of h(k), the resolution will
ratedfromthebands. ThelowerrowofFig.S1showsthe
berestricted. Ifdataisavailableonlyonadiscretemesh
joint occupation of the two modes that come closest to
{k } of the BZ, one would typically approximate the
m
zero energy (for L = 30 for better visibility). These are
integral in Eq. (4) of the main text by a discrete sum,
thetwoedgemodesattheoppositeendsofthesample,if
and the derivatives occurring in the integrand by a dis-
any edge states exist. In the absence of edge modes, the
crete difference. While this procedure gives reasonable
plottedmodesarethebulkmodesatthebandboundary.
approximationstotheChernnumberdeepinsideagiven
Asthesefiguresshow,atanyJ(cid:48) (cid:54)=J,abandgapopens.
phase(seeFig.S3),itisunreliableclosetothetopological
EdgestatesappearonbothboundarieswhenJ(cid:48) >J,i.e.,
phase transition (see also Ref. [36]). As seen in Fig. S3,
when the edge is connected to the bulk by a weak link.
this shortcoming can be partially remedied by a scaling
Associated to each Dirac cone and boundary, there is
with increasing resolution.
exactlyoneedgemodewithfinitek (notethatonlyhalf
y A more robust method has been developed by Fukui
of the first BZ, k ∈ [0,π], is shown). The edge modes
y and coworkers [48]. In their approach, one defines link
belonging to the two Dirac cones propagate in opposite
variables between points in k space,
directions, leading to a vanishing Chern number. When
∆ > 2J, the Dirac cones merge at k = 0, and as a
y U (k)=(cid:104)n(k)|n(k+µˆ)(cid:105)/N (k), (S1)
µ µ
result the edge modes do not cross the band gap. They
approach the bulk energy bands with increasing ∆.
where |n(k)(cid:105) is the eigenstate of the n’th band at quasi-
We now turn to the question how these topological
momentum k and µˆ the displacement vector by a dis-
transitions can be observed using the proposed scheme.
crete step in the BZ in direction of µ = x,y. N (k) =
µ
We proceed analogous to Fig. 2 of the main text. First,
|(cid:104)n(k)|n(k+µˆ)(cid:105)| is a normalization. Using these link
weuseEq.(2)tocomputen(k,t)onafinitegridintime
variables, one can define the lattice field strengths
andquasimomentumspacewithinthefirstBZ.Tomimic
a realistic experimental situation, we again add noise F˜ (k)=lnU (k)U (k+xˆ)U (k+yˆ)−1U (k)−1 (S2)
xy x y x y
withstandarddeviationof10%oftheaveragevalue. We
=lnU (k)U (k+xˆ)U (k+xˆ+yˆ)U (k+yˆ)
useEq.(1)toextractfromthisdatathecharacteristicsof x y −x −y
hˆ(k) along a path in k-space with fixed ky0. The results (wherethelogarithmisdefinedontheprincipalbranch).
are summarized in Fig. S2. TheformulationinthesecondlinecorrespondstoaWil-
AccordingtoRef.[40],thesystemsupportszero-energy son loop of the U’s around a plaquette in k space, which
edge modes if the path hˆ (k ) is (i) closed and (ii) en- immediately shows its gauge invariance. From Eq. (S2),
k0 x
y
6
FIG.S1. Edgestatesintheπ-fluxHarpermodel,foracylindricgeometry(openboundariesinx-directionwithevennumber
of sites, periodic in y-direction). Upper row: Spectrum. Lower row: Occupation probability as a function of k . Shown is the
y
combinedprobabilityforthetwomodesclosestto(cid:15)=0(forincreasedvisibility,thecolormapisrescaledforeachfigure). Edge
states appear when J(cid:48)/J > 1. At ∆ = 2J, the two points of minimal direct gap (the Dirac cones if J(cid:48) = J) merge, and as a
result possible edge modes do not lie at zero energy for any ∆>2J.
a)

mi1n.0jk p c)

mi1n.0jk p e)

mi1n.0jk p 1.2
0.8 0.8 0.8 1
0.H6» »Lê 0.H6» »Lê 0.H6» »Lê 0.8
0.4 0.4 0.4 0.6
0.2 0.2 0.2 0.4
0.8 1.0 1.2 1.4 1.6 1.8 2.0J'J 0.2 0.4 0.6 0.8 1.0k0yp 1 2 3 4D 0.2
b)

1.0h`z êd)1

.0h`z ê f)

1.0h`z -0.02
0 0.1 0.2 0.3 0.4 0.5
0.5 0.5 0.5
0.8 1.0 1.2 1.4 1.6 1.8 2.0J'J 0.2 0.4 0.6 0.8 1.0k0yp 1 2 3 4D
-0.5 -0.5 -0.5
FIG. S3. Chern number of Haldane model [Fig. 3(a)], com-
-1.0 ê-1.0 ê -1.0
putedfromdifferencesumsoverah(k)withfiniteresolution
(J(cid:48) =0.3J). Dashingbecomeslongerwithincreasingnumber
FIG. S2. Topological phase transitions in the π-flux Harper
model. Upper row (a,c,e): Edge modes at k0 are supported ofgridpointsinthefirstBZ(2M×2M points,withM =1..5).
y
if hˆ (k ) encloses a circle around the origin. Whether the The solid line is the extrapolation for M → ∞ and the dots
ky0 x aretheexactresultfromanintegrationoverthefirstBZ.Suf-
circle is closed can be quantified by min|ϕ |, even for
kx,ky0 ficientlyfarawayfromthephasetransition,forafinitegridof
noisy ‘data’ similar to Fig. 2 of the main text. Bottom row roughly32×32sites,w canbereliablyextracted. However,
(b,d,f): Theedgestatesareonlyatzero-energyifhˆz =0(i.e., the result is not as reli−able as the one using the method of
the parametric curve hˆky0(kx) encloses hˆ =0). The solid line Ref. [48] [see main text, Fig. 3(f)].
represents the ideal prediction and the dots simulate a noisy
experiment (which measures only the absolute value of hˆ ).
z
(a,b) As a function of J(cid:48)/J, at ∆ = 0 and k0 = π/2: zero- Now, we want to show how the lattice field strengths
y
energy edge modes appear for J(cid:48)/J >1. (c,d) As a function (S2) can be computed using only the Hamiltonian vec-
of ky0, at J(cid:48)/J =2 and ∆=0: although edge modes may be tor hˆ(k) as input. To do this, we rewrite Eq. (S2)
supportedatanyk0,theirenergyvanishesonlyattheposition
y in terms of the projector onto the eigenstate |n(k)(cid:105),
of the Dirac cone. (e,f) As a function of ∆, at J(cid:48)/J =2 and
ky0 at the position of one of the Dirac points: when ∆>2J, Pk(n) =|n(k)(cid:105)(cid:104)n(k)|, namely
no zero-energy edge modes are supported.
(cid:20) (cid:21)
Tr(P P P P )
F˜ (k)=ln k k+xˆ k+xˆ+yˆ k+yˆ , (S4)
xy N (k)N (k+xˆ)N (k+yˆ)N (k)
x y x y
the Chern number can be approximated as
(cid:112)
withN (k)= |Tr(P P )|. Thisformulaisgenerally
w˜ = 1 (cid:88)F˜ (k ). (S3) valid foµr non-AbeliankBke+rrµˆy connections. It has the
2πi xy m
m advantage that the projectors can be easily expressed
in terms of the original Hamiltonian. For lattices
An advantage of this discretization is that its result is
with two sublattice states as they are considered here,
always an integer [48]. Hence, it converges much faster
towardstherealChernnumberthanalternativemethods. the projectors are given by Pk(±) = 12(I ± hˆ(k) · σ).
Infact,itcanbeshownthatthenumberofk-spacepoints Therefore, knowledge of hˆ(k) allows to immediately
neededinthefirstBZtoobtaintheexactresultisonthe compute the Chern number. The result is presented in
order of 2|w| [48]. Fig. 3(f) of the main text (where we assumed that the
7
additionalinformationofthesignchangeofhˆ isknown). Eq. (S5) to
z
1 (cid:88)
(cid:104)cˆ† cˆ (cid:105) = (cid:104)(cid:96)(cid:48)s(cid:48)|k(cid:48)−(cid:105)(cid:104)k(cid:48)−|(cid:96)s(cid:105) (S8)
(cid:96)s (cid:96)(cid:48)s(cid:48) tr N
u
k(cid:48)
C. Influence of a harmonic trap θ(µ(r )−(cid:15) (k(cid:48)))θ(µ(r )−(cid:15) (k(cid:48))) ,
(cid:96)s − (cid:96)(cid:48)s(cid:48) −
where θ(µ(r )−(cid:15) (k(cid:48))) is the Heavyside function. Ad-
To date, most optical-lattice experiments are per- (cid:96)s −
ditionally, the local energy is now modified to ν =
formed in the presence of a harmonic confinement (but (cid:96)s
ν +µ(r ).
see also the recent developments as in Ref. [49]). Such s (cid:96)s
For a shallow trap, it is justified to approximate
a harmonic confinement can reduce the contrast of the
µ(r ) ≈ µ(r ). We can then carry out the sums over
scheme, but the essentials of the method still work, as (cid:96)s (cid:96)
the unit cells (cid:96) separately, and we obtain
we discuss now.
n(k,t) =(cid:88)|I(k,k(cid:48))|2 (S9)
tr
k(cid:48)
1. Momentum-distribution in local density approximation
×[1−sin(ϑ )cos((k−k(cid:48))(r −r )+ϕ +ω(t−t ))] .
k(cid:48) A B k(cid:48) m
In general, the interference pattern after a time of Here, we defined r as the position of the basis atom s
s
flight, if we neglect the Wannier envelope, is given by within a unit cell. As we will see, |I(k,k(cid:48))|2 is strongly
peaked at k =k(cid:48), so that we obtain the original expres-
(cid:88)(cid:88)
n(k,t)= e−ik(r(cid:96)s−r(cid:96)(cid:48)s(cid:48))ei(ν(cid:96)s−ν(cid:96)(cid:48)s(cid:48))t(cid:104)cˆ†(cid:96)scˆ(cid:96)(cid:48)s(cid:48)(cid:105) s|Iio(nk)w|2i.thTooutseteratph,ism, wodeifiheadvebtyoaevtaimluea-tdeeIp(ekn,dke(cid:48)n)t, fwahcticohr
(cid:96),(cid:96)(cid:48) s,s(cid:48)
(S5) is given by
In the absence of a trap, one has ν = ν and
tΠhec†co|r0r(cid:105)e,ladteonrsotihnagvea ctoompbleeteelyvafilullaetdedlo(cid:96)swiner tbhaensdst(afoter I(k,k(cid:48))= N1u (cid:88)e−i(k−k(cid:48))r(cid:96)e−i21mωt2rr(cid:96)2θ(µ(r(cid:96))−(cid:15)−(k(cid:48)))
k k− (cid:96)
simplicity, we assume a band insulator at vanishing 1 (cid:90)
temperature). Employing the Fourier transformation ≈ πR2 d2re−i(k−k(cid:48))re−i21mωt2rr2θ(µ(r)−(cid:15)−(k(cid:48))) .
cˆ† = √1 (cid:80) (cid:16)(cid:104)k−|(cid:96)s(cid:105)cˆ† +(cid:104)k+|(cid:96)s(cid:105)cˆ† (cid:17), max (S10)
(cid:96)s Nu k∈1stBZ k− k+
where N is the number of unit cells, one obtains
u In the second line, we assured correct normalisation
(cid:104)cˆ†(cid:96)scˆ(cid:96)(cid:48)s(cid:48)(cid:105)= N1 (cid:88) (cid:104)(cid:96)(cid:48)s(cid:48)|k(cid:48)−(cid:105)(cid:104)k(cid:48)−|(cid:96)s(cid:105) . (S6) bmyaxdke2fi(nµin0g−t(cid:15)h−e(kr)a)d/imusωt2orf=th2e(µe0n−tir(cid:15)emcinl)o/umd,ωt2Rr,m2waxher≡e
u k(cid:48)∈1stBZ (cid:15)min = mink(cid:15)−(k). Using the series expansion of the
second exponential, we can evaluate the integral to
Using
1 (cid:18)ϑ (cid:19) I(k,k(cid:48))= i (cid:88)∞ (i((cid:15)−(k(cid:48))−µ0)t)nC (k,k(cid:48)).
(cid:104)(cid:96)A|k−(cid:105)= √ e−ik·r(cid:96)Asin k , (S7a) (µ −(cid:15) )t n! n
N 2 0 min n=1
u
(cid:18) (cid:19) (S11)
1 ϑ
(cid:104)(cid:96)B|k−(cid:105)=−√Nue−ik·r(cid:96)Beiϕkcos 2k , (S7b) Hnoetrees, tChne(kge,nke(cid:48))ra=lize1dF2h(ynp;e1r,gneo+m1e;t−ricξ)f,uwnchteioren,p1aFnpd2ξd≡e-
1(qa)2µ0−(cid:15)−(k(cid:48)), with q ≡|k−k(cid:48)|, a the lattice spacing,
one arrives at the formula (1) of the main text. 4 Eosc
In the presence of a trap, however, Eq. (1) has to be andEosc = 12mωt2ra2 anintrinsicenergyscaleofthetrap.
modified. If we assume a shallow trap, so that we can Equation (S9) together with (S11) is the central result
employ a local density approximation, then its princi- of this section. We will now proceed to providing some
pal effect will be a local decrease of the Fermi energy furtheranalyticalinsightbyinvokingsomeweakapprox-
as µ(r ) = µ(r ) = µ − 1mω2r2 . Here, m is the imations to I(k,k(cid:48)). Afterwards, we will repeat some of
(cid:96)s (cid:96)s 0 2 tr (cid:96)s
atom mass and ω is the frequency of the trap, which thenumericalexperimentsofthemaintext,demonstrat-
tr
we assumed cylindrically symmetric (r = |r |). Fur- ingthattheproposedmeasurementschemeworksalsoin
(cid:96)s (cid:96)s
ther, µ is the Fermi energy in the centre of the trap. the presence of a trap.
0
Then, the mode with energy (cid:15) (k) will be occupied
−
within a radius of the trap given by µ(r (k))=(cid:15) (k),
max −
r2 (k) = 2(µ −(cid:15) (k))/mω2. In other words, parti- 2. Zero-momentum-transfer and short-time approximations
max 0 − tr
cleswithlowerenergyoccupyalargerextentofthetrap.
Formaximalcontrast,itisthusconvenienttochoosethe Themixtureofdifferentk-modesbyI(k,k(cid:48))istypical
Fermi energy in the trap centre just below the upper for experiments carried out in traps. For shallow traps,
band, µ ≈ min (cid:15) (k). We can incorporate the spa- however, such momentum transfers can typically be ne-
0 k +
tiallyvaryingFermilevelbymodifyingthecorrelatorsin glected. InEq.(S11),thisbecomesmanifestthroughthe
8
small denominator of ξ (the generalised hypergeometric sultsshowninFig.S4. Ascanbeseenintheupperrowof
function decreases fast as a function of its argument). Fig. S4, the signal is deteriorated due to the population
Thus, the integral I(k,k(cid:48)) is strongly peaked at k = k(cid:48) decrease close to the edges of the sample, especially for
and it is justified to set I(k,k(cid:48))=I(k)δ . The exper-
k,k(cid:48)
imental signature will therefore be
n(k,t) =|I(k)|2n(k,t), (S12)
tr
with n(k,t) given by Eq. (1) of the main text, and
(cid:18)µ −(cid:15) (k)(cid:19)2 (cid:18)(µ −(cid:15) (k))t(cid:19)
|I(k)|2 = 0 − sinc2 0 − .
µ −(cid:15) 2
0 min
(S13)
Thus, the momentum distribution without trap obtains
a multiplicative correction, with an intuitive interpreta-
tion: The first term in |I(k)|2 is simply the reduction FIG. S4. As Fig. 2 of the main text, but in the presence
of a harmonic trapping potential. (a-d) Time-resolved TOF
in contrast due to the reduced density at the borders
images for the π-flux Harper model (∆ = 0 and k0 = π/2).
of the trap (high-energy modes are only occupied in the y
We assumed the experimental parameters given at the end
centre of the trap where the Fermi energy is sufficiently
of Sec. C2, and took a realistic lattice size of 40×40 oc-
high). The second term describes a dephasing of differ- cupied sites. Again, we added normal-distributed noise with
entregionsofthetrap,becauseduringthetimeevolution a standard deviation of 0.1 of the average ideal signal. In a
afterthequenchthephaseateachsiteoscillateswiththe trap, the signal quality decreases due to the decrease of the
site offset ν plus a locally varying contribution from the numberofparticlestowardstheedgeofthesample. However,
s
local chemical potential. despite the smaller signal, the presence (closed circle in e,f)
Another useful limit of I(k,k(cid:48)) that one can express or absence (open circle in g,h) of edge states is still visible.
in a simple analytical formula is the limit of short times,
t(µ −(cid:15) (k))(cid:28)1. KeepingtermsuptoO(t2),weobtain
0 −
|I(k,k(cid:48))|2 =(cid:18) µ0−(cid:15)k(cid:48) (cid:19)2(cid:26)1J (2(cid:112)ξ)2 (S14)
µ −(cid:15) ξ 1 the modes around k = π/2, which have higher energy.
0 min x
−(cid:18)(µ0−ξ(cid:15)k(cid:48))t(cid:19)2(cid:104)J2(2(cid:112)ξ)2−(cid:112)ξJ3(2(cid:112)ξ) Hofohwˆ(ekve)rc,aansstthilelbboetteoxmtrarcotweddreemliaobnlsyt,raattesl,eatshtefwarinadwinayg
from the topological transition [53]. We now turn to a
(cid:112) (cid:112) (cid:105)(cid:111)
+(2−ξ)J (2 ξ)J (2 ξ) , more detailed analysis of the topological transition.
1 3
where J (X) denote Bessel functions. Again, the same In Fig. S5, we repeat the analysis of Fig. S2 for the
γ
twoeffectsasabovearefound,theoveralllossofcontrast topologicaltransitionasafunctionofJ(cid:48)/J [54]. Theblue
due to the overall prefactor in front of the curly brackets bulletsaredatacomputedforthetrapwiththeabovepa-
and a gradual decrease in contrast due to the dephasing rameters, under the addition of normal-distributed noise
between different trap regions. with a standard deviation of 0.1 of the average ideal sig-
In the numerics presented in the following section, we nal. The extracted curve lies close to the dashed line,
userealisticparameters,appearinginstate-of-the-artex- whichrepresentsthecleandatainthepresenceofatrap,
periments [13, 15], J/(cid:126)=2π·0.26kHz, ω =2π·10kHz, i.e., without the added noise. The trap also slightly
ω = 2π·50Hz, a = 380nm. For these values, we have shifts the topological transition from the trap-free case
tr
E ≈2π·1.6Hz(cid:28)(µ −(cid:15) (k))=O(J)and,duringone (solid line), but the general behaviour is clearly pre-
osc 0 −
period, t(µ −(cid:15) (k)) ≤ O(2πJ) (cid:28) 1. Therefore, either served. The overall behavior is similar to what happens,
0 − ω
approximation, Eq. (S13) or Eq. (S14), is well justified e.g., to the Mott-insulator–Superfluid transition, which
in a typical experimental setting with such parameters, also becomes less sharp in the presence of a trap. These
especially the short-time approximation. results demonstrate that, even though a trap diminishes
the quality of the results, the proposed scheme allows
a reliable tomography of the Hamiltonian hˆ(k). Let us
3. The π-flux Harper model in a trap also stress that the problems associated to a trap can be
avoided altogether in a new generation of optical-lattice
Using the above parameter values, we recalculated experiments, which work in box potentials without har-
Fig.2ofthemaintextforthecaseofatrap,withthere- monic confinement Ref. [49].
9
min(cid:72)(cid:200)(cid:106)(cid:200)(cid:76)(cid:144)Π
k
1.0
0.8
0.6
0.4
0.2
J'(cid:144)J
0.8 1.0 1.2 1.4 1.6 1.8 2.0
FIG. S5. Topological phase transitions in the π-flux Harper
model in the presence of a trap. Edge modes at k0 = π/2
y
are supported if hˆ (k ) encloses a circle around the origin,
ky0 x
quantified by a vanishing min|ϕ |. The dots simulate a
kx,ky0
noisyexperimentinthepresenceofatrap,with40×40lattice
sites,41timesteps,andanaddednoiseof0.1ofthemeanof
theidealcase. Theappearanceofzero-energyedgemodesfor
J(cid:48)/J > 1 is less clear than in the trap-free case (Fig. S2a),
butcanstillbedistinguished. Thedashedlinerepresentsthe
idealised prediction for an experiment in a trap but without
any noise. It is slightly shifted from the ideal, trap-free case
(solid line).
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[41] Our measurement scheme itself is robust also when ap- [53] To obtain the form of hˆ in these plots, we simply nor-
proachingthelimitofanenhancedtranslationalsymme- malise the oscillation of n(k,t) by its mean value over
tr
trywherebothsublatticestatesbecomeequivalent.How- one time period, and then proceed similar as before,
ever, in this trivial limit the band gap closes, impeding i.e., we identify 2sin(ϑ ) with the amplitude of the nor-
k
the preparation of the band insulator. malisedtime-periodicoscillationandϕ withtheshiftof
k
[42] N. Goldman, G. Juzeliunas, P. Ohberg, and I. B. Spiel- the first maximum.
man, arXiv:1308.6533 [cond-mat.quant-gas] (2013). [54] To obtain the form of hˆ for these data, we fit a cosine
[43] See the Supplemental Material for useful details for ob- plusconstantton(k,t) ,andnormalisetheresultsagain
tr
serving topological phase transitions in the Harper and by the mean value over one period.
Haldane models, and for a discussion of the influence of