Exploring competing density order in the ionic Hubbard modelwith ultracold fermions

Michael Messer,1 Remi Desbuquois,1 Thomas Uehlinger,1

Gregor Jotzu,1 Sebastian Huber,2 Daniel Greif,1 and Tilman Esslinger1

1Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland2Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

We realize and study the ionic Hubbard model using an interacting two-component gas of fermionicatoms loaded into an optical lattice. The bipartite lattice has honeycomb geometry with a stag-gered energy-offset that explicitly breaks the inversion symmetry. Distinct density-ordered phasesare identified using noise correlation measurements of the atomic momentum distribution. For weakinteractions the geometry induces a charge density wave. For strong repulsive interactions a Mottinsulator forms and we observe a restoration of the inversion symmetry. The local density distri-butions in different configurations are characterized by measuring the number of doubly occupiedlattice sites as a function of interaction and energy-offset. We further probe the excitations of thesystem using direction dependent modulation spectroscopy and discover a complex spectrum, whichwe compare with a theoretical model.

PACS numbers: 05.30.Fk, 03.75.Ss, 67.85.Lm, 71.10.Fd, 71.30.+h, 73.22.Pr

Changes in the fundamental properties of interactingmany-body systems are often determined by the com-petition between different energy scales, which may in-duce phase transitions. A particularly intriguing situa-tion arises when the geometry of a system sets an en-ergy scale that competes with the scale given by the in-teraction of its constituents. The importance of geom-etry becomes already apparent in reduced dimensions,where the evolution from one phase to another changesits nature [1]. Exceedingly complex situations arise inthe interplay between magnetic order and the underlyinggeometry, which leads to geometric frustration and ex-otic phases in various models [2]. A tractable approachto generic questions is provided by the ionic Hubbardmodel, which captures key aspects of the physics of acompeting geometry and interactions in the charge sec-tor. The Hamiltonian has a staggered energy-offset on abipartite lattice, such that geometry supports a band in-sulating charge density wave (CDW). Conversely, strongrepulsive on-site interactions favour a Mott insulatingstate (MI) at half-filling, thereby restoring the brokeninversion symmetry. The model was introduced in thecontext of charge-transfer organic salts [3, 4] and hasbeen proposed to explain strong electron correlations inferroelectric perovskite materials [5]. Ultracold atomsin optical lattices are an excellent platform for studyingcompeting energy scales, as they allow for tuning variousparameters and the geometry of the Hamiltonian [6–17].Here we explore the ionic Hubbard Model using ultra-cold fermions loaded into a tunable optical honeycombpotential.

The ionic Hubbard model has been studied theoreti-cally in 1D chains [18–23] and on the 2D square lattice[24–27]. More recently, these studies have been extendedto a honeycomb lattice, motivated by possible connec-tions to superconductivity in layered nitrides [28] and

strongly correlated topological phases [29]. We considerthe ionic Hubbard model on a honeycomb lattice:

H = −t∑〈ij〉,σ

c†iσ cjσ + U∑i

ni↑ni↓ + ∆∑i∈A,σ

niσ, (1)

where c†iσ and ciσ are the creation and annihilation op-erators of one fermion with spin σ = ↑, ↓ on site i andniσ = c†iσ ciσ. The system is characterized by three en-ergies: the kinetic energy denoted by the tunnelling am-plitude t and summed over nearest neighbours 〈ij〉, theon-site interaction U and the staggered energy-offset be-tween sites of the A and B sub-lattice ∆, with ∆ > 0.All parameters of the Hamiltonian are computed usingWannier functions [30].

The interplay between the interaction energy U , theenergy-offset ∆ and tunnelling t leads to quantum phaseswhich differ by their density ordering. The two limitingcases can be qualitatively understood in the atomic limitat half-filling. For U � ∆ the system is described by aMI state. For a large energy-offset ∆ � U , we expect aband insulator with staggered density and two fermionson lattice site B [24]. The resulting CDW pattern re-flects the broken inversion symmetry of the underlyinggeometry. We can characterize the transition by an or-der parameter NA − NB, which is zero in the MI stateor when ∆ = 0, with NA(B) the total number of atomson sub-lattice A(B). Fig. 1a provides a schematic viewof the different scenarios.

In order to realize the ionic Hubbard model we create aquantum degenerate cloud of 40 K as described in previ-ous work [30] and detailed in [31]. We prepare a balancedspin mixture, using two Zeeman states of the F = 9/2hyperfine manifold, with total atom numbers between1.5×105 and 2.0×105, with 10% systematic uncertainty.The fermions with temperatures between 16(2)% and13(2)% of the Fermi-temperature, are then loaded into

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U=25.3(5)t, Δ=20.3(5)t

A B

FIG. 1. Noise correlations (a) Schematic view of the ionicHubbard model on a honeycomb lattice at half-filling. Circlesdenote lattice sites A and B, where larger circles indicatelower potential energy. The phase diagram exhibits two lim-iting cases: For ∆ � U, t a CDW ordered state is expectedwith two fermions of opposite spin (red, blue) on lattice sitesB, and empty sites A. In the other limit (U � ∆, t) a MI withone fermion on each lattice site should appear. (b) Measurednoise correlation pictures obtained from absorption images ofthe atomic momentum distribution. Comparing panel 1 withpanel 2, additional correlations appear due to broken inver-sion symmetry in the CDW ordered phase. When introducingstrong interactions, these correlations are not observed any-more (panel 3), indicating the restoration of inversion sym-metry. Below each panel horizontal and diagonal cuts of thenoise correlation image are shown. For the three differentratios of ∆ and U , between 165 and 201 measurements weretaken each. We show the average of C(dx, dy) and C(dx,−dy),which reflects the symmetry of the system.

a three-dimensional optical lattice within 200 ms. Us-ing interfering laser beams at a wavelength λ = 1064 nmwe create a honeycomb potential in the xy-plane, whichis replicated along the z-axis [14, 30]. All tunnellingbonds are set to t/h = 174(12) Hz. The tunable lat-tice allows us to independently adjust the energy-offset∆ = [0.00(4), 41(1)]t between the A and B sub-lattice[31]. Depending on the desired interaction strength we ei-ther use the Feshbach resonance of the mF = −9/2,−7/2mixture or the mF = −9/2,−5/2 mixture.

We probe the spatial periodicity of the density distri-bution in the interacting many-body state by measuringcorrelations in the momentum distribution obtained aftertime-of-flight expansion and absorption imaging [33–38].After preparing the system in a shallow honeycomb lat-tice with a given U and ∆, we rapidly convert the lattice

geometry to a deep simple cubic lattice. This ensuresthat we probe correlations of the underlying density orderrather than a specific lattice structure. The atoms are re-leased from the lattice and left to expand ballistically for10 ms. We then measure the density distribution, whichis proportional to the momentum distribution of the ini-tial state n(q). From this, we compute the correlator ofthe fluctuations of the momentum distribution [32],

C(d) =

∫〈n(q0 − d/2) · n(q0 + d/2)〉dq0∫〈n(q0 − d/2)〉〈n(q0 + d/2)〉dq0

− 1, (2)

where the 〈〉 brackets denote the statistical averaging overabsorption images taken under the same experimentalconditions.

Owing to the fermionic nature of the particles, thisquantity exhibits minima when d = m2π/λ, with m avector of integers [31]. This is illustrated by the anti-correlations of a repulsively interacting, metallic statewith U = 4.85(9)t and ∆ = 0.00(4)t, shown in Fig. 1b,left panel. There, the spatial periodicity of the atomicdensity follows the structure of the lattice potential, andminima in the correlator are observed for m = (0,±2)and m = (±2, 0). For ∆ = 39.8(9)t, additional minimaare observed at m = (±1,±1), see Fig. 1b, central panel.For a simple cubic lattice potential of periodicity λ/2, theamplitude of these minima is given by [31]

C

(±2π

λ,±2π

λ

)∝ (NA −NB)2

(NA +NB)2. (3)

Thus, the observation of additional peaks confirms thepresence of CDW-ordering with NA 6= NB. Finally, for∆ = 20.3(5)t and U = 25.3(5)t, these additional minimaare not observed any more (see Fig. 1b, right panel),signalling the restoration of the externally broken inver-sion symmetry by repulsive on-site interactions. In thiscase the interactions suppress the CDW-order, despitethe presence of a large ∆.

Based on these measurements we expect the local dis-tribution of atoms on each lattice site to depend onthe exact values of U and ∆. We measure the fractionof atoms on doubly occupied sites D using interaction-dependent rf-spectroscopy [9]. The number of doubly oc-cupied sites compared to the number of singly occupiedsites is directly related to the nature of the insulatingstates [39, 40]: the MI state is signaled by a suppresseddouble occupancy while the CDW order is formed byatoms on alternating doubly occupied sites.

In the experiment we set an energy-offset ∆ and mea-sure D for different attractive and repulsive interactionsU = [−24.6(13),+29.1(7)]t. Fig. 2a shows D as a func-tion of U at constant ∆ = 16.3(4)t. For strong attrac-tive interactions we observe a large fraction of doublyoccupied sites, which continuously decreases as U is in-creased. When tuning from attractive to weak repulsiveinteractions (∆ � U), we still observe a large D as ex-pected for the CDW. For strong repulsive interactions

3

a b

−20 −10 0 10 20 30U/t

0.0

0.2

0.4

0.6

0.8Do

uble

occu

panc

yD

Δ/t = 16.3(4)

−60 −40 −20 0 20 40(U−Δ)/t

Δ/t41(1)32.3(8)24.4(6)16.3(4)8.5(2)0.00(4)

FIG. 2. Double occupancy measurement (a) The measureddouble occupancy D as a function of the on-site interactionU for a fixed energy-offset ∆ = 16.3(4)t. (b) For differentvalues of ∆ (different colors) we obtain the double occupancyfor a range of interactions U = [−24.6(13), 29.1(7)]t. Hol-low (full) circles represent attractive (repulsive) interactions.Vertical error bars show the standard deviation of 5 measure-ments and horizontal error bars the uncertainty on our latticeparameters.

(U � ∆) the measured double occupancy vanishes, con-firming the restoration of inversion symmetry and thesuppression of the CDW ordering. Fig. 2b shows D asa function of the energy scale U − ∆, which is the en-ergy difference of a doubly occupied site neighbouring anempty site compared to two singly occupied sites in theatomic limit. For the largest negative value of U − ∆we observe the highest D for all ∆. For positive valuesof U − ∆ the double occupancy continuously decreasesand vanishes for the largest positive U − ∆, consistentwith a MI state. In contrast, for the intermediate regimethe measured D depends on the individual values of Uand ∆, as now the finite temperature, chemical poten-tial and the tunnelling itself play an important role anda detailed analysis would be required for a quantitativeunderstanding.

A characteristic feature of the MI and band insulatingCDW state is a gapped excitation spectrum, which weprobe using amplitude-modulation spectroscopy [9, 41].We sinusoidally modulate the intensity of the latticebeam in y-direction by ±10% for 40 ms. Since the hon-eycomb lattice is created from several beams interferingin the xy-plane [14], this leads to a modulation in tunnelcoupling ty of 20% and tx of 8%, as well as a modulationof U by 4% and ∆ by up to 6%. The interlayer tun-nelling tz is not affected meaning that excitations onlyoccur in the honeycomb plane. We set U = 24.4(5)t andmeasure D after the modulation for frequencies up toν = 11.6 kHz (≈ 67 t). All measurements are performedin the linear-response regime [42].

Fig. 3a shows the measured spectra for different valuesof ∆. The MI state exhibits a gapped excitation spec-trum, which is directly related to a particle-hole excita-tion with a gap of size U [9, 30, 42]. In the limit of ∆ = 0we detect this gap as a peak in the excitation spectrum

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(U)

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0.67(2)

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Peak

posi

tion

(U)

FIG. 3. Modulation spectroscopy measurement (a) Excita-tion spectra observed by measuring the double occupancy Dfrom amplitude modulation spectroscopy of the lattice beamin y-direction for different energy-offsets ∆ at repulsive on-site interaction U = 24.4(5)t. Solid lines are multiple Gaus-sian fits to the modulation spectra. (b) Schematics for therelevant energy scales |U − ∆| and U + ∆ as a response tothe lattice modulation. (c) Modulation spectroscopy of thelattice beam in z-direction. The measured excitation frequen-cies are shown as a function of ∆ and compared to the valueof U = 24.4(5)t (horizontal line). The inset shows the spa-tially dependent excitation spectrum. (d) Comparison of themeasured excitation resonances (points) with the values of|U − ∆|, U + ∆ (lines). The area of the marker indicatesthe strength of the response (peak height) to the lattice mod-ulation. Full (empty) circles represent a positive (negative)response in double occupancy. Error bars as in Fig. 2, verticalerror bars in (c),(d) show the fit error for the peak position.

at ν = U/h. With increasing ∆ the single excitationpeak splits into two peaks corresponding to different ex-citation energies [43]. The nature of the excitations canbe understood as follows: The transfer of one particlecosts approximately an energy of U −∆ if a double oc-cupancy is created on a B site and U + ∆ if it is createdon an A site (see Fig. 3b). The excitation of additional

4

half filling quarter filling

0.0 0.5 1.0 1.5Δ/U

0.5

1.0

1.5

2.0

Ener

gyhº

(U)

FIG. 4. Theoretical result for the kinetic energy responsefunction χ(ν) of the double occupancy on a modulated foursite model as a function of ∆ at constant U = 25t. Circu-lar (diamond) data points represent the response for the halffilled (quarter filled) case. The area of the marker shows therelative size of the calculated response, whereas full (empty)data points have a positive (negative) response signal.

double occupancies shows that atoms were initially pop-ulating both sub-lattices, as expected in the MI regime.For small ∆/U the system shows a clearly identifiablecharge-gap, which vanishes if U ∼ ∆. For large ∆ thecharge gap reappears, and a minimum in the spectra re-veals the breaking of double occupancies as a response toamplitude modulation. This is in agreement with the ex-pected band insulating CDW, where double occupanciesare on the B sub-lattice and A sites are empty.

The situation changes for amplitude modulation of thez lattice beam intensity by ±10%. In this case excitationsare created along links perpendicular to the honeycombplane. Since the honeycomb lattice is replicated along thez-axis, we observe a single peak at ν = U/h, independentof the energy-offset ∆ (see Fig. 3c). The inset of Fig.3c shows the direction dependent modulation spectrumfor ∆ = 8.5(2)t, which allows us to independently deter-mine the energy scales of the system in different spatialdirections.

We extract the excitation energies by fitting multipleGaussian curves to our experimental data and compareour results with the values of |U −∆|, U + ∆ and U inFig. 3d. We observe a vanishing peak at U + ∆ for thelargest ∆. This is expected as there are fewer and feweratoms on A sub-lattice in the system for an increasingenergy-offset. Our measurements are in good agreementwith a picture based on nearest-neighbour dynamics.

However, we observe additional peaks at ν ≈ U/hif U ∼ ∆, which can not be understood in a two-sitemodel. To rule out any higher-order contribution, weverified that the response signal has a quadratic de-pendence on the modulation parameters, as expectedfor linear response [42]. Furthermore, we strongly sup-pressed the tunnelling perpendicular to the honeycombplanes tz (corresponding to a 2D ionic Hubbard model).

Even for tz/h = 2 Hz we observe the creation of double-occupancies at ν ≈ U/h [31].

To interpret the nature of the response at hν ≈ U wecalculate the kinetic energy response function

χ(ν) =∑m

〈m|δD|m〉|〈m|K|0〉|2δ(hν − εm0), (4)

where the sum runs over all excited states m > 0,δD = D− 〈0|D|0〉 is the induced change in double occu-pancy, K the kinetic energy operator and εm0 denotes theexcitation energy measured above the ground state |0〉.We evaluate χ(ν) in exact diagonalization of a cluster offour sites for varying filling fractions.

The result shown in Fig. 4 for U/t = 25 clearly in-dicates that the peak at hν ≈ U around U = ∆ origi-nates from regions of the lattice where the filling deviatesfrom one particle per site. In particular, for a configu-ration with two particles on four sites, the ground stateat U = ∆ is a configuration with negligible double occu-pancy and only the lower sub-lattice sites are filled. Thelattice modulation at hν ≈ U then moves one particle toan energetically costly site. For U = ∆, this configura-tion is resonantly coupled to a state where both particlesare on the same, low-energy site. Hence, this processleads to an increase in the measured double occupancy.The analysis of such a four-site cluster explains the pos-itive signal at energy U in the intermediate (U ≈ ∆)regime.

In conclusion, we have realized and studied the ionicHubbard model with ultracold fermions in an optical hon-eycomb lattice. Our observations show that increasinginteractions suppress the CDW order and restore inver-sion symmetry. Additionally, we probed correlations be-yond nearest-neighbour, which had not been accessibleso far [44]. Future work can address open questions con-cerning the nature of the intermediate regime betweenthe two insulating phases, which is theoretically debatedand should depend on the dimensionality of the system[25, 45]. Furthermore, we can extend our studies of theionic Hubbard model to include topological phases byintroducing complex next nearest neighbour tunnelling[29, 46, 47].

We thank Ulf Bissbort, Frederik Gorg, Diana Prychy-nenko, Vijay Shenoy, Leticia Tarruell, Evert vanNieuwenburg and Lei Wang for insightful discussions. Weacknowledge support from the SNF, NCCR-QSIT, QUIC(EU H2020-FETPROACT-2014) and SQMS (EU ERCadvanced grant). R.D. acknowledges support from ETHZurich Postodoctoral Program and Marie Curie Actionsfor People COFUND program.

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6

SUPPLEMENTAL MATERIAL

Preparation and optical honeycomb lattice

To simulate the ionic Hubbard model we create a quan-tum degenerate cloud of 40 K by using a balanced spinmixture of the |F,mF 〉 = |9/2,−9/2〉 and |9/2,−7/2〉Zeeman states, which is evaporatively cooled in an op-tical dipole trap. Depending on the desired interac-tion strength we either use the Feshbach resonance ofthe mF = −9/2,−7/2 mixture to access an interactionrange of U = [−24.6(13), 4.91(9)]t or a mF = −9/2,−5/2mixture to reach strongly repulsive interaction strengthsU = [11.7(2), 29.1(7)]t. The fermions, with temperaturesbetween 16(2)% and 13(2)% of the Fermi-temperature,are then loaded into a three-dimensional optical latticewithin 200 ms. The honeycomb lattice with staggeredenergy-offset ∆ is created by interfering laser beams ata wavelength λ = 1064 nm that give rise to the followingpotential [14]:

V (x, y, z) = −VX cos2(kx+ θ/2)− VX cos2(kx)

−VY cos2(ky)− VZ cos2(kz)

−2α√VXVY cos(kx) cos(ky) cosϕ, (5)

where VX,X,Y,Z are the single beam lattice depths in

each spatial direction, k = 2π/λ, and the visibilityα = 0.90(5). We interferometrically stabilize the phaseϕ = 0.00(3)π and control the energy-offset ∆ betweenthe A and B sub-lattice by varying the value of θ aroundπ. This is achieved by changing the frequency detuningbetween the X and the X (which has the same frequencyas Y ) beam. In the case of ∆ = 0 the lattice depths areset to VX,X,Y,Z = [14.0(4), 0.79(2), 6.45(20), 7.0(2)]ERto prepare isotropic tunnelling bonds in the honeycomblattice (t/h = 174(12) Hz), where ER is the recoil en-ergy. When breaking inversion symmetry (∆ 6= 0) weadjust the final lattice depths in order to keep t on alllattice bonds constant. Owing to the harmonic con-finement of the lattice beams and the remaining dipoletrap our system has harmonic trapping frequencies ofνx,y,z = [86(2), 122(1), 57(1)] Hz. The lattice depths areindependently calibrated using Raman–Nath diffractionon a 87Rb Bose–Einstein condensate. For the measure-ment of the double occupancy D, both the independentlydetermined offset in D of 2.2(3)% due to an imperfectinitial spin mixture as well as the calibrated detection ef-ficiency of 89(2)% for double occupancies are taken intoaccount [44].

Noise correlations

The anti-correlations in the fluctuations of the momen-tum distribution stem from the fermionic nature of theparticles. To illustrate this, let us consider the simple

case of two identical fermions, occupying the lowest bandof a one-dimensional optical lattice of periodicity λ/2.Their wavefunctions can be expressed as Bloch waves,with a different quasi-momentum q for each particle, suchthat they obey the Pauli principle. If the momentum ofthe first particle is measured to be q1, then the secondcannot be detected in any of the momenta q1 − n~4π/λwith n an integer.

In the experiment, the momentum distribution n(q, t =0) is accessed by suddenly releasing the atomic cloud fromits confinement. After an expansion time sufficient to ne-glect the initial size of the cloud, the momentum distri-bution has been converted to the spatial atomic densityn(x, t) which is then directly measured by taking an ab-sorption image. These two quantities are related by

n(x, t) = n(q =mx

~t, t = 0) (6)

In the following we use n to designate the atomic densityin momentum space. In general, we are interested in theprobability to detect two particles at momenta q1 and q2simultaneously, which is given by

P (q1, q2) = 〈n(q1)n(q2)〉 − 〈n(q1)〉〈n(q2)〉

=∑α,β

〈Ψ†α(q1)Ψα(q1)Ψ†β(q2)Ψβ(q2)〉

− 〈Ψ†α(q1)Ψα(q1)〉〈Ψ†β(q2)Ψβ(q2)〉, (7)

where Ψ†α(q) creates and Ψα(q) annihilates a particlewith internal state α at momentum q. Representing theoperators by their underlying Bloch-wave structure wehave

P (q1, q2) = |W (q1)|2|W (q2)|2∑α,β

∑k,l,m,n

ei λ/2(q1(k−m)+q2(l−n))

×(−〈b†α,k b

†β,l bα,m bβ,n〉 − 〈b

†α,k bα,m〉〈b

†β,l bβ,n〉

)+ δ(q1 − q2)

∑α

〈Ψ†α(q1)Ψα(q2)〉, (8)

where b†α,k (bα,k) creates (annihilates) a particle at thesite k with an internal state α. Here, W (q) is the slowlyvarying envelope of the Bloch wave. The last term con-cerns the correlation of an atom with itself, which we arenot considering here. To calculate the four-operator ex-pectation value, we assume that the atomic distributionis well described by Fock states (〈b†α,k bβ,l〉 = δk,lδα,βnk),where nk is the number of particles on site k. Thus, wehave

P (q1, q2) = −|W (q1)|2|W (q2)|2

×∑α

∑k,l

nα,knα,lei λ/2(k−l)(q1−q2) (9)

P (q0, d) = −|W (q0 + d/2)|2|W (q0 − d/2)|2

×∑α

∑k,l

nα,knα,lei λ/2(k−l)d, (10)

7

0 1 2 3Frequency hº/U

0.1

0.2

Doub

le oc

cupa

ncy

FIG. 5. Modulation spectroscopy measurement for the two-dimensional ionic honeycomb potential. Excitation spectraobserved by measuring the double occupancy D after si-nusoidal modulation of the lattice depth Vy for an energy-offset ∆ = 24.3(6)t at constant repulsive on-site interac-tion U = 24.1(4)t, thereby realizing the intermediate regime|U − ∆| ∼ t. Although the tunnelling perpendicular to thehoneycomb planes is tuned below 2 Hz we still observe theresponse at energy ≈ U . Error bars show the standard devi-ation of at least 3 measurements.

where we have introduced the center of mass q0 = (q1 +q2)/2 and the relative position d = q1 − q2. The slowlyvarying dependence in q0 can be rewritten in terms of themomentum distribution

〈n(q1)〉〈n(q2)〉 = |W (q0 + d/2)|2|W (q0 − d/2)|2N2 (11)

with N the total atom number. Thus, the correlations inmomentum are fully characterized by

C(d) =

∫dq0P (q0, d)∫

dq0〈n(q0 + d/2)〉〈n(q0 − d/2)〉

= −∑α

∑k,l

nα,knα,lN2

ei λ/2(k−l)d (12)

We now show that this quantity is not only sensitiveto the periodicity imposed by the lattice, but can alsoreveal underlying order in the density distribution. Con-sider the case where, for each internal state, the densitytakes the value nA/M on even-numbered sites and nB/Mon odd numbered sites, with M the number of internalstates. The correlation signal then takes on the form

C(d) = −n2A + n2B + 2nA nB cos(λ/2 d)

M N2

∑k,l

ei λ(k−l)d

(13)The sum is equal to N2 if d = m 2π/λ with m an integerand zero otherwise, and the correlation signal is thensimply

C(m 2π/λ) = − (nA + (−1)m nB)2

M(14)

Thus, anti-correlations always appear at momenta 2m×2π/λ, corresponding to the reciprocal lattice vector,

while anti-correlations at momenta (2m+1)×2π/λ signala staggering of the atomic density between the even- andodd-numbered sites. While this derivation was carriedout for a one-dimensional lattice, it can readily be gener-alized to higher dimensions, provided the full informationon the momentum density can be accessed.

In the experiment, we measure the momentum distri-bution of the absorption images following a three stepdetection protocol. After preparing the system in a shal-low honeycomb lattice with a given U and ∆, the latticedepth is suddenly increased in 1 ms, which prevents anyfurther evolution of the atomic density distribution. Sub-sequently, the lattice geometry is converted to a simplecubic lattice within 1 ms to ensure our observable probescorrelations of the underlying density order rather thana specific lattice structure. Finally, the strength of theinteractions is reduced within 50 ms, the atoms are re-leased from the lattice and left to expand ballistically for10 ms after which we measure the density distribution byabsorption imaging.

From this measurement, we compute the quantity C,which is shown in Fig. 1. As our imaging technique in-tegrates the density along the line of sight, we do nothave access to the full information, but rather to thecolumn density. Thus, the derivation presented aboveshould be generalized to two dimensions, while the oc-cupancy along the third direction can be treated as aninternal degree of freedom. Accordingly, the depth of theanti-correlation minima for a two-components Fermi gaswill be divided by 2Nz, where Nz is the typical num-ber of sites populated along the integrated direction. Toachieve optimal signal-to-noise, we only consider atomicdensities above 20 % of the maximum density. Further-more, we remove short-range correlations induced by thereadout noise of the CCD chip of the camera by convo-luting the density distribution with a Gaussian of width∆q = k/25. Finally, we take advantage of the reflectionsymmetry of the momentum distribution, and averagetogether C(dx, dy) with C(dx,−dy). Note that, by defi-nition, C(dx, dy) = C(−dx,−dy).

Modulation spectroscopy

The tunnelling tz between sites of adjacent layers canbe controlled via the lattice depth VZ . To realize theionic Hubbard model in two dimensions we suppress thetunnelling tz below 2 Hz by setting the lattice depthalong the z-direction to VZ = 30(1)ER thereby decou-pling the honeycomb planes. For the modulation spec-troscopy measurement we follow the same procedure asdescribed in the main text and sinusoidally modulate theamplitude of the lattice depth in y-direction by ±10%.As a result we exclude possible contributions to the re-sponse signal, which may result from a residual couplingto the orthogonal direction. This ensures that the linear

8

response measurement only probes energy scales that arerealized within the xy-honeycomb plane. Fig. 5 showsthe excitation spectrum for the two-dimensional case.

Even with suppressed tunnelling tz we observe a clearpeak for modulation frequencies hν = U . As a result wecan exclude that our response signal is resulting from animperfect orthogonality of the lattice beams.