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Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms

A. V. Gorshkov1∗, M. Hermele2, V. Gurarie2, C. Xu1, P. S. Julienne3,J. Ye4, P. Zoller5, E. Demler1,6, M. D. Lukin1,6, and A. M. Rey4

1Physics Department, Harvard University, Cambridge, MA 021382Department of Physics, University of Colorado, Boulder, CO 80309

3Joint Quantum Institute, NIST and University of Maryland, Gaithersburg, MD 20899-84234JILA, NIST, and Department of Physics, University of Colorado, Boulder, CO 80309

5Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck,Austria and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria

6Institute for Theoretical Atomic, Molecular and Optical Physics,Harvard-Smithsonian Center of Astrophysics, Cambridge, MA 02138 and

∗e-mail: gorshkov@post.harvard.edu(Dated: September 26, 2009)

Fermionic alkaline-earth atoms have unique properties that make them attractive candidates for therealization of novel atomic clocks and degenerate quantum gases. At the same time, they are attractingconsiderable theoretical attention in the context of quantum information processing. Here we demon-strate that when such atoms are loaded in optical lattices, they can be used as quantum simulators ofunique many-body phenomena. In particular, we show that the decoupling of the nuclear spin from theelectronic angular momentum can be used to implement many-body systems with an unprecedented de-gree of symmetry, characterized by the SU(N) group with N as large as 10. Moreover, the interplay ofthe nuclear spin with the electronic degree of freedom provided by a stable optically excited state allowsfor the study of spin-orbital physics. Such systems may provide valuable insights into strongly correlatedphysics of transition metal oxides, heavy fermion materials, and spin liquid phases.

The interest in fermionic alkaline-earth atoms [1, 2, 3, 4, 5,6, 7, 8] stems from their two key features: (1) the presence ofa metastable excited state3P0 coupled to the ground1S0 statevia an ultranarrow doubly-forbidden transition [1] and (2)thealmost perfect decoupling [1] of the nuclear spinI from theelectronic angular momentumJ in these two states, since theyboth haveJ = 0. This decoupling implies that s-wave scat-tering lengths involving states1S0 and3P0 are independentof the nuclear spin, aside from the restrictions imposed byfermionic antisymmetry. We show that the resulting SU(N)spin symmetry (whereN = 2I + 1 can be as large as 10) to-gether with the possibility of combining (nuclear) spin physicswith (electronic) orbital physics open up a wide field of ex-tremely rich many-body systems with alkaline-earth atoms.

In what follows, we derive the two-orbital SU(N)-symmetric Hubbard model describing alkaline-earth atomsin 1S0 and 3P0 states trapped in an optical lattice. We fo-cus on specific parameter regimes characterized by full orpartial atom localization due to strong atomic interactions,where simpler effective spin Hamiltonians can be derived.The interplay between orbital and spin degrees of freedomin such effective models is a central topic in quantum mag-netism and has attracted tremendous interest in the condensedmatter community. Alkaline earth atoms thus provide, onthe one hand, a unique opportunity for the implementationof some of these models for the first time in a defect-freeand fully controllable environment. On the other hand, theyopen a new arena to study a wide range of models, many ofwhich have not been discussed previously, even theoretically.We demonstrate, in particular, how to implement the Kugel-Khomskii model studied in the context of transition metal ox-ides [9, 10, 11, 12, 13], the Kondo lattice model [14, 15, 16,17, 18, 19, 20, 21, 22, 23, 24, 25, 26] studied in context of

manganese oxide perovskites [20] and heavy fermion materi-als [25], as well as various SU(N)-symmetric spin Hamilto-nians that are believed to have spin liquid and valence-bond-solid ground states [27, 28, 29, 30, 31, 32, 33, 34]. For ex-ample, we discuss how, by appropriately choosing the initialstate, a single alkaline-earth atom species withI = 9/2 (suchas87Sr) can be used to study experimentally such a distinc-tively theoretical object as the phase diagram as a functionofN for all N ≤ 10.

Before proceeding, we note that, while an orthogonal sym-metry group SO(5) can be realized in alkali atoms [35],proposals to obtain SU(N>2)-symmetric models with alkaliatoms [36, 37] and solid state systems [11, 38] are a sub-stantial idealization due to strong hyperfine coupling and acomplex solid state environment, respectively. In this context,alkaline-earth-like atoms make a truly exceptional systemtostudy models with SU(N>2) symmetry.

2

FIG. 1: Interaction parameters between g (green) and e (yel-low) atoms loaded in the lowest vibrational state of the corre-sponding optical lattice. Here we assumedI = 1/2, and the ar-rows indicate themI = ±1/2 spin states.|s, t〉 denote the singletand triplet nuclear spin states of the two atoms (only one of threetriplet states -| ↑↑〉 - is shown). The dashed circle represents anti-symmetrization of the nuclear spin state (i.e.|s〉). The interaction en-ergyUX (X = gg, ee, eg+, eg−) is proportional to the correspond-ing scattering lengthaX .

Many-body dynamics of alkaline-earth atoms in an opticallattice

We begin with the Hamiltonian describing cold fermionicalkaline-earth atoms in an external trapping potential:

H =∑

αm

∫

d3r؆

αm(r)(− ~2

2M∇2 + Vα(r))Ψαm(r) (1)

+~ω0

∫

d3r(ρe(r) − ρg(r)) +

g+eg + g−eg

2

∫

d3rρe(r)ρg(r)

+∑

α,m<m′

gαα

∫

d3rραm(r)ραm′(r)

+g+

eg − g−eg

2

∑

mm′

∫

d3r؆

gm(r)Ψ†em′(r)Ψgm′ (r)Ψem(r).

HereΨαm(r) is a fermion field operator for atoms in internalstate|αm〉, whereα = g (1S0) or e (3P0) denotes the elec-tronic state andm = −I, . . . , I denotes one of theN = 2I+1nuclear Zeeman states. The density operators are defined asραm(r) = Ψ†

αm(r)Ψαm(r) andρα(r) =∑

m ραm(r). Theterm Vα(r) describes the external trapping potential, whichwe will assume to be an optical lattice independent of the nu-clear spin: even for a relatively deep lattice with a 100 kHztrap frequency, tensor and vector light shifts should be wellbelow 1 Hz [1].~ω0 is the transition energy between|g〉 and|e〉. Extra lasers can be used to drive transitions between|g〉and|e〉 levels [1, 2]. Since we will only need these extra lasersfor system preparation, we have not included the correspond-ing terms in the Hamiltonian.

The interaction is characterized by fours-wave scatteringlengthsaX , X = gg, ee, eg+, eg−, which define four interac-tion parametersgX = 4π~

2aX/M , whereM is atomic mass.agg, aee, anda±

eg are the scattering length for two atoms inthe electronic state|gg〉, |ee〉, and|±〉 = (|ge〉 + |eg〉)/

√2,

respectively. As shown in Fig. 1, the fermionic antisymmetrythen forces the nuclear state to be symmetric for the only anti-symmetric electronic state|−〉 and antisymmetric otherwise.Very few aX are known at the moment (see Supplementary

Information).

The independence of each of the four scattering lengthsfrom the nuclear spin state is essential to the fulfillment ofthe SU(N) symmetry of our model (see next Section). Thisindependence is a consequence of the decoupling betweennuclear and electronic degrees of freedom exhibited duringthe course of a collision involving any combination ofg ore states, which both haveJ = 0. While for the |e〉 ≡ 3P0

atom, the decoupling is slightly broken by the admixture withhigher-lying P states withJ 6= 0, this admixture is very small[1] and the resulting nuclear-spin-dependent variation ofthescattering lengths is also expected to be very small, on the or-der of10−3 (see Supplementary Information). Foragg, whichdoes not involve state|e〉, this variation should be even smaller(∼ 10−9).

The interaction terms in Eq. (1) describe the most generals-wave two-body interaction consistent with elastic collisionsas far as the electronic state is concerned and with the inde-pendence of the scattering length from the nuclear spin. Whilethe assumption of elasticity forg-g ande-g collisions is welljustified, since no inelastic exit channels exist,e-e collisionsare likely to be accompanied by large losses, which means thatthe magnitudes of the imaginary and real parts of thee-e scat-tering length are likely to be comparable (see SupplementaryInformation). Therefore, we focus below on those situationswhere twoe atoms never occupy the same site.

We assume that only the lowest band in bothe andg lat-tices is occupied and expand the field operators in terms ofthe corresponding (real) Wannier basis functionsΨαm(r) =∑

j wα(r − rj)cjαm, wherec†jαm creates an atom in internalstate|αm〉 at sitej (centered at positionrj). Eq. (1) reducesthen to a two-orbital single-band Hubbard Hamiltonian

H = −∑

〈j,i〉α,m

Jα(c†iαmcjαm + h.c.) +∑

j,α

Uαα

2njα(njα − 1)

+V∑

j

njenjg + Vex

∑

j,m,m′

c†jgmc†jem′cjgm′cjem. (2)

HereJα = −∫

d3rwα(r)(− ~

2

2M∇2 + Vα(r))wα(r − r0) are

the tunneling energies,r0 connects two nearest neighbors,h.c. stands for Hermitian conjugate,njαm = c†jαmcjαm, andnjα =

∑

m njαm. The tunneling is isotropic, which is a cru-cial difference between this model and its analogues in solidstate systems with orbital degeneracy [9]. The sum〈j, i〉 isover pairs of nearest neighbor sitesi, j. V = (U+

eg + U−eg)/2

andVex = (U+eg − U−

eg)/2 describe the direct and exchangeinteraction terms. The onsite interaction energies areUαα =gαα

∫

d3rw4

α(r) andU±eg = g±eg

∫

d3rw2

e(r)w2g(r). Constant

terms, proportional to∑

j njα, are omitted in Eq. (2). Exper-imental control over the parameters in Eq. (2) will allow us tomanipulate the atoms (see Methods).

3

Symmetries of the Hamiltonian

To understand the properties of the Hamiltonian in Eq. (2),we consider its symmetries. We define SU(2) pseudo-spin al-gebra via

T µ =∑

j

T µj =

1

2

∑

jmαβ

c†jαmσµαβcjβm, (3)

whereσµ (µ = x, y, z) are Pauli matrices in thee, g basis.We further define nuclear-spin permutation operators

Smn =

∑

j

Smn (j) =

∑

j,α

Smn (j, α) =

∑

j,α

c†jαncjαm, (4)

which satisfy the SU(N) algebra[Smn , Sp

q ] = δmqSpn−δpnSm

q ,and thus generate SU(N) rotations of nuclear spins (N = 2I+1).

In addition to the obvious conservation of the total numberof atomsn =

∑

j(nje+njg), H exhibitsU(1)×SU(N) sym-metry (see Methods for the discussion of enhanced symme-tries), whereU(1) is associated with the elasticity of collisionsas far as the electronic state is concerned ([T z, H ] = 0) andSU(N) is associated with the independence of scattering andof the trapping potential from the nuclear spin ([Sm

n , H ] = 0for all n, m). The two-orbital SU(N)-symmetric HubbardHamiltonian in Eq. (2) is a generalization toN > 2 of itsSU(2)-symmetric counterpart [9] and to two orbitals of itssingle-orbital counterpart [28]. The SU(N) symmetry and thelargely independent spin and orbital degrees of freedom aretwo unique features present in alkaline-earths but absent inalkalis due to strong hyperfine interactions.

One important consequence of SU(N) symmetry is the con-servation, for anym, of Sm

m , the total number of atoms withnuclear spinm. This means that an atom with largeI, e.g.87Sr (I = 9/2), can reproduce the dynamics of atoms withlower I if one takes an initial state withSm

m = 0 for somem values. To verify SU(N) symmetry of the interaction ex-perimentally, one could, thus, put two atoms in one well inspinsm andm′ and confirm that collisions do not populateother spin levels. This feature of SU(N) symmetry is in starkcontrast to the case of weaker SU(2) symmetry, where the de-pendence of scattering lengths on the total spin of the two col-liding particles allows for scattering into spin states other thanm andm′. We note that although collisions are governed byelectronic interactions and obey the nuclear-spin SU(N) sym-metry, the nuclear spins still indirectly control the collisionsvia fermionic statistics and give rise to effective spin-orbitaland spin-spin interactions.

One can alternatively implement the two-orbital Hubbardmodel with two ground-state species of alkaline-earth atoms(e.g. 171Yb and 173Yb, or 173Yb and 87Sr). If we still re-fer to them as|g〉 and |e〉, the nuclear distinguishability andthe fact that both atoms are in the ground state will resultin a+

eg = a−eg, corresponding to an enhanced symmetry (see

Methods). While experimentally more challenging, the useof two different ground state species will solve the prob-lem of losses associated with collisions of two excited state

FIG. 2: Young diagrams describing the irreducible representa-tions of SU(N) on individual sites. a, A general diagram consistsof nj boxes arranged into at most two columns (to satisfy fermionicantisymmetry with only two orbital states) whose heights wewilldenote byp andq, such thatN ≥ p ≥ q andp + q = nj . SeeSupplementary Information for a brief review of Young diagrams.b,The Young diagrams for the two special cases discussed in themaintext: (1) (p, q) = (1, 0) and(2) (p, q) = (p, 0) on a bipartite lattice.

atoms and will reduce the (already very weak) nuclear-spin-dependence ofaee andaeg.

Spin Hamiltonians

One of the simplest interesting limits of Eq. (2) is thestrongly interacting regime (J/U ≪ 1) where the Hilbertspace is restricted to a given energy manifold of theJg =Je = 0 Hamiltonian (with a fixed number of atoms on eachsite), and tunneling is allowed only virtually, giving risetoan effective spin (and pseudo-spin) Hamiltonian. Single-siteenergy manifolds can be classified according to the numberof atomsnj = njg + nje, the pseudo-spin componentT z

j ,and the spin symmetry (SU(N) representation) described bya Young diagram. As shown in Fig. 2a, each diagram con-sists ofnj boxes and at most two columns of heightsp andq,representing two sets of antisymmetrized indices.

TheU(1)×SU(N) symmetry of Eq. (2) restricts the orderJ2 spin Hamiltonian to the form

H(p,q) =∑

〈i,j〉,α

[

κijα niαnjα + λij

α Snm(i, α)Sm

n (j, α)]

+∑

〈i,j〉

[

κijgenignje + λij

geSnm(i, g)Sm

n (j, e)

+κijgeS

emgm(i)Sgn

en (j) + λijgeS

engm(i)Sgm

en (j) + i ↔ j]

,(5)

where the sum overn andm is implied in all but theκ termsandSαm

βn (j) = c†jβncjαm. i ↔ j means that all 4 precedingterms are repeated withi andj exchanged. The coefficientsκ,λ, κ, andλ are of orderJ2/U with the exact form determinedby what single-site energy manifolds we are considering.κterms describe nearest neighbor repulsion or attraction, whileλ, κ, andλ terms describe nearest neighbor exchange of spins,pseudo-spins, and complete atomic states, respectively. With-out loss of generality,κij

α = κjiα andλij

α = λjiα . In many

cases (e.g. case (2) below), the Hilbert space, whichH(p,q)

4

a

(0,0)0

b

FIG. 3: The ground-state phase diagram for the SU(N=2) Kugel-Khomskii model restricted to two wells, left (L) and right (R). a,The phase diagram forTz = −1 (two g atoms). |gg〉 = |gg〉LR.|s〉 and |t〉 are spin singlet and triplet states, respectively.b, Thephase diagram forTz = 0 (one g atom and onee atom). |Σ〉 =1√2(|eg〉LR − |ge〉LR) and |τ 〉 = 1√

2(|eg〉LR + |ge〉LR) are anti-

symmetric and symmetric orbital states, respectively. SeeSupple-mentary Information for a detailed discussion of both of these dia-grams.

acts on, hasnie and nig constant for alli, which not onlyforcesκij

ge = λijge = 0 but also allows one to ignore the con-

stantκijα andκij

ge terms. We now discuss two special cases ofH(p,q) shown in Fig. 2b. A third case,(p, q) = (1, 1), whichreduces forN = 2 to the spin-1 Heisenberg antiferromagnetis discussed in the Supplementary Information.

(1) In the case of one atom per site,(p, q) = (1, 0). H(p,q) isthen a generalization to arbitrary N of the SU(N = 2) Kugel-Khomskii model [9, 13], and we rewrite it as (see Supplemen-tary Information)

H(1,0) =∑

〈i,j〉

[

2(κge + λgeS2ij)(T

xi T x

j + T yi T y

j ) + λgeS2ij

−[A + BS2ij ](T

zi T z

j +1

4) + h(1 − S2

ij)(Tzi + T z

j )]

, (6)

whereS2ij =

∑

mn Snm(i)Sm

n (j) is +1 (−1) for a symmetric(antisymmetric) spin state,A = 2κge − κe − κg, B = 2λge +κe + κg, andh = (κe − κg)/2. TheN = 2 Kugel-KhomskiiHamiltonian is used to model the spin-orbital interactions(notto be confused with relativistic spin-orbit coupling) in transi-tion metal oxides with perovskite structure [13]. Our imple-mentation allows to realize clean spin-orbital interactions un-altered by lattice and Jahn-Teller distortions present in solids[13].

To get a sense of the competing spin and orbital orders[10, 11, 12] characterizingH(1,0), we consider the simplestcase of only two sites (L andR) andN = 2 (with spin statesdenoted by↑ and↓). To avoid losses ine-e collisions, we setUee = ∞ (see Supplementary Information). The double-wellground-state phase diagram forT z = 1 (two e atoms) is thentrivial, while theT z = −1 (two g atoms) andT z = 0 (onegatom and onee atom) diagrams are shown in Fig. 3. One cansee that, depending on the signs and relative magnitudes of theinteractions, various combinations of ferromagnetic (triplet)and antiferromagnetic (singlet) spin and orbital orders are fa-

ba

1

654321

65432

- Neel - VBS

- possible critical

spin liquid

FIG. 4: Probing the phases of the SU(N) antiferromagnet on a 2Dsquare lattice. a shows the phase diagram for the casenA + nB =N . Some points on this diagram have been explored in earlier nu-merical studies [29, 30, 31] and are marked according to the groundstate obtained: Neel (circles), columnar-valence-bond solid (VBS)[shown schematically inb] (squares), and possibly critical spin liq-uid (triangle) [30, 31]. Since for sufficiently largeN quantum fluc-tuations tend to destabilize long-range magnetic ordering, it is likelythat VBS ordering characterizes the ground state for allN > 4 (i.e.above the wavy line).

vored. In the Methods, we propose a double-well experimentalong the lines of Ref. [39] to probe the spin-orbital interac-tions giving rise to theT z = 0 diagram in Fig. 3b. Multi-wellextensions of this experiment may shed light on the model’smany-body phase diagram, which has been studied forN = 2and mostly at mean-field level or in special cases, such as inone dimension or in the presence of enhanced symmetries (seee.g. [10, 11, 12]).

(2) In order to study SU(N) spin physics alone, we con-sider the case ofg atoms only. On a bipartite lattice withsublattices A and B, we choose A sites to havenA < Natoms [(p, q) = (nA, 0)] and B sites to havenB < N atoms[(p, q) = (nB , 0)]. This setup can be engineered in cold atomsby using a superlattice to adjust the depths of the two sublat-tices favoring a higher filling factor in deeper wells.H(p,q)

then reduces to

H(p,0) =2J2

g Ugg

U2gg − (Ugg(nA − nB) + ∆)2

∑

〈i,j〉S2

ij , (7)

where∆ is the energy offset between adjacent lattice sites.The coupling constant can be made either positive (antifer-romagnetic) or negative (ferromagnetic) depending on thechoice of parameters [39]. Three body recombination pro-cesses will likely limit the lifetime of the atoms whennj ≥ 3(see Supplementary Information).

We focus on the 2D square lattice in the antiferromag-netic regime. The casenA + nB = N shares with theSU(2) Heisenberg model the crucial property that two ad-jacent spins can form an SU(N) singlet, and has thus beenstudied extensively as a large-N generalization of SU(2) mag-netism [27, 28]. Fig. 4a shows the expected phase diagram

5

for the casenA + nB = N , which features Neel (circles),valence-bond-solid (VBS) (squares) [Fig. 4b], and possiblecritical spin liquid (triangle) [30, 31] ground states. To ac-cess various ground states of the system, the initial state mustbe carefully prepared so that the conserved quantitiesSm

m takevalues appropriate for these ground states. Another interestingand experimentally relevant case,nA = nB 6= N/2, whichcan also exhibit spin liquid and VBS-type ground states, isdiscussed in the Supplementary Information and in Ref. [34].

Since one can varyN just by choosing the number of ini-tially populated Zeeman levels (e.g. via a combination of opti-cal pumping and coherent manipulation), alkaline-earth atomsoffer a unique arena to probe the phase diagram ofH(p,0), in-cluding exotic phases such as VBS [Fig. 4b], as well as com-peting magnetically ordered states. We propose to load a bandinsulator ofN g atoms per site, then slowly split each well intotwo to form an array of independent SU(N) singlets in a pat-tern shown in Fig. 4b. The intersinglet tunneling rate shouldthen be adiabatically increased up to the intrasinglet tunnelingrate. AsN increases, the magnetic or singlet nature of thestate can be probed by measuring the Neel order parameter(see the description of the Kugel-Khomskii double-well ex-periment in the Methods) and spin-spin correlations via noisespectroscopy in the time of flight [40] (which directly mea-sures

∑

i,j〈Smn (i, g)Sn

m(j, g)〉eIQ(i−j)).

The Kondo lattice model (KLM)

The SU(N) Kondo lattice model (KLM) [15, 17] is anotherexample of the rich physics, beyond the Mott regime, whichcould be simulated with alkaline-earth atoms. The KLM isone of the canonical models used to study strongly corre-lated electron systems, such as manganese oxide perovskites[20] and rare earth and actinide compounds classed as heavyfermion materials [25].

For its implementation with cold atoms (forN = 2, seealso Refs. [23, 24]), we propose to put onee atom (localizedspin) per site in a deep lattice such thatJe ≪ Uee, so thatwe can setJe = 0 andnje = 1 for all j in Eq. (2). We alsosuppose that we can setUgg = 0, e.g. by taking a very shallowg-lattice (see Fig. 5a). The resulting Hamiltonian is the SU(N)KLM [15, 17]

HKLM =−∑

〈j,i〉mJg(c

†igmcjgm+h.c.)+Vex

∑

j,m,m′

c†jgmc†jem′cjgm′cjem.(8)

The magnitude ofVex can be adjusted by shifting thee andglattices relative to each other [7].

The properties of the SU(N) KLM depend crucially on thesign of the exchange interaction. For concreteness, we focuson the antiferromagnetic (AF) case (Vex < 0), which favorsformation of spin-antisymmetric states (singlets, forN = 2)between mobile fermions and localized spins. This regimedescribes the physics of heavy fermion materials [25], and,in the case of a single localized spin, gives rise to the Kondoeffect.

In the limit |Vex| ≪ Jg, g atoms mediate long-range RKKY

a

b

FIG. 5: Kondo lattice model for the case N = 2. a, The schematicof the setup.g atoms are green;e atoms are yellow; the spin ba-sis is ↑, ↓. b, Schematic representation of the competition be-tween RKKY magnetism vs Kondo singlet formation in the SU(2)AF KLM (see [16, 25, 26] and references therein). In this model,the localized spin-1/2 e atoms couple antiferromagnetically to thedelocalizedg atoms, via an on-site exchange interaction propor-tional toVex. This coupling favors the formation of localized Kondosinglets betweene and g atoms, with characteristic energy scalekBTK ∼ Jg exp(−cJg/|Vex|), with c a dimensionless constant oforder one [25]. On the other hand, theg atoms can mediate long-range RKKY interactions between thee atoms, giving rise to mag-netic order (which can be antiferromagnetic (AF) or ferromagneticdepending on the density ofg atoms), where the characteristic en-ergy iskBTRKKY ∼ V 2

ex/Jg . The competition between Kondo ef-fect and RKKY magnetism leads to very rich physics. For smallval-ues of|Vex|/Jg , the RKKY interaction is dominant and the systemorders magnetically. At intermediate values of|Vex|/Jg , the energyscalesTK andTRKKY are of comparable strength, and a variety ofnovel quantum phenomena are expected to arise, including quantumcriticality and non-Fermi liquid (NFL) physics [25, 26]. With furtherincrease of the|Vex|/Jg coupling, magnetic order is suppressed, thelocalizede atoms become screened into singlet states and melt intotheg-atom Fermi sea, forming the so called heavy Fermi liquid state(HFL). The large Fermi volume [21], which is set by the total num-ber ofg atoms pluse atoms, can be directly probed by measuring themomentum distribution via time of flight imaging.

interactions [14] between localized spins and tend to inducemagnetic ordering (antiferromagnetic or ferromagnetic de-pending on the density ofg atoms) of the latter, at least forN = 2. The engineering of RKKY interactions can be testedin an array of isolated double wells (see Methods). At in-termediate and large|Vex|, the formation of Kondo singletsdominates the RKKY interaction and favors a magneticallydisordered heavy Fermi liquid (HFL) ground state with a sig-

6

nificantly enhanced effective quasiparticle mass (see Fig.5b).The competition between RKKY interactions and the Kondoeffect in the regime where both are comparable is subtle, andthe resulting phases and phase transitions [25, 26] are notwell-understood. Ultracold alkaline-earth atoms provideapromising platform to study these phases and phase transi-tions.

In the large-N limit [15, 17], the SU(N) HFL can be con-trollably studied, and1/N expansions have successfully re-produced the experimentally observed properties of the HFL.However, very little is known about the SU(N) model outsidethe HFL regime. Several very interesting parameter regimesin this domain can be directly probed with our system, as dis-cussed in the Methods.

Experimental Accessibility

The phenomena described in this manuscript can be probedwith experimental systems under development. Indeed, weshow in the Methods that SU(N)-breaking terms are suffi-ciently weak, and here we discuss the temperature require-ments.

The key energy scale in the spin Hamiltonians [Eq. (5)]is the superexchange energyJ2/U , while the RKKY en-ergy scale iskBTRKKY ∼ V 2

ex/Jg. In their region of va-lidity (J < U and |Vex| < Jg, respectively), these en-ergy scales are limited from above by the interaction energy(U and |Vex|, respectively), which typically corresponds totemperaturesT . 100nK [39]. Thanks to the additionalcooling associated with certain adiabatic changes [41, 42],T ∼ 10nK and the Mott insulating regime have already beenachieved with fermionic alkali atoms [43], and are thereforeexpected to be achievable with fermionic alkaline-earths,aswell (a bosonic alkaline-earth Mott insulator has already beenachieved [44]). Furthermore, the requirement to reachkBTsmaller thanJ2/U or V 2

ex/Jg can often be relaxed. First,the double-well experiments, such as the ones discussed inthe Methods in the contexts of the Kugel-Khomskii and theKondo lattice models, are performed out of thermal equilib-rium, and can, thus, access energy scales far below the tem-perature of the original cloud [39]. Second, for SU(N) anti-ferromagnets, the energy range betweenJ2/U andNJ2/Umay also exhibit intriguing physics: in this regime, SU(N)singlets, which requireNJ2/U energy to break, stay intactbut can diffuse around. Finally, in theVex < 0 Kondo latticemodel, exotic heavy Fermi liquid behavior is expected whenJg . |Vex| and the temperature is below the Kondo tempera-ture, i.e.kBT . Jg exp(−cJg/|Vex|) with c is a dimension-less constant of order one [25]. Thus, withJg chosen to be onthe order of|Vex|, kBT as high as∼ |Vex| may be sufficient.

Outlook

The proposed experiments should be regarded as bridgesaiming to connect well-understood physics to the complex andpoorly understood behavior of strongly correlated systems. It

is important to emphasize that, except for the one dimensionalcase, the phase diagram of most of the models considered isonly known at mean field level or numerically in reduced sys-tem sizes. Therefore, their experimental realization in cleanand controllable ultracold atomic systems can provide majoradvances.

Our proposal motivates other new lines of research. Ultra-cold bosonic or fermionic diatomic molecules [45] may giverise to similar SU(N) models with largeN and with the possi-bility of long-range interactions. Ions with alkaline-earth-likestructure, such as Al+ could also be considered in this context.It would also be interesting to explore the possibility of real-izing topological phases with SU(N) models for applicationsin topological quantum computation [34]. Beyond quantummagnetism, the fact that the formation of SU(N) singlets re-quiresN partners might give rise to novel exotic types of su-perfluidity and novel types of BCS-BEC crossover [37]. Prac-tical applications of our Hubbard model, such as the calcula-tion of the collisional frequency shift in atomic clocks [46],can also be foreseen.

Note added in proof. After the submission of this paper, atheoretical study of the SU(6)-symmetric173Yb system wasreported [50].

Methods

Experimental tools available for alkaline-earth atoms

Many experimental tools, such as tuning the interactionstrength by adjusting laser intensities [39], are common toboth alkali and alkaline-earth atoms. There are, however,some experimental tools specific to alkaline earths; we reviewthem in this Section.

First, a combination of optical pumping [2] and direct co-herent manipulation of the|g〉 − |e〉 transition in the presenceof a magnetic field [1, 2] can be used [8] to prepare any desiredsingle-atom state within the 2 (2 I + 1)-dimensional manifoldwith basis|αm〉, whereα = g or e andm = −I, . . . , I. Thiscoherent manipulation can also be used to exchange quantuminformation between nuclear spin states and electronic states.Second, by using far-detuned probe light or a large magneticfield to decouple the electronic angular momentumJ and thenuclear spinI, the electronic|g〉 − |e〉 degree of freedom canbe measured by collecting fluorescence without destroying thenuclear spin state [8]. Fluorescence measurement of the nu-clear spins can be achieved by mapping nuclear spin statesonto electronic states [7, 8]: for example, for a spin-1/2 nu-cleus, aπ pulse between|g, m = 1/2〉 and |e, m = −1/2〉allows one to accomplish a swap gate between the nuclear1/2,−1/2 qubit and the electronice, g qubit. Single-sitespatial resolution during the coherent manipulation and fluo-rescence measurement can be achieved using magnetic fieldgradients [7] or dark-state-based techniques [8, 47] that relyon an auxiliary laser field whose intensity vanishes at certain

7

locations. Third, an appropriate choice of laser frequenciesallows one to obtain independent lattices for statesg and e[7]. Finally, optical Feshbach resonances [48] may be used tocontrol scattering lengths site-specifically and nearly instanta-neously.

Enhanced Symmetries

While in the general case, our Hubbard model [Eq. (2)]satisfiesU(1) × SU(N) symmetry, for particular choices ofparameters, higher symmetry is possible. In particular, ifJg = Je and the interaction energies for all states within thepseudo-spin triplet are equal (Ugg = Uee = U+

eg), the fullSU(2) symmetry (not just U(1)) in the pseudo-spin space issatisfied. Alternatively, ifVex = 0, then bothSm

n (i, g) andSm

n (i, e) generate SU(N) symmetries resulting in the overallU(1) × SU(N) × SU(N) symmetry. Finally, if both condi-tions are satisfied, i.e. all fourUX are equal andJg = Je, thenH satisfies the full SU(2N) symmetry (2N can be as high as20) generated by

Sαmβn =

∑

j

Sαmβn (j) =

∑

j

c†jβncjαm, (9)

in which case the interaction reduces toU2

∑

j nj(nj − 1),wherenj = njg + nje.

In the case when|e〉 and|g〉 correspond to two ground statesof two different atoms (with nuclear spinIe andIg, respec-tively), we will havea+

eg = a−eg (i.eVex = 0), which is equiva-

lent to imposingU(1)×SU(Ng = 2Ig+1)×SU(Ne = 2Ie+1) symmetry, whereSU(2Iα + 1) is generated bySm

n (i, α).While for Ig 6= Ie, them index in cjαm will run over a dif-ferent set of values depending onα, the Hubbard Hamiltonianwill still have the form of Eq. (2) (except withVex = 0). Ifone further assumes thatJg = Je andUgg = Uee = Ueg,the interaction satisfies the fullSU(Ng + Ne) symmetry. Itis worth noting that for the case of two different ground stateatoms, this higher symmetry is easier to achieve than for thecase of two internal states of the same atom, sincea+

eg = a−eg

automatically. Thus, in particular, it might be possible toob-tainSU(18) with 87Sr (I = 9/2) and43Ca (I = 7/2) simplyby adjusting the intensities of the two lattices (to setJg = Je

andUgg = Uee) and then shifting the two lattices relative toeach other (to setUeg = Ugg).

Enhanced symmetries of the Hubbard model [Eq. (2)] areinherited by the spin Hamiltonian [Eq. (5)]. In particular,im-posingSU(2) × SU(N) instead ofU(1) × SU(N) forcesκij

ge = κjige, κij

ge = κjige, κij

g = κije = κij

ge + κijge ≡ κij ,

λijge = λji

ge, λijge = λji

ge, λijg = λij

e = λijge + λij

ge ≡ λij .Alternatively, imposingU(1) × SU(N) × SU(N) forcesκij

ge = λijge = 0. Finally, imposing the full SU(2N) forces

the satisfaction of both sets of conditions, yielding

H =∑

〈i,j〉

[

κijninj + λijSβnαm(i)Sαm

βn (j)]

, (10)

which is, of course, equivalent to restricting Eq. (5) tog-atoms

only and extending labelsm andn to run over2N states in-stead ofN .

Double-well Kugel-Khomskii and RKKY experiments

In the main text and in the following Methods Section, wediscuss the open questions and previously unexplored regimesassociated with the SU(N) Kugel-Khomskii and Kondo latticemodels (KLM) that can be studied with ultracold alkaline-earth atoms. As a stepping stone toward these many-body ex-periments, we propose in this Section two proof-of-principleexperiments in an array of isolated double wells withN = 2(with the spin basis↑, ↓): one to probe the spin-orbital in-teractions of the Kugel-Khomskii model and one to probe theRKKY interactions associated with KLM.

We first propose an experiment along the lines of Ref. [39]to probe the spin-orbital interactions giving rise to theT z = 0diagram in Fig. 3b. In the Supplementary Information, we de-scribe how to prepare an array of independent double wells inthe state|e, ↑〉L|g, ↓〉R, which is a superposition of the foureigenstates featured in Fig. 3b. The energies of these foureigenstates [Eqs. (S4-S7)] can be extracted from the Fourieranalysis of the population imbalance as a function of time:

∆N(t) = neR + ngL − ngR − neL = − cos[

4tJeJg

~U−

eg

]

−

cos[

4tJeJg

~U+eg

]

. ∆N can be measured by combining the dump-

ing technique, band mapping, and Stern-Gerlach filtering ofRef. [39] with the use of two probe laser frequencies to distin-guish between|g〉 and|e〉.

We now turn to the double-well experiment aimed at prob-ing RKKY interactions. After preparing the state1√

2(|g, ↓

〉L + |g, ↓〉R)|e, ↓〉L|e, ↑〉R (see Supplementary Informationfor how to prepare this state), we propose to monitor theNeel order parameter for thee atoms,Nez = 1

2 [ne↑L −ne↓L − (ne↑R − ne↓R)]. In the limit |Vex| ≪ Jg, Nez(t) =

− 13 cos

(

Vext~

)

− 23 cos

(

Vext2~

− 3V 2ex

t

8Jg~

)

[in the Supplemen-

tary Information, we present the plot ofNez(t) for Vex =−Jg/10]. It exhibits fast oscillations with frequency∼ Vex,modulated by an envelope of frequency∼ V 2

ex/Jg inducedby RKKY interactions. In order to probe RKKY interactionsonly, it is important to suppress super-exchange∼J2

e /Uee andthus to chooseJe/Uee small. To study the full spatial depen-dence of RKKY interactions, one must of course go beyondthe double-well setup. We also note that recent experimentsusing alkali atoms populating the lowest two vibrational levelsof a deep optical lattice have measured the local singlet-tripletsplitting induced byVex [49].

Physics accessible with the alkaline-earth Kondo lattice model

The alkaline-earth atom realization of the AF KLM is well-suited to access a number of parameter regimes that are outof reach in solid state materials. One example is the onedimensional (1D) limit, since, to our knowledge, real solidstate materials exhibiting KLM physics are restricted to 2Dor

8

3D. Another example is the regime of large Kondo exchange(|Vex| ≫ Jg), which is interesting even forN = 2. In thislimit the system is well described by theU → ∞ Hubbardmodel [18] by identifying the Kondo singlets with empty sites(holes) and the unpaired localized spins with hard core elec-trons. From this mapping, possible ferromagnetic orderingisexpected at small hole concentration (smallng), however thestability of this phase for increasing hole concentration andfinite |Vex| values remains unknown. For generalN , in theextreme limitJg = 0, the ground state is highly degenerate:for any distribution of theg atom densitynjg < N , there isa ground state (with further spin degeneracy), where on eachsite the spins combine antisymmetrically to minimize the ex-change interaction. Lifting of such extensive degeneracies of-ten leads to novel ground states; this will be addressed in fu-ture studies using degenerate perturbation theory inJg/Vex.For N > 2, AF SU(N) spin models have a different kind ofextensive degeneracy, which was argued to destroy antiferro-magnetism and to lead to non-magnetic spin liquid and VBS-like ground states [34]. Similar expectations are likely toap-ply to the KLM at small|Vex|/Jg, where theN = 2 antifer-romagnetism may give way to situations where the localizedspins form a non-magnetic state that is effectively decoupledfrom the mobile fermions [22].

Even though we have setUgg to zero in Eq. (8), it can betuned, for example, by adjusting theg-lattice depth and cangive rise to interesting physics. For example, theng = 1case, which is known to be forN = 2 either an antifer-romagnetic insulator or a Kondo insulator depending on theratio |Vex|/Jg [19], will become for large enoughUgg andN > 2 a Mott insulator, because the two atoms on each sitecannot combine to form an SU(N) singlet. Ifng is reducedfrom unity, the doping of this Mott insulator can be studied,and it will be interesting to understand how this physics, usu-ally associated with cuprate superconductors, is related to theother ground states of the KLM, usually associated with heavyfermion compounds.

Experimental Accessibility

Immediate experimental accessibility makes our proposalparticularly appealing. Having shown in the main text that the

temperature requirements of our proposal are within reach ofcurrent experimental systems, here we show that the nuclear-spin dependence of interaction energies is sufficiently weak tokeep the SU(N) physics intact.

In the Supplementary Information, nuclear-spin-dependentvariation in the interaction energies is estimated to be∆Ugg/Ugg ∼ 10−9 and∆Uee/Uee ∼ ∆U±

eg/U±eg ∼ 10−3.

Since the scale of SU(N) breaking is at most∆U , a very con-servative condition for the physics to be unaffected by SU(N)breaking is that all important energy scales are greater than∆U . In particular, in the spin models with more than one atomper site, the condition is∆U ≪ J2/U , which can be satisfiedsimultaneously withJ ≪ U even for∆U/U ∼ 10−3. Withone atom per site, the SU(N) breaking scale is not∆U butrather(J/U)2∆U , which relaxes the condition to the imme-diately satisfied∆U/U ≪ 1. Similarly, in the Kondo latticemodel, the conditions∆Vex ≪ J, |Vex| can be satisfied for∆Vex/|Vex| ∼ 10−3.

Acknowledgments

We gratefully acknowledge conversations withM. M. Boyd, A. J. Daley, S. Folling, W. S. Bakr, J. I. Gillen,L. Jiang, G. K. Campbell, and Y. Qi. This work was supportedby NSF, CUA, DARPA, AFOSR MURI, NIST.

Author contributions

All authors contributed extensively to the work presented inthis paper.

Additional information

Supplementary information accompanies this pa-per on www.nature.com/naturephysics. Reprintsand permissions information is available online athttp://npg.nature.com/reprintsandpermissions. Corre-spondence and requests for materials should be addressed toA.V.G.

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10

SUPPLEMENTARY ONLINE MATERIALS

Known Scattering Lengths

Very few scattering lengthsaX (X = gg, ee, eg+, eg−) be-tweeng (1S0) ande (3P0) states of alkaline-earth-like atomsare known at the moment.agg is known for all isotopic com-binations of Yb [51] and Sr [52]. Estimates ofaee for 88Sr[53] and ofa−

eg for 87Sr [54] also exist. Finally, there is aproposal describing how to measurea+

eg via clock shifts [55].

Nuclear-Spin Independence of the Scattering Lengths

Independence of scattering lengths from the nuclear spin isa key assumption of the paper. This feature allows us to ob-tain SU(N)-symmetric models withN as large as 10 and dis-tinguishes alkaline-earth atoms from alkali atoms, which canexhibit at most an SO(5) symmetry [56, 57, 58, 59], a symme-try that is weaker than SU(4). The assumption of nuclear-spinindependence of scattering lengths is consistent with recentexperiments, where - within experimental precision - the clockshift does not depend on how the Zeeman levels are populated[60, 61]. In this Section, we present the theoretical justifica-tion of this assumption.

Direct magnetic dipole-dipole coupling between the nu-clear spins of two atoms sitting on the same site of an op-tical lattice is negligible: even for two magnetic dipole mo-ments as large as 10 nuclear magnetons at a distance of 10 nm(which is significantly smaller than the confinement typicallyachieved in optical lattices [62]), the interaction energystillcorresponds to a frequency smaller than one Hertz. Therefore,nuclei can affect the collisions only via the electrons. Allfourscattering lengths (agg, a±

eg, andaee) are, thus, expected tobe independent of the nuclear spin because bothg ande havetotal electronic angular momentumJ equal to zero, which re-sults in the decoupling between nuclear and electronic degreesof freedom during the course of a collision. The decouplingduring a collision is a consequence of the fact that each of thefour molecular electronic states that correlate with theJ = 0separated atom pair has zero projectionΩ of total electronicangular momentum on the molecular axis. The nuclear spinsin this case can only couple very weakly to other molecularstates, even if there is a molecular curve crossing.

While the short-range potential energy structure for amolecule like Sr2 is very complex for the excited states[63, 64], we will now show that scattering length differencesamong different combinations of nuclear spin projections forthe sameΩ = 0 potential are expected to be very small. Thescattering lengtha can be computed asa = a[1 − tan(Φ −π/8)], wherea is the average scattering length governed bythe asymptotic behavior of the potential andΦ is the semiclas-sical phase computed at zero energy from the classical turn-ing pointR0 to infinity: Φ =

∫ ∞R0

dR√

M [−V (R)]/~, where−V (R) is the (positive) depth of the interaction potential atseparationR andM/2 is the reduced mass [65]. DefiningR(t) as the classical trajectory from timet = −∞ to timet =

∞ of a particle of massM/2 at zero energy in the potentialV (R), we can rewrite the phase asΦ = −

∫ ∞−∞ dtV (R(t))/~.

The order of magnitude of the changeδΦ in the phase asso-ciated with different nuclear spin projections can, thus, be es-timated asδΦ ∼ ∆tδV/~, where∆t is the total time in theshort-range part of the collision andδV is the typical energydifference associated with different nuclear spin projectionsduring this time. SinceδV vanishes atR → ∞, only the shortrange molecular region contributes to the phase difference.Therefore, assumingδΦ ≪ 1, a ∼ a, and| cos(Φ − π/8)| ∼1, the nuclear-spin-dependent variationδa in the scatteringlength can be estimated asδa/a ∼ δΦ ∼ ∆tδV/~.

Turning to the actual numbers,∆t can be estimated fromthe depth (∼ 103cm−1hc) and the range (∼ 10 Bohr radii)of the appropriate interatomic potential (see e.g. [63, 64]) tobe ∆t ≈ 1 ps. Forg-g collisions, δV/h can be estimatedby the second-order formulaE2

hf/(hEopt) ∼ 200 Hz, whereEhf/h ∼ 300MHz is the approximate value for the hyper-fine splittings in3P1 in 87Sr andEopt/h ∼ 400 THz is theoptical energy difference between1S0 and3P1 in 87Sr. Thisyields the following estimate for the dependence ofagg on thenuclear spin:δagg/agg ∼ δΦ ∼ 10−9. Fore-e ande-g colli-sions, an analogous second-order formula would use the finestructure splitting between3P1 and3P0 in 87Sr (Ef/h ∼ 6THz) instead ofEopt to yield δΦ ∼ 10−7. However, the latterestimate (δΦ ∼ 10−7) is too optimistic since molecular statesthat are split byEf at large interatomic separations may comeorders of magnitude closer at short range [66]. Therefore, amore realistic conservative estimate would use the first-orderformula δV ∼ Ehf to yield δaee/aee ∼ δa±

eg/a±eg ∼ δΦ ∼

10−3. It is important to note, however, that these are all onlyvery rough estimates. For example, hyperfine coupling in amolecule will differ from the hyperfine coupling in separatedatoms. In fact, since it is very difficult to predictδa/a accu-rately, these values would need to be measured. To concludethis Section, we would like to emphasize that, as mentionedin the main text, if the small nuclear-spin dependence ofaee

anda±eg is not negligible for some applications, one can use

two different ground state atomic species instead of a groundand an excited state of one species.

Likelihood of Lossy e-e Collisions and Possible Solutions

Collisions of twoe atoms are likely to be accompanied bylarge loss [53]. This can occur if the molecular0+

g potentialthat correlates with thee-e atoms undergoes an avoided cross-ing with a potential curve that correlates with a lower energypair of separated atoms (see, for example, Ref. [64]). Simi-lar crossings that result in inelastic energy transfer collisionswere examined for1P1+

1S0 collisions of alkaline earth atomsin Ref. [67]. The likelihood of a relatively high probability ofan inelastic event during such a crossing with species suchas Sr or Yb means that the imaginary partbee of the scatter-ing length is expected to be large. However, just likeaee, bee

can not be calculated accurately from the potentials but wouldneed to be measured.

The possible effects ofbee on the four examples we discuss

11

FIG. S1:A general Young diagram.

[Eqs. (6-8) and Eq. (S1)] are as follows.H(p,0) [Eq. (7)] is,of course, not affected because it involves onlyg atoms. InH(1,1) [Eq. (S1)] andHKLM [Eq. (8)], thee lattice is assumedto be so deep thatJe is negligible compared toUee + Vex andUee, respectively, or to the experimental timescale, thus, fullysuppressing tunneling ofe atoms and occupation of one siteby more than onee atom. The presence of an imaginary partbee of thee-e scattering length will give an effective nonzerowidth to the state with more than onee atom per site and can,therefore, only further suppress this tunneling by a Zeno-likeeffect [68, 69, 70].

Therefore,H(1,0) [Eq. (6)] is the only example that can beaffected by largebee. In order forH(1,0) to contain a non-negligible term proportional toJ2

e /Uee, the ratio |bee/aee|would need to be very small [71]. Several approaches toavoid the losses associated withbee in H(1,0) are possible.First, the large variety of stable atoms with two valence elec-trons (which includes not only alkaline-earths, but also Zn,Cd, Hg, and Yb) may have coincidentally an isotope withsmall |bee/aee|, which is more likely for lighter atoms [67].Second, while obtaining a good optical Feshbach resonance[51, 72, 73, 74, 75] to reduce|bee/aee| might not be possible,it should be possible to use optical Feshbach resonances toenhancebee and, thus, suppress [68, 69, 70] the virtual occu-pation of one site by twoe atoms;H(1,0) would then have thesame form as in Eq. (6), except withUee effectively set to in-finity. Notice that here we suggest to use optical Feshbachresonances to affecte-e scattering, which is different fromthe typical application tog-g scattering [51, 72, 73, 74, 75].Third, one can consider using a different ground state atom torepresent state|e〉, which would setVex = 0 in H(1,0). Fi-nally, one could simply use ane-lattice that is deep enough tomakeJe negligible, which would, however, lead to the loss ofterms inH(1,0) that exchange the pseudospin between neigh-boring sites.

Brief Review of Young Diagrams

Irreducible representations of SU(2) are classified accord-ing to the total half-integer angular momentumJ and have di-mension2J + 1. On the other hand, a (semistandard) Youngdiagram, instead of a single valueJ , is used to describe anirreducible representation of SU(N) for a generalN [76, 77].As shown in the example in Fig. S1, a Young diagram has allits rows left-aligned, has the length of rows weakly decreasingfrom top to bottom, and has at mostN rows. The dimensionof the representation corresponding to a given diagram is the

FIG. S2:(p,q) = (1,1) Young diagram.

number of ways to fill the diagram with integers from1 to Nsuch that the numbers weakly increase across each row andstrictly increase down each column. For our purposes, thenumber of boxes in the diagram is the number of atoms on thesite, and the diagram describes the (nuclear) spin symmetryofthe particular chosen single-site energy manifold. In particu-lar, columns represent antisymmetrized indices, while rowsare related to (but do not directly represent) symmetrized in-dices. It is the relation between antisymmetrized indices andthe columns that limits the number of rows toN . On the otherhand, since the full wavefunction (spin and orbital) on eachsite must satisfy complete fermionic antisymmetry, the rela-tion between rows and symmetrized indices and the fact thatwe have only two orbital states (g ande) force all our diagramsto have at most two columns.

The (p, q) = (1, 1) spin Hamiltonian and the spin-1 Heisenbergantiferromagnet

In the main text, we discussed two special cases of thespin HamiltonianH(p,q), both of which had a single-columnSU(N) representation on each site (i.e.q = 0). In this Sec-tion, we discuss the simplest SU(N) representation with twocolumns,(p, q) = (1, 1) [see Fig. S2]. It can be obtainedwhen there is oneg and onee atom per site in the electronicsinglet|ge〉 − |eg〉 configuration. SettingJe = 0 to avoide-ecollisions,H(p,q) reduces to

H(1,1) =J2

g

2(Ugg + Vex)

∑

〈i,j〉S2

ij . (S1)

The case ofN = 2 is the spin-1 antiferromagnetic Heisen-berg model. This model has a 1D ground state with hiddentopological structure [78]. Recently, applications of relatedmodels in one-way quantum computation have been proposed[79, 80]. Models with more complicated two-column repre-sentations may have exotic chiral spin liquid ground statesthatsupport non-Abelian anyons and that might thus be used fortopological quantum computation [81].

The Kugel-Khomskii model and the double-well phase diagram

In the main text, we omitted the values of the parame-ters in H(p,q) that characterize the Kugel-Khomskii modelH(1,0) [Eq. (6)]. In this Section, we present these parame-ters. We also present a detailed discussion of the double-wellcase phase diagram.

12

The parameters inH(p,q) that characterize the Kugel-

Khomskii modelH(1,0) [Eq. (6)] areλijg = −κij

g =2J2

g

Ugg

≡

−κg, λije = −κij

e =2J2

e

Uee

≡ −κe, κijge = −J2

e+J2

g

2U+eg

− J2e+J2

g

2U−

eg

≡

κge, λijge =

J2e+J2

g

2U+eg

− J2e+J2

g

2U−

eg

≡ λge, κijge =

JeJg

U−

eg

− JeJg

U+eg

≡ κge,

λijge =

JeJg

U−

eg

+JeJg

U+eg

≡ λge. To avoid loss ine-e collisions,

we assume for the rest of this Section thatUee = ∞ (seeSupplementary Information for a discussion of losses ine-ecollisions).

The nontrivial orbital-orbital, spin-spin, and spin-orbital in-teractions inH(1,0) [Eq. (6)] result in competing orders, withthe actual ground-state order dependent on the parameters ofthe HamiltonianH(1,0). To get a sense of the possible orders,we consider the caseN = 2 (with the spin states denoted by↑and↓) and discuss the double-well problem, with the wells de-noted byL (left) andR (right). Due to the large optical energyseparatinge andg, which we have ignored after Eq. (1), thethree manifolds of constantT z = T z

L + T zR (T z = −1, 0, 1)

should each be considered separately.The four states in theT z = 1 manifold, the subspace of

two e atoms, are|ee〉|s〉 and |ee〉|t〉. Here|ee〉 = |ee〉LR isthe orbital (or pseudo-spin) state, while|t〉 = | ↑↑〉LR, | ↓↓〉LR, 1√

2(| ↑↓〉LR + | ↓↑〉LR) and|s〉 = 1√

2(| ↑↓〉LR − | ↓↑

〉LR) are the triplet and singlet spin states. SinceUee = ∞,all four of these states have zero energy and the ground-statephase diagram is trivial.

The four states in theT z = −1 manifold (twog atoms) aresplit byH(1,0) into two energy manifolds:

|gg〉|t〉, E = 0, (S2)

|gg〉|s〉, E = −4J2

g

Ugg

. (S3)

Only |gg〉|s〉 can take advantage of the virtual tunneling sincetwo g atoms in the triplet spin states cannot sit on the samesite. Which of the two manifolds is the ground manifold de-pends on the sign ofUgg, as shown in the ground-state phasediagram in Fig. 3a. It is important to emphasize that forUgg < 0, the subspace of oneg atom per site may be sub-ject to extra loss down to the lower energy states that havebothg atoms in the same well. It is also worth noting that thediagram is only valid forJg ≪ |Ugg|.

Finally, the eight states in theT z = 0 manifold (oneg atomand onee atom) are split byH(1,0) into four energy manifolds:

|Σ〉|t〉, E = − (Jg + Je)2

U−eg

, (S4)

|τ〉|s〉, E = − (Jg + Je)2

U+eg

, (S5)

|τ〉|t〉, E = − (Jg − Je)2

U−eg

, (S6)

|Σ〉|s〉, E = − (Jg − Je)2

U+eg

, (S7)

where|Σ〉 = 1√2(|eg〉LR − |ge〉LR) and|τ〉 = 1√

2(|eg〉LR +

|ge〉LR) are anti-symmetric and symmetric orbital states, re-spectively. The denominatorsU−

eg andU+eg in the energies of

the |t〉 and|s〉 states, respectively, reflect the fact that tunnel-ing preserves the nuclear spin. At the same time, the± signsin the numerators can be understood by considering the caseJg = Je, when all states with overall symmetry under par-ticle exchange must have zero energy since for these statestunneling is forbidden due to the Pauli exclusion principle.The corresponding ground-state phase diagram as a functionof the signs and relative magnitude ofU+

eg andU−eg is shown in

Fig. 3b. As in the case of theT z = 1 phase diagram, negativeinteraction energies may lead to increased losses.

Effects of Three-Body Recombination

Three-body recombination [70, 82, 83, 84, 85] is a pro-cess during which three atoms come together to form a di-atomic bound state and a single atom, and both final productshave enough kinetic energy to leave the trap. While in certaincases, three-body recombination can be an asset [70], usuallyit results in the loss of atoms and, thus, limits the durationofthe experiment. For our purposes, we can describe three-bodyrecombination by a decay rateγ3 [70] resulting in a loss ofthree particles from one site. This rate will likely depend onwhat atomic states are involved and, to the best of our knowl-edge, has not yet been measured or calculated for fermionicalkaline-earth atoms.

Out of the four examples [Eqs. (6-8) and Eq. (S1)] that wediscuss, onlyH(1,1) [Eq. (S1)] andH(p,0) [Eq. (7)] may beaffected by three-body recombination (HKLM [Eq. (8)] as-sumes negligibleg-g interactions, such as in a very shallowglattice or with a low density ofg atoms). In the case ofH(1,1),two g atoms and onee atom occupy the same site virtuallyin the intermediate state that gives rise to the second orderspin Hamiltonian with interaction strength∝ J2

g /(Ugg+Vex).Thinking of γ3 as an effective linewidth for the intermediatestate,H(1,1) will be valid and losses small provided thatγ3 issmaller than the effective ”detuning”Ugg + Vex. Since scat-tering lengths for alkaline-earth atoms [51, 52, 54] are compa-rable to those for alkali atoms,Ugg + Vex can be on the orderof several kHz [62]. At the same time,1/γ3 for bosonic al-kali atoms in deep traps can be on the order of 1 s [86]. Ifγ3

were the same in our case,γ3 ≪ Ugg + Vex would be satis-fied. Ways of controlling the interactions via optical Feshbachresonances [51, 72, 73, 74, 75] may also be envisioned.

In the case ofH(p,0) [Eq. (7)], (nA, nB) = (1, 1) does notsuffer from three-body recombination.(nA, nB) = (1, 2) and(2, 2) may have three atoms per site virtually. As in the dis-cussion ofH(1,1), providedγ3 associated with threeg atomsper site is smaller thanUgg, these configurations should beaccessible. For the case(nA, nB) = (1, 2), γ3 ≫ Ugg isalso acceptable, since it will effectively prohibit the tunnel-ing of the atoms to the state with 3 atoms on a site [70], butthe interaction can still take place through the intermediatestate, in which an atom from aB site tunnels to anA siteand back. One can also envision ways to use optical Feshbach

13

FIG. S3:Square lattice valence plaquette solid for N = 4.WhenN = 4 andnA = nB = 1, four sites are required toform an SU(4) singlet; these singlets can in turn form theschematically shown plaquette-ordered state or a disorderedphase made of resonant plaquette states [87].

resonance techniques [51, 72, 73] to induce largeγ3. To beable to resolve the superexchange coupling∼ J2

g /Ugg in caseswherenA or nB is equal to 3, one must haveγ3 < J2

g /Ugg.Given that superexchange coupling can be as high as 1 kHz[62], this condition should also be achievable. AlthoughnA

or nB greater than 3 will result in even shorter lifetimes [84],there is a good chance that relatively largenA andnB can beachieved: at least, for bosonic alkali atoms in ann = 5 Mottinsulator state, the lifetime can still be as long as0.2 s [86].

The (p, 0) spin Hamiltonian with nA = nB 6= N/2

In the main text, we focused on one special case of the anti-ferromagnetic(p, 0) spin Hamiltonian on a square lattice, thatwith nA + nB = N (wherenA andnB denote the numberof atoms per site on the two sublattices). In this Section, wedescribe another interesting and experimentally relevantcase,nA = nB 6= N/2 [81, 87, 88, 89, 90, 91, 92, 93]. Poten-tial ground states include states built from valenceplaquettes(Fig. S3) [88, 89], resonantplaquette states [87], and topo-logical spin liquids [81, 90]. Valence plaquette states andres-onant plaquette states are the natural generalization of VBSstates and resonant valence bond states (RVB) [94], respec-tively; for example, whennA = nB = 1, N lattice sites areneeded to form a SU(N) singlet. Fig. S3 depicts a square lat-tice valence plaquette solid fornA = nB = 1 andN = 4.Techniques for detecting some of these phases are discussedin Ref. [81]. The experiment described in the main text forthe casenA + nB = N can also be generalized to probe thenA = nB 6= N/2 phase diagram including exotic phases suchas valence plaquette solids [Fig. S3], as well as competingmagnetically ordered states. The main difference is that afterpreparing a band insulator ofN g atoms per site, each siteshould be split not necessarily into two sites but into the num-ber of sites that is appropriate for the case being considered(e.g. 4 for the case shown in Fig. S3).

L R RL

FIG. S4:A schematic diagram describing the preparationof the double-well state |e, ↑〉L|g, ↓〉R.

Double-well Kugel-Khomskii and RKKY experiments

In the Methods, we have omitted the description of howto prepare the initial states for the proof-of-principle double-well Kugel-Khomskii and RKKY experiments. We presentthis description in this Section.

We first describe how to prepare an array of independentdouble wells in the state|e, ↑〉L|g, ↓〉R, which we use for theproof-of-principle experiment to probe the spin-orbital inter-actions in the Kugel-Khomskii model [95, 96, 97, 98, 99, 100,101]. After loading a band insulator of|g, ↓〉 atoms in a deepoptical lattice, an additional lattice for bothg (green) ande(yellow) atoms with twice the spacing of the first lattice isturned on in one direction to create an array of independentdouble wells [62]. Then, as shown in Fig. S4, in the presenceof ane-lattice bias,σ+ polarized light on resonance with the|g, ↓〉L → |e, ↑〉L transition can be used to prepare the state|e, ↑〉L|g, ↓〉R. For examples of earlier orbital physics stud-ies with ultracold atoms, where the orbitals are distinguishedonly by the different motional states of the atoms, we refer thereader to Refs. [102, 103, 104, 105, 106, 107] and referencestherein.

We now describe how to prepare the initial state1√2(|g, ↓

〉L + |g, ↓〉R)|e, ↓〉L|e, ↑〉R (see Fig. S5a) for the double-well proof-of-principle RKKY experiment, whose expectedNeel order parameterNez(t) for Vex = −Jg/10 is show inFig. S5b. The first step to prepare the initial state1√

2(|g, ↓

〉L + |g, ↓〉R)|e, ↓〉L|e, ↑〉R is to load a band insulator withthree|g, ↓〉 atoms per site on the long lattice and then slowlyramp up the short lattice with a bias so that it is energeticallyfavorable to have two atoms in the left well and one in theright well. Next one can change the state of the right atomfrom |g, ↓〉R to |e, ↑〉R by applying aπ pulse ofσ+ polar-ized light resonant with this single-atom transition. The leftwell will be unaffected because the spectrum is modified bythe interactions (if interactions alone do not provide the de-sired selectivity, one could, for example, change the bias ofthe e-lattice). The next step is to change the state of theleft well from two |g, ↓〉L atoms populating the lowest twovibrational states to|e, ↓〉L|g, ↓〉L both populating the low-est vibrational state. This can be accomplished by usingπ-polarized traveling wave laser light to apply aπ pulse reso-nant with the transition between these two many-body states[106]. This results in|e, ↓〉L|g, ↓〉L|e, ↑〉R. One can then

14

b

a

RL

FIG. S5:Proof-of-principle experiment to probe RKKYinteractions in an array of isolated double wells. a,Schematic representation of the initial state1√2(|g, ↓〉L + |g, ↓〉R)|e, ↓〉L|e, ↑〉R. b, In the limit

|Vex| ≪ Jg, the Neel order parameter for thee atoms[Nez(t) = 1

2 [ne↑L − ne↓L − (ne↑R − ne↓R)]] is

Nez(t) ≈ − 13 cos

(

Vext~

)

− 23 cos

(

Vext2~

− 3V 2ex

t

8Jg~

)

, which is

shown in red forVex = −Jg/10. It exhibits fast oscillationswith frequency∼ Vex, modulated by an envelope offrequency∼ V 2

ex/Jg induced by RKKY interactions

(− 13 − 2

3 cos(

3V 2ex

t

8Jg~

)

shown in blue).

temporarily shift theg and e lattices relative to each otherto setU±

eg interactions to zero, then makeJg nonzero, andwait until theg atom evolves into the desired superposition1√2(|g, ↓〉L + |g, ↓〉R) via tunneling. This yields the desired

state 1√2(|g, ↓〉L + |g, ↓〉R)|e, ↓〉L|e, ↑〉R.

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