Superfluid Phases of Dipolar Fermions in Harmonically Trapped Optical Lattices
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t Superfluid Phases of Dipolar Fermions in Harmonically Trapped Optical Lattices
Doga Murat Kurkcuoglu, Li Han, and C. A. R. S´a de Melo
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA (Dated: November 4, 2018)We describe the emergence of superfluid phases of ultracold dipolar fermions in optical latticesfor two-dimensional systems. Considering the many-body screening of dipolar interactions at in-termediate and larger filling factors, we show that several superfluid phases with distinct pairingsymmetries naturally arise in the singlet channel: local s -wave ( sl ), extended s -wave ( se ), d -wave( d ) or time-reversal-symmetry breaking ( sl + se ± id )-wave. We obtain the temperature versus fillingfactor phase diagram and show that d -wave pairing is favored near half-filling, that ( sl + se )-waveis favored near zero or full filling, and that time-reversal-breaking ( sl + se ± id )-wave is favored inbetween. The inclusion of a harmonic trap reveals that a sequence of phases can coexist in the clouddepending on the filling factor at the center of the trap. Most notably in the spatial region wherethe ( sl + se ± id )-wave superfluid occurs, spontaneous currents are generated, and may be detectedusing velocity sensitive Bragg spectroscopy. PACS numbers: 03.75.Hh, 03.75.Kk, 03.75.Ss, 67.85.-d
Ultracold heteronuclear molecules are very interest-ing quantum systems to study because they possesselectric dipole moments. This internal degree of free-dom adds richness to the nature of interactions betweenmolecules in comparison to interactions between atomsin purely atomic systems. Dipolar molecules can be ei-ther fermionic or bosonic in nature depending on theirconstituent atoms, and dipolar interactions allow for theemergence of quantum phases which may be extremelydifficult to be realized in condensed matter. Recently,ultracold dipolar molecules were produced optically [1]followed by their production from Bose-Fermi mixturesof ultracold atoms first in the vicinity of Feshbach res-onances [2], and later brought into their rovibrationalground state [3]. The production of these heteronuclearmolecules in harmonic traps has paved the way for stud-ies of the quantum phases of interacting dipolar bosonicof fermionic molecules. In the case of trapped clouds,a few quantum phases have been proposed for dipo-lar bosons including ferroelectric superfluids [4], Wignercrystals [5], while for dipolar fermions phases such asferroeletric [4] or ferro-nematic [6] Fermi liquids andBerezinskii-Kosterlitz-Thouless [7] or j -triplet [8] super-fluids have been suggested. In the case of optical lat-tices, additional phases such as supersolids [9] or micro-emulsions [10] have been proposed for dipolar bosons,while studies for dipolar fermions in optical lattices arejust beginning.A natural next step for experiments is the loading ofdipolar (heteronuclear) molecules in optical lattices. Wehave particularly in mind fermionic dipolar molecules,such as Na K, which seem to be stable against chem-ical reactions in their electronic-roto-vibrational ground-state [9]. Such molecules also have hyperfine structuredue to the nuclear spins, and a mixture of two hyperfinestates with sufficient long lifetimes may be created [12].In anticipation of these experiments, we discuss here thequantum phases of dipolar fermions in harmonically con- fined optical lattices, paying particular attention to theemergence of superfluid phases that break time reversalsymmetry spontaneously, as there are no confirmed ana-logues in condensed matter physics [13]. By includingthe effects of screening, we show that quantum phasesof dipolar fermions in harmonically confined optical lat-tices (two-dimensional geometry) can be approximatelydescribed by an extended Hubbard model for interme-diate and high filling factors, where only local on-siteand a few neighbor interactions are required. For attrac-tive local and nearest neighbor interactions, we derivethe phase diagram and establish all accessible phases inthe singlet channel. The most important phases corre-spond to d-wave superfluidity and to superfluid phasesthat involve a superposition of s -wave and d -wave com-ponents of the order parameter and that break time re-versal symmetry spontaneously. These phases naturallyarise due to the non-local nature of dipolar interactionsbetween heteronuclear molecules. In the broken time-reversal symmetry phases, spontaneous currents flow andcan be detected using experimental techniques such asvelocity sensitive Bragg spectroscopy [14, 15].The bare Hamiltonian for dipolar fermions in opticallattices for a two-dimensional system ( xy -plane) is H BA = − t X h ij i σ c † iσ c jσ + U X i n i ↑ n i ↓ + X i
4, the interactions are V x = V y = V = D (1 − α/ / [ a ǫ L ( a )], and become negative whenthe condition sin α > / a s toproduce U = −| U | <
0. We choose the electric field to beparallel to the lattice plane with angles α = π/ , φ = π/ H = − t X h ij i σ c † iσ c jσ − | U | X i n i ↑ n i ↓ − | V | X h ij i σσ ′ n iσ n jσ ′ (2)on a square lattice, with first few neighbors hopping andinteractions. In order to establish the quantum phasesas a function of filling factor ν , we start by constructingthe partition function Z = R D c † D ce S for the action S = Z β dτ "X iσ c † iσ ( τ )( − ∂ τ + µ ) c iσ ( τ ) − H ( c † , c ) . (3)By symmetry, the singlet order parameters for super-fluidity correspond to local s -wave ∆ sl , extended s -wave∆ se and d -wave ∆ d pairing [17]. Upon a simple and stan-dard integration of the fermionic degrees of freedom the action becomes S = − N s T X q,α | ∆ α ( q ) | V α + T r ln (cid:18) G − T − V T (cid:19) + µN s T , (4)where α = sl, se, d ; the interactions are V sl = | U | , V se = V d = | V | ; and the four-vector q = ( iν n , q ). The inversefree fermion propagator matrix is G − ( k, k ′ ) = (cid:18) iω n − ξ k iω n + ξ k (cid:19) δ k,k ′ (5)with kinetic energy ξ k = ǫ k − µ , band dispersion ǫ k = − t [cos( k x a ) + cos( k y a )], chemical potential µ ,four-vector k = ( iω n , k ), and unit cell length a . Thebandwith of the dispersion is w = 8 t . The additionalmatrix appearing in Eq. (4) is V ( k, k ′ ) = (cid:18) α ( k − k ′ )∆ ∗ α ( − k + k ′ ) 0 (cid:19) λ α ( k , k ′ ) , (6)where the Einstein summation over α is understood, and λ α ( k , k ′ ) are the symmetry factors for the order param-eters, which in the limit of zero momentum pairing ( k = k ′ ) become λ sl ( k , k ) = 1, λ se ( k , k ) = cos( k x a )+cos( k y a ), λ d ( k , k ) = cos( k x a ) − cos( k y a ).In terms of the quasiparticle ( γ = 2) or quasihole ( γ =1) energies E k ,γ = ( − ) γ p ξ k + | ∆ α λ α ( k ) | , where thesymmetry function λ α ( k ) = λ α ( k , k ), the effective actionbecomes S = − N s | U | T | ∆ sl | − N s | V | T ( | ∆ se | + | ∆ d | )+ S + µN s T . (7)Here, the second term in the action is S = P k ,γ ln [1 + exp( − E k ,γ /T )]. Notice that there are threepossible pure phases: local s -wave ( sl ); extended s -wave( se ) and d -wave ( d ). In addition, there are several possi-ble binary mixed phases sl ± se , sl ± d , and se ± d , whichdo not break time-reversal symmetry, and there are alsothose that do, such as sl ± ise , sl ± id , and se ± id . Lastly,several possible ternary mixed phases involving all threesymmetries sl , se and d may also exist. General Case:
The order parameter equations can beobtained by minimization of the action with respectiveto each order parameter. By taking δS/δ ∆ ∗ α = 0, with α = sl, se, d , we obtain∆ α = V α N s X k tanh( E k , / T )2 E k , Λ α ( k ) , (8)with symmetry factors Λ α ( k ) = λ α ( k ) (cid:2) ∆ α ′ λ α ′ ( k ) (cid:3) , where repeated indices α ′ indicate summation.The number equation that fixes the chemical potentialis obtained through the thermodynamic relation N = − ∂ Ω /∂µ , where Ω = − T ln Z is the thermodynamic po-tential. In the present approximation Ω = − T S , and thenumber equation reduces to ν = 1 N s X k (cid:20) − ξ k E k , tanh( E k , / T ) (cid:21) , (9)where ν = N/N s is the filling factor.Using the amplitude-phase representation, we writethe order parameters as ∆ sl = | ∆ sl | e iφ sl for the local s -wave symmetry, ∆ se = | ∆ se | e iφ se for the extended s -wavesymmetry, and ∆ d = | ∆ d | e iφ d for the d -wave symmetry.The critical temperature can be obtained by setting theorder parameters ∆ sl = 0, ∆ se = 0, and ∆ d = 0 inEqs. (8) and (9). In this case, the filling factor depen-dence of the critical temperature T c ( ν ) and the criticalchemical potential µ c ( ν ) can be obtained for pure sl -, se - and d -wave symmetries. The solutions for T c ( ν ) areshown in Fig. 1 for two cases | U | /w = 0 and | V | /w = 3 / | U | /w = 1 / | V | /w = 3 /
8, where thecorresponding superfluid phases are also indicated. Thephase diagram obtained within the saddle point approxi-mation is very accurate provided that both | V | /w ≪ | U | /w ≪
1, but it is only semi-quantitative when | V | /w < ∼ | U | /w < ∼
1. The phase diagram issymmetric about ν = 1, since the Helmholtz free en-ergy F = Ω + µN is invariant under the global particle-hole transformation µ → − µ and ν → − ν . Noticethat s -wave phases are favored at lower filling factors,while the d -wave phase is favored near half-filling, thisis directly correlated with the higher effective densityof states in this vicinity. The time-reversal-symmetry-breaking phases occur at filling factors between the s -wave and d -wave phases. Generally, when | U | /w = 0, theonly accessible phases are se -, d - and ( se ± id )-wave, andwhen | U | /w = 0 the only accessible phases are ( sl + se )-, d and ( sl + se ± id )-wave. se dse + id Ν T c (cid:144) w H a L se dse + sl + id Ν T c (cid:144) w H b L FIG. 1: Critical temperature T c /w versus filling factor ν atfixed interaction | U | /w = 0 in (a) or | U | /w = 1 / | V | /w = 3 /
8. Notice the tetracritical point where the normaland all superconducting phases meet.
Given that on-site interactions can be experimentallycontrolled, we focus our discussion at | U | /w = 0, whichalready contains the essential physics of superfluid phasesthat spontaneously break time reversal symmetry andhave a d -wave component. The Ginzburg-Landau theorynear T c is obtained by expanding the action of Eq. (7)in terms of the order parameters ∆ se , ∆ d and theircomplex conjugates. From the thermodynamic potentialΩ = − T S , we can calculate the Helmoltz Free energy F = Ω + µN . The free energy per site F = F/N s takes the simple form F = a se | ∆ se | + a d | ∆ d | + b se | ∆ se | + b d | ∆ d | +2 b sd (cid:2) cos(2 δφ ) (cid:3) | ∆ se | | ∆ d | + µ ( ν − , (10)when the thermodynamic potential Ω is expanded tofourth order in the order parameters using the action S defined in Eq. (8). The coefficients a and b dependexplicitly on the parameters of the model used. In thepresent case the possible phases se ± d are not accessible,and a tetracritical point exists where the normal and su-perconducting phases with se , d and se ± id symmetriesmeet. In addition, the free energy depends only on 2 δφ and does not distinguish between the phases se + id and se − id , which are thus degenerate. In the se ± id phases,time-reversal symmetry is broken but not chirality. Harmonic Trap:
The essential effect of an underly-ing harmonic trap V h ( r ) = kr / se -, d - and ( se ± id )-wave for | U | /w = 0 and sl + se , d - and ( sl + se ± id )-wave for | U | /w = 0. Within the local density approximation, wesolve the order parameter Eq. (8) and number Eq. (9)with µ → µ − V h ( r ). We obtain the profiles of the fillingfactor and order parameters, shown in Fig. 2, as a func-tion of dimensionless position from the center of the trap η = [ w/ (8 ǫ h )] / ( r/a ) , where ǫ h = ka /
2, and for param-eters | U | /w = 0, | V | /w = 3 / T /w = 0 . ν = 1 (half-filling) at the center of the trap. Noticethat as the filling factor decreases from the center of thetrap to its edge, all accessible phases emerge: d -wave su-perfluid at the center of the trap, followed sequentially byregions of ( se ± id )- an se -wave superfluid, and the normalstate. Similarly, in the case of | U | /w = 0 at low temper-atures and assuming that ν = 1 at the center of the trap,the sequence of phases from the center of the trap is d -,( sl + se ± id )-, ( sl + se )-wave superfluid followed by a nor-mal region at the edge. The interesting qualitative aspecthere is the emergence of regions where time-reversal sym-metry is spontaneously broken: ( se ± id ) for | U | /w = 0and sl + se ± id for | U | /w = 0. This is very important ina very broad sense, because there are no confirmed ex-amples in condensed matter physics of superfluids thatspontaneously break-time-reversal symmetry. [13] Spontaneous Currents:
In order to keep the discussionsimple, we continue to focus on the case of | U | /w = 0,and discuss the spontaneous current flow in the shell cor-responding to the ( se ± id )-wave superfluid. Considerfor example that either the se + id phase or the se − id phase is realized in the example of Fig. 2. Given that ei-ther chiral phase spontaneously break time-reversal sym-metry, it is expected that within the boundaries of the se + id ( se − id ) phase spontaneous currents circulateclockwise (counter-clockwise) near the outer boundary,and counter-clockwise (clockwise) near the inner bound-ary. To visualize the spontaneously generated currents, d sese + id Η Ν H a L d sese + id Η È D Α È (cid:144) w H b L FIG. 2: Spatially resolved filling factors ν in (a) and super-fluid order parameters ∆ α ( α = se, d ) in (b) as a functionof η = [ w/ (8 ǫ h )] / ( r/a ), for ν (0) = 1 (half-filling) at cen-ter of trap and parameters | U | /w = 0, | V | /w = 3 /
8, and
T /w = 0 . we perform a long-wavelength expansion of the action inEq. (4), which leads to the effective Free energy density F eff = F di + F nd + F h + F . The first term is F di = ∇ ∆ ∗ se c se,se m ∇ ∆ se + ∇ ∆ ∗ d c d,d m ∇ ∆ d , the second term is non-diagonal in the indices se and d F nd = h ∂ x ∆ ∗ se c se,d m ∂ x ∆ d − ∂ y ∆ ∗ se c se,d m ∂ y ∆ d + C.C. i , the third term is F h = γ se V h ( r ) | ∆ se | + γ d V h ( r ) | ∆ d | , while the last term F is given in Eq. (10). Addinga current source term − i∂ m − a m and consideringthe phase difference δφ = φ d − φ se = ± π/
2, weobtain within the se ± id phase the particle currentdensity J i = J i,φ + J i, | ∆ | , in Cartesian representation( i = x, y ). Here, J i,φ = m (cid:2) | ∆ d | c d,d + | ∆ se | c se,se (cid:3) ∂ i φ d is a phase-related contribution and J i, | ∆ | = m χ [ | ∆ se | c i,se,d ∂ i | ∆ d | − | ∆ d | c i,se,d ∂ i | ∆ se | ] is anamplitude-related component, where χ = sin( δφ ) = ± se ± id phase. In addition, the coeffi-cients c i,se,d satisfy the relation c x,se,d = − c y,se,d = c se,d .Given the existence of the harmonic potential, wetransform the currents to polar coordinates ( r, θ ), andrequire the radial current J r to vanish ( J r = ˆr · J = 0)at the boundaries between the se ± id and se occurringat r = R se and at the boundaries between se ± id and d phases occurring at r = R d . At these boundariesspontaneous currents flow only within the se ± id phase limits, since these are the only phases that breakspontaneously time-reversal symmetry. Under theseconditions, non-trivial solutions for φ d = χ [ π/ f ( r ) θ ]and φ se = χf ( r ) θ are possible with boundary con-ditions f ( r = R se ) = +1 , , f ( r = R d ) = − , and df ( r ) /dr | R se = df ( r ) /dr | R d = 0 . The spontaneouscurrents at the interface boundaries are tangential,having the forms J θ ( r = R se ) ≈ (2 χ/m ) | ∆ d | c d,d /R se and J θ ( r = R d ) ≈ − (2 χ/m ) | ∆ se | c se,se /R d , which canbe detected via Bragg spectroscopy as discussed next. Detection of time-reversal-symmetry-breaking:
A de-tection scheme of spontaneous currents using velocity sensitive Bragg spectroscopy [14, 15] is shown in Fig. 3with right- (left-) going beam of frequency ω ( ω ′ ) andlinear momentum k ( k ′ ). In Fig. 3a, circulating currentsare shown at the boundaries of the region for ( se + id )superfluidity, due to spontaneous breaking of time re-versal symmetry at lower temperatures. The case of anormal region (higher temperatures), where no sponta-neous currents exist, is shown in Fig. 3b for comparison.In Fig. 3a, the two dark spots making angles θ and π − θ with the horizontal satisfy the Bragg conditions due tothe Doppler shift creating by circulating currents in the( se + id ) region. Such Bragg spots are inexistent whenthere are no-circulating currents present, as is the case ofthe normal-state shown in Fig. 3b. FIG. 3: Schematic plots of a velocity sensitive Bragg spec-troscopy scheme to detect circulating currents of superfluid re-gions that break time reversal symmetry spontaneously. Thecase illustrated corresponds to | U | /w = 0 with filling factor ν = 1 at the center of the trap. Chemical and Collisional Stability:
The chemical andcollisional stability of candidate molecules is a veryimportant issue. Currently is known that fermionicmolecules such as LiCs and KRb are not chemically sta-ble [9], and tend to decay through collisions into Li and Cs or K and Rb , respectively, and thus are notideal candidates for the effects proposed here. How-ever, fermionic NaK is chemically stable, has a hyper-fine structure, and the hyperfine states in electronic-roto-vibrational ground state may have sufficiently long life-times [12], thus making it an ideal candidate for the ef-fects proposed here [16]. Conclusions:
We discussed screened dipolar fermionsin harmonically confined optical lattices modeled byan extended attractive Hubbard model, where both in-teractions and filling factors can be controlled. Wehad in mind particularly the fermionic dipolar molecule Na K which is chemically stable in its electronic-roto-vibrational ground state, but presents a hyperfine struc-ture allowing for the creation of two-mixed spin states.We established the superfluid phases in the singlet chan-nel and indicated that accessible phases have not onlypure s -wave or d -wave characters, but also mixed ( s ± id )-wave character which breaks time reversal symmetryspontaneously. We calculated the spatially-dependentprofiles of filling factor and order-parameter for varioussuperfluid phases, and proposed a Bragg spectroscopy ex-periment to detect the time-reversal symmetry breakingphase, which contains spontaneously circulating super-currents.We thank ARO (Grant No. W911NF-09-1-0220) forsupport. [1] J. M. Sage et. al., Phys. Rev. Lett. , 203001 (2005).[2] C. Ospelkaus et. al., Phys. Rev. Lett. , 120402 (2006).[3] S. Ospelkaus et. al., Phys. Rev. Lett. , 030402 (2010).[4] M. Iskin and C. A. R. S´a de Melo, Phys. Rev. Lett. ,110402 (2007).[5] H. P. B¨uchler et. al., Phys. Rev. Lett. , 060404 (2007).[6] B. M. Fregoso and E. Fradkin, Phys. Rev. Lett. ,205301 (2009).[7] G. M. Bruun and E. Taylor, Phys. Rev. Lett. , 245301(2008).[8] Y. Li and C. Wu, Scientific Reports , 392 (2012), see alsoarXiv:1005.0889 (2010). [9] I. Danshita and C. A. R. S´a de Melo, Phys. Rev. Lett. , 225301 (2009).[10] L. Pollet et. al., Phys. Rev. Lett. , 125302 (2010).[11] P. S. Zuchowski and J. M. Hutson, Phys. Rev A ,060703(R) (2010).[12] T. A. Schulze, I. I. Temelkov, M. W. Gempel, T. Hart-mann, H. Knckel, S. Ospelkaus, and E. Tiemann, Phys.Rev. A , 023401 (2013).[13] Although Strontium Ruthenate has been conjectured tobe a triplet p x ± ip y superconductor which breaks time re-versal symmetry spontaneously, experimental evidence isnot conclusive since no spontaneously circulating currentshave been observed.[14] S. R. Muniz, D. S. Naik, and C. Raman, Phys. Rev. A , 041605(R) (2006).[15] G. Veeravalli et. al. Phys. Rev. Lett. , 250403 (2008).[16] The suplemental material contains a derivation of thelattice Hamiltonian, a discussion about screening effects,and an analysis of the collisional properties of dipolarmolecules.[17] The inclusion of next-nearest neighbor terms represent-ing the screened dipolar interactions reveal its anisotropicnature even in the cases of φ = ± π/ , ± π/
4, and thefour-fold symmetric superfluid phases discussed here mayacquire a subdominant two-fold component.
Supplemental Materials: Superfluid Phases of Dipolar Fermions in HarmonicallyTrapped Optical Lattices
In this supplemental material, we provide details of the construction of the effective lattice hamiltonian in thepresence of long-range dipolar interactions, the effects of screening and a discussion of collisional properties of dipolarmolecules.To describe the superfluid phases of ultracold dipolar fermions in optical lattices for two-dimensional systems,first we derive in this supplemental material the lattice Hamitonian used to obtain distinct pairing symmetries thatnaturally arise in the singlet channel: local s -wave ( sl ), extended s -wave ( se ), d -wave ( d ) or time-reversal-symmetrybreaking ( sl + se ± id )-wave. Second, we discuss the screening effects of the dipolar interactions within the randomphase approximation. Finally, we comment on the effects of chemical and collisional stability of dipolar molecules,and suggest that fermionic NaK molecules are potentially a very good candidate for the emergence of the superfluidphases discussed in the main text. Effective Lattice Hamiltonian
To obtain the effective lattice Hamiltonian described in this manuscript, we start with dipolar molecules confinedto move in two-dimensions and described by the following continuum Hamiltonian H C = H SP + H SR + H LR , (S1)where the first term represents the single particle energy H SP = Z d r ψ † σ ( r ) h ˆ K + V P ( r ) i ψ σ ( r ) , (S2)where ˆ K = −∇ / (2 m ) is the kinetic energy operator (¯ h = 1) and V P ( r ) is a periodic potential that produces a squarelattice pattern. The second term represents the short-ranged (local) attractive contact interaction H SR = − g Z d r ψ †↑ ( r ) ψ †↓ ( r ) ψ ↓ ( r ) ψ ↑ ( r ) (S3)and the last term represents the long-range interactions H LR = 12 Z d r d r ′ V LR ( r , r ′ ) ψ † σ ( r ) ψ † σ ′ ( r ′ ) ψ σ ′ ( r ′ ) ψ σ ( r ) . (S4)Here, the long-range interaction is described by the term V LR ( r , r ′ ) = 14 πǫ (cid:20) q | r − r ′ | − q | r − r ′ + d | − q | r − r ′ − d | (cid:21) , which represents the Coulombic interaction between dipoles with effective charges + q and − q , separated by thecharacteristic distance | d | . All the dipoles are assumed to be aligned along the same direction of a large externalelectric field E , such that d k E . The position vectors r and r ′ reside on the xy plane.Noticing that the minima of the optical lattice potential V P ( r ) define the lattice sites, we can write the lattice-fermion creation operators as ψ † σ ( r ) = P i ϕ ∗ iσ ( r ) c † iσ and the anihilation operators as ψ σ ( r ) = P i ϕ iσ ( r ) c iσ . Here, theWannier functions ϕ iσ ( r ) obey the orthonormality condition R d r ϕ ∗ iσ ( r ) ϕ jσ ′ ( r ) = δ ij δ σσ ′ . In the local Wannier basis,each contribution to the Hamiltonian becomes H SP = − X iσ ǫ i c † iσ c jσ − X i = jσ t ijσ c † iσ c jσ , (S5)where the total on-site (local) energy is ǫ i = − R d r ϕ ∗ iσ ( r ) (cid:2) −∇ / (2 m ) + V P ( r ) (cid:3) ϕ iσ ( r ) is independent of the site due totranslational invariance ( ǫ i = ǫ ) , and the hopping matrix elements are t ijσ = R d r ϕ ∗ iσ ( r ) (cid:2) −∇ / (2 m ) + V P ( r ) (cid:3) ϕ jσ ( r ) . We take ǫ to be our reference energy and set the on-site energy ǫ = 0.The short-range interaction is written as H SR = X ijkℓ U ijkℓ c † i ↑ c † j ↓ c k ↓ c ℓ ↑ , (S6)where U ijkℓ = − g R dr ϕ ∗ i ↑ ( r ) ϕ ∗ j ↓ ( r ) ϕ k ↓ ( r ) ϕ ℓ ↑ ( r ) . Due to the orthonormality of the Wannier functions, the maincontribution to H SR comes from U iiii = U s , while otherwise U ijkℓ = 0. This implies that the contribution fromshort-ranged s-wave interactions is described by the on-site interaction H SR = X i U s c † i ↑ c † i ↓ c i ↓ c i ↑ , (S7)with U s = − g R d r | w ( r ) | , with g = 4 π ¯ h a s /m , and where we used the simplification ϕ i ↑ ( r ) = ϕ i ↓ ( r ) = w ( r ) . Similarly the long-range part of the Hamiltonian can be written as H LR = X ijkℓ V σσ ′ ijkℓ c † iσ c † jσ ′ c kσ ′ c ℓσ , (S8)where the general matrix element has the form V σσ ′ ijkℓ = 12 Z d r d r ′ V LR ( r , r ′ ) ϕ ∗ iσ ( r ) ϕ ∗ jσ ′ ( r ′ ) ϕ kσ ′ ( r ′ ) ϕ ℓσ ( r ) . Due to the orthonormality of the Wannier functions, the dominant contributions are those corresponding to V σσ ′ iiii = (1 / R d r d r ′ V LR ( r , r ′ ) | ϕ iσ ( r ) | | ϕ iσ ′ ( r ′ ) | which is effectively spin-independent, since the simplification ϕ i ↑ ( r ) = ϕ i ↓ ( r ) = w ( r ) holds in the present case, leading to V σσ ′ iiii = V / / R d r d r ′ V LR ( r , r ′ ) | w ( r ) | | w ( r ′ ) | . The on-site contribution of the long-range interactions can be written in Fourier space as V = R d k V LR ( k ) | w F ( k ) | , where V LR ( k ) is the Fourier transform of V LR ( r , r ′ ) = V LR ( r − r ′ ) , and w F ( k ) is the Fourier transform of | w ( r ) | .The last contribution to the lattice Hamiltonian is V σσ ′ ijji = (1 / R d r d r ′ V LR ( r , r ′ ) | ϕ iσ ( r ) | | ϕ jσ ′ ( r ′ ) | , which is alsoeffectively spin-independent, and corresponds to a density-density interaction with V σσ ′ ijji ≈ V LR ( r i − r j ) = V LR ( i, j ) . All the other terms from the tensor V σσ ′ ijkℓ are comparatively small due to the orthonormality of the Wannier functions,leading to the simplified expression H LR = X i V c † i ↑ c † i ↓ c i ↓ c i ↑ + X i Screening effects It is well stablished that screening effects are very important at sufficiently large densities for electronic materialswhich interact via long-ranged Coulomb forces [S1]. In such systems, the effective interactions between electrons canbe reduced to purely on-site or to on-site and nearest neighbors. Screening effects serve as the basis for the justificationof simplified lattice models in condensed matter physics, such as the Hubbard model, where only on-site interactionsare considered, or the extended Hubbard model with on-site and nearest neighbor interactions.In the case of dipolar interactions, screening can also be important if the density of dipoles is sufficiently large.If in real space the bare interactions V BA ( ρ ) depend only on the separation ρ = r − r ′ between particles located atpositions r and r ′ , then in momentum space the screened interactions, in their simplest description, can be expressedas a ladder sum of repeated interaction events [S1] leading to V SC ( q ) = V BA ( q )1 − V BA ( q ) P ( q ) , (S11)where P ( q ) is the zero-frequency polarization function for fermions, producing static screening of the bare interaction.At the first level of approximation P ( q ) can be replaced by the non-interacting polarization at zero frequency P ( q ) = X k n F ( ξ k + q ) − n F ( ξ k ) ξ k + q − ξ k . The standard approach used here is called the static random phase approximation (RPA) for screening [S1].The expression for the screened interaction in momentum space becomes V SC ( q ) = V BA ( q ) /ǫ ( q ) = V BA χ ( q ) , where ǫ ( q ) = 1 − V BA ( q ) P ( q ) is the dielectric function and χ ( q ) is the electric permittivity. The screening interactionpotential in real space becomes V SC ( ρ ) = 1 V D Z d r ′′ V BA ( r ′′ ) ǫ − NL ( ρ , r ′′ ) (S12)where the non-local screening function is ǫ NL ( ρ , r ′′ ) = χ − ( ρ − r ′′ ) . The screened interaction can be finaly written as V SC ( ρ ) = V BA ( ρ ) /ǫ L ( ρ ) , (S13)where the local screening function is defined to be ǫ L ( ρ ) = V BA ( ρ ) / (cid:2) V − D R d r ′′ V BA ( r ′′ ) χ ( ρ − r ′′ ) (cid:3) . In solids, for electrons interacting only via Coulomb repulsion, it is well established that screening plays a veryimportant role and leads to an effective lattice Hamiltonian that includes only local Coulomb (on-site Hubbard) inter-actions and nearest neighbors screened Coulomb (extended Hubbard) interactions, as can be inferred from standardmany-body textbooks [S1]. Such effective Hamiltonian is meant to describe quite accuratelly electrons interacing viaCoulomb forces in crystal structures at nearly any filling factor of the electronic band, with the sole exception ofvery low filling factor, where screening is not effective and the long-ranged nature of the Coulomb forces needs tobe taken into account. We performed a similar analysis here for long-ranged dipolar interactions using the randomphase approximation described above, and find that the long-range dipolar interactions in an optical lattice are weaklyscreened for filling factors ν < . 05, but they are strongly screened beyond ν > . 1. This indicates that for fillingfactors larger ν = 0 . 1, we need to consider at most interaction between the first few neighbors, which thus justifiesthe use of the screened Hamiltonian that produces the phase diagrams shown in Figs. 2 and 3 of the main text. Chemical and Collisional Stability Much of the experimental effort involving dipolar molecules has been devoted to heteronuclear dimers consistingof alkali atoms [S2–S5]. It is known experimentally that fermionic K Rb molecules are chemically unstable [S6]towards the formation of dimers K and Rb . In addition, theoretical work have shown that all heteronuclear Li dimerswill be subject to reactive trap losses, but all the remainder bi-alkali heteronuclear molecules should be stable withrespect to atom exchange collisions in their ground rovibronic state [S9]. Of the remaining stable bi-alkali heteronuclearmolecules (NaK , NaRb , NaCs , RbCs) , one of the best candidates for the observation of many-body effects caused bylong-ranged dipolar interactions is NaK, which can be fermionic Na K or bosonic Na K in nature. These systemsare currently being pursued by some groups [S7, S8]. Another serious candidate is bosonic RbCs, which is also beingexplored experimentally [S5]. The formation of trimers of the remaining stable heteronuclear molecules was alsotheoretically found to be highly endoenergic for ground robrational singlet states [S9], and it is also very likely to beendoenergic for the first few excited robrational singlet states. Nevertheless, additional experimental and theoreticalstudies of few body effects like dimer, trimer and tetramer formation and stability need to be performed. Althoughlosses are expected due to attractive interactions necessary for pairing and superfluidity of fermionic dipolar molecules,it is not yet known theoretically or experimentally how big losses will be. However, preliminary theoretical [S9] andexperimental [S7] work seem to suggest that fermionic Na K molecules are arguably the best candidate for the manybody effects that lead to the emergence of fermionic dipolar superfluidity with breaking of time reversal symmetry,as discussed in the main text. [S1] G. D. Mahan, Many Particle Physics , 3rd Ed., Kluwer Academic/Plenum Publishers (2000).[S2] J. Deiglmayr, M. Repp, A. Grochola, O. Dulieu, R. Wester, and M. Weidem¨uller, J. Phys. (Conf. Ser.) , 853 (2010).[S4] K.-K. Ni, S. Ospelkaus, D. Wang, G. Que’me’ner, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin,Nature , 1324 (2010).[S5] T. Takekoshi, L. Reichsllner, A. Schindewolf, J. M. Hutson, C. R. Le Sueur, O. Dulieu, F. Ferlaino, R. Grimm, H.-C.N¨agerl, e-print: arXiv1405.6037v1 (2014).[S6] D. S. Jin and J. Ye, Physics Today, May issue, pp 27-31, (2011)[S7] T.A. Schulze, I.I. Temelkov, M.W. Gempel, T. Hartmann, H. Knckel, S. Ospelkaus, and E. Tiemann Phys. Rev. A ,023401 (2013).[S8] I.Bloch, private communication (2014).[S9] P. S. Zuchowski and J. M. Hutson, Phys. Rev A81