Mott insulating phases and magnetism of fermions in a double-well optical lattice
MMott insulating phases and magnetism of fermions in a double-well optical lattice
Xin Wang, Qi Zhou, , and S. Das Sarma , Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, Maryland 20742, USA Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA (Dated: November 3, 2018)We theoretically investigate, using non-perturbative strong correlation techniques, Mott insulatingphases and magnetic ordering of two-component fermions in a two-dimensional double-well opticallattice. At filling of two fermions per site, there are two types of Mott insulators, one of whichis characterized by spin-1 antiferromagnetism below the Neel temperature. The super-exchangeinteraction in this system is induced by the interplay between the inter-band interaction and thespin degree of freedom. A great advantage of the double-well optical lattice is that the magneticquantum phase diagram and the Neel temperature can be easily controlled by tuning the orbitalenergy splitting of the two-level system. Particularly, the Neel temperature can be one order ofmagnitude larger than that in standard optical lattices, facilitating the experimental search formagnetic ordering in optical lattice systems.
PACS numbers: 05.30.Fk, 67.85.-d, 71.27.+a, 03.75.-b
Introduction .- There are currently worldwide efforts instudying collective properties of cold atoms either in asingle trap or in an optical lattice[1]. A central goalof these studies is to explore novel many-body quantumphases in both bosonic and fermionic systems. Whileboth bosonic and fermionic Mott insulators have beenrealized in laboratories[2–5], the experimental search formagnetism in optical lattices is currently on-going. Mostof these studies have been focusing on the single-bandphysics. For example, it is known that two-componentfermions in the lowest band can be used to study spin-1 / t /U is even smaller since U (cid:29) t typically. Asa result, the Neel temperature of antiferromagnetism inordinary optical lattices is far too low for experimentalobservation. Meanwhile, it is also challenging for the sys-tem to reach equilibrium because of the long relaxationtime. A scheme to enhance the relevant energy scales istherefore very desirable, particularly in the context of theexperimental study of many-body magnetism in optical lattice systems.In this Rapid Communication, we theoretically studyquantum magnetism of fermions in a double-well opticallattice. Instead of the usual spin-1 / / n s , n p ) = (1 , u (2 , − v (0 , Model .- We consider the Hamiltonian containing atight-binding-band part and an on-site interaction part,characterizing the two lowest bands (labeled by s and p respectively) in a symmetric double-well lattice, H = H band + H int . In the real space, the band part can be a r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov (a) s orbitalsp orbitals ε s ε p t sx t px (b) + ) (c) E+ EE T E S ) u v+ )) u v ! ! FIG. 1: (Color online) (a) Profile of the double-well potential(the green line) along the x -direction. The s (red lines) and p (blue lines) orbitals are schematically shown. The hoppingintegrals and energies for s ( p ) orbitals, t sx ( t px ) and ε s ( ε p ), areindicated. Note that the potential is not double-welled alongthe y -direction (not shown), where the hopping integrals are t sy and t py correspondingly. (b) ( n s , n p ) = (1 ,
1) eigenstatesin atomic limit, including triplet states with energy E T anda singlet state with energy E S . (c) Linear combinations of(2 ,
0) and (0 ,
2) states as eigenstates for the Hamiltonian inatomic limit, whose energies are denoted by E ± . See the textfor details. written as H band = (cid:88) r σ (cid:104) ( ε s − µ ) s † σ, r s σ, r + ( ε p − µ ) p † σ, r p σ, r (cid:105) + (cid:88) r σ (cid:16) − t sx s † σ, r s σ, r + x − t sy s † σ, r s σ, r + y + t px p † σ, r p σ, r + x − t py p † σ, r p σ, r + y + h.c. (cid:17) , (1)where s † σ, r ( p † σ, r ) creates a fermion with spin σ on the s ( p )orbital of site r , ε s and ε p are the energies for s and p orbitals, and µ is the chemical potential. The hoppingamplitude for s and p orbitals may differ in x and y direc-tions, thus we label them by t sx , t sy , t px , t py respectively.The interacting part of the Hamiltonian can be writtenas H int = (cid:88) r (cid:34) U s n s ↑ , r n s ↓ , r + U p n p ↑ , r n p ↓ , r + U sp ( n s ↑ , r n p ↓ , r + n s ↓ , r n p ↑ , r ) − U sp (cid:16) s †↓ , r p †↑ , r p ↓ , r s ↑ , r + p †↑ , r p †↓ , r s ↑ , r s ↓ , r + h.c. (cid:17) (cid:35) , (2)where n ασ, r = α † iσ, r α iσ, r ( α = s, p ) is the number oper-ator for orbital α at site r , U α = π (cid:126) a s M (cid:82) d xW α ( x )denotes the intra-band interaction, while U sp = π (cid:126) a s M (cid:82) d xW s ( x ) W p ( x ) denotes the inter-band inter-action, where a s is the scattering length, M is the massof the fermion, and W α ( x ) is the Wannier wave functionfor each band. The inter-orbital terms in Eq.(2) char-acterized by U sp are referred as density-density, spin-exchange and pair-hopping interaction. This model isessentially the rotationally invariant Slater-Kanamori in-teraction widely studied in transitional metal oxides [17].The main difference here is that the spin-exchange andpair-hopping are as strong as the inter-orbital density-density interaction. An important parameter which con-trols the multi-band physics is the energy level splittingbetween the two levels, defined as ∆ ≡ ε p − ε s . When∆ is small or intermediate, interactions between the twoorbitals give rise to interesting phenomena, as we dis-cuss below. When ∆ becomes very large the physicsreduces to that of the single-band model. The Hamil-tonian in Eq. (2) has been previously considered forfermions at resonance in an ordinary optical lattice inone-dimension[18]. In our case, the reduced band gapmakes the realization of a two-band system more practi-cal in current experiments. Moreover, higher dimension-ality of our system gives distinct physical phenomena notaccessible in one dimension.We start from the atomic limit, where the tunnelingterms are absent. We are interested in the states at fill-ing of two fermions per site, the schematics of which areshown in Fig. 1(b) and 1(c). When there is one fermionin each orbital, they form triplets which is denoted as( n s , n p ) = (1 , p †↑ s †↑ | (cid:105) , √ (cid:16) p †↑ s †↓ + p †↓ s †↑ (cid:17) | (cid:105) ,and p †↓ s †↓ | (cid:105) , with degenerate energy E T = 2( ε s − µ ) + ∆.The singlet state √ (cid:16) p †↑ s †↓ − p †↓ s †↑ (cid:17) | (cid:105) has a higher en-ergy E S = 2( ε s − µ ) + ∆ + 2 U sp . E S > E T sim-ply because two fermions interact with each other by s -wave short-range interaction. In the spirit of the Hub-bard model, atoms with different spins repel with eachother and atoms with the same spin do not interact.On the other hand, the two fermions can also form ad-mixtures u (2 , ± v (0 , E ± = 2( ε s − µ )+∆+ U p + U s ± (cid:114)(cid:16) ∆ + U p − U s (cid:17) + U sp . Bycontrolling ∆ in the double-well lattice, E − can be madeeither smaller or larger than E T . Throughout the paperwe fix parameters U s = 12 t , U p = 14 t , U sp = 12 t , andvary ∆ and the temperature T . Straightforward algebrareveals that at the critical value ∆ c = 4 t , E − = E T . Forlatter use we note that the hopping integrals are chosenas t sx = t sy = t py = t , t px = 2 t . The large t px stemsfrom the fact that p bands are spatially more extendedalong the x -direction. However, our solution to the lat-tice model as well as the physics therein does not dependon this particular set of parameters in any important way.When the hopping terms are switched on, we employthe single-site DMFT [16] to solve the strongly-correlatedinteracting lattice fermion problem. The key approxima- n s - n p µ /t (a)(b) ∆ /t=4 ∆ /t=4.5 n s + n p (a)(b) ∆ /t=0 ∆ /t=2.5 FIG. 2: (Color online) (a) Total occupancy ( n s + n p ) versuschemical potential µ , calculated for three different values of∆. The calculation is done without magnetic order at tem-perature T = 0 . t . ε s = 0 in this plot. The points (the circle,the square the diamond and the triangle) indicate the loca-tion at approximately the center of the gap for the line withcorresponding color, where we study magnetic ordering. (b)The difference in occupancy n s − n p plotted at the same µ scale. tion is the neglecting of momentum dependence of theself-energy: Σ ( k , ω ) → Σ ( ω ), which is solved iterativelyfrom an auxiliary quantum impurity problem plus a self-consistency condition. We use the matrix representationof the continuous-time hybridization-expansion quantumMonte Carlo impurity solver [19] prescribed specificallyfor multi-band interactions. This is a state-of-the-arthighly demanding numerical solution of the strongly in-teracting multi-band lattice Hubbard model in the con-text of our double-well optical lattice system. Mott physics .- There are multiple choices to fill a singlelattice site with two fermions, forming different types ofMott insulator. To distinguish them, we have calculatedboth n s + n p and n s − n p as functions of µ for differentvalues of ∆, as shown in Fig. 2(a). A Mott insulatinggap at filling two is evident for all cases, and there is noqualitative difference in the value of n s + n p between dif-ference cases. However, the difference in occupancy forthe two orbitals ( n s − n p ) shows distinct behaviors. Atvery large level splitting ∆ = 4 . t , n s (cid:29) n p . This is con-sistent with the analysis in the previous section for theatomic limit, where each lattice site is filled by the state u (2 , − v (0 ,
2) and u (cid:29) v . In contrast, at small ∆, e.g.as seen in Fig. 2(b) for ∆ = 2 . t and ∆ = 0, n s ≈ n p inthe Mott insulating regime. This indicates that on eachsite the triplet states dominate the ground state. This isalso consistent with the atomic limit where the energy of(1,1) states E T continuously decreases and eventually the(1,1) triplet becomes the ground state with decreasing ∆,as discussed in the previous section. The transition be- tween the two types of insulator is a crossover. Since inthis paper we focus on properties at non-zero tempera-ture, we shall not discuss the nature of this transition atzero temperature.For ∆ → ∞ , the magnetism is manifestly absent andthe ground state continuously connects to the trivialband insulator in the lowest band of an ordinary opti-cal lattice. For small and intermediate ∆, however, amagnetization of spin-1 may arise from the triplet stateson a single lattice site. As a result, the physics of mag-netic ordering in double-well lattices at filling two is farricher than that in standard optical lattices. Magnetic order .- When the (1,1) states dominate theon-site Fock states for small ∆, the interacting Hamilto-nian can be mapped to a spin-1 Heisenberg model, whichcan be written as H eff = (cid:88) r ( J x S r · S r + x + J y S r · S r + y ) , (3)where J x = 2 t sx U s + U sp + 2 t px U p + U sp , J y = 2 t sy U s + U sp + 2 t py U p + U sp , (4) S r = A † Σ A is a spin-1 operator, Σ is spin-1 Pauli matri-ces, and A = (Ψ † , Ψ † , Ψ †− ) T are creation operators fortriplet states p †↑ s †↑ | (cid:105) , √ (cid:16) p †↑ s †↓ + p †↓ s †↑ (cid:17) | (cid:105) , and p †↓ s †↓ | (cid:105) .Physically, the spin-exchange terms in Eq. (3) come fromthe exchange of fermions with different spins between thenearest-neighbor sites in either of the two orbitals. Bothorbitals contribute to the spin-exchange terms in the ef-fective Hamiltonian.In the one-dimensional case, Eq. (3) has previouslybeen derived in Ref. 18. For that case, it has been knownthat the one-dimensional spin-1 chain does not have anymagnetic order, rather the Haldane phase. Nevertheless,a two-dimensional spin-1 system can develop antiferro-magnetic ground states[20–22]. Therefore, one expectsto see antiferromagnetic spin ordering in a double-welloptical lattice when the gap ∆ is small and the temper-ature is low[23].To characterize the magnetization, we define m = | n s ↑ − n s ↓ + n p ↑ − n p ↓ | and solve the full Hamiltonian H = H band + H int . The results for m as a function ofthe temperature for different ∆ are shown in Fig. 3(a).Clearly, the magnetization arises below the Neel temper-ature (denoted by T Neel ) and saturates to its maximumvalue as the temperature approaches zero. For ∆ = 0, theantiferromagnetic ordering is most pronounced: it hasthe highest Neel temperature T Neel (cid:39) . t . However,this must be interpreted with caution because experi-mentally it is very difficult to tune the two bands overlapwith each other while keeping the tight-binding modelvalid. We therefore focus on the cases with nonzero ∆.As ∆ is increased, the magnetization drops faster as T increases, and the Neel temperature decreases. For a rel-atively large ∆ = 4 . t , T Neel (cid:39) . t . Note that this m T/t (a) ∆ /t=0 ∆ /t=2.5 ∆ /t=4 ∆ /t=4.5 ! /t T /t (b) FIG. 3: (Color online) (a) Total magnetization m = | n s ↑ − n s ↓ + n p ↑ − n p ↓ | versus temperature, for four selectivevalues of ∆. The chemical potential is selected at approx-imate center of the gap (see Fig. 2). (b) Color plot of themagnetization on the ∆- T plane. In the white regime there isno magnetic order, while for low T the magnetization reachesthe maximum value, indicated as dark colors. The Neel tem-perature is shown as blue dashed lines seperating colored andwhite regimes. Note that the y -axis does not start from zero:it starts from the lowest temperature T = 0 . t reached inthe DMFT calculation. value of ∆ is already above the critical value ∆ c = 4 t in the atomic limit where u (2 , − v (0 ,
2) is the groundstate. This indicates that many-body effects, such as thecorrelation between nearest-neighbor sites, enhance thethreshold of ∆ c for a finite m to emerge.To give a broader picture, we show in Fig. 3(b) a colorplot of the magnetization on a plane, of which the axesare the energy level splitting ∆ and the temperature T . The blue dashed line, separating colored and whiteregimes, indicates the Neel temperature, above which nomagnetic order is present. The dark color shown nearthe lowest accessible temperature T = 0 . t in our simu-lations characterizes the saturation of the magnetizationto its maximum value. As the temperature is increasedto intermediate values, the color turns to red, indicat-ing a moderate drop of the magnetization. When the temperature is close to the Neel temperature, the mag-netization drops rapidly, as can be seen from the narrowyellow edge. A close examination of the Neel tempera-ture reveals that it drops relatively slowly for ∆ < ∆ c ,but very rapidly for ∆ > ∆ c . This is consistent with thequalitative atomic picture. For ∆ → ∞ , the magneticordering and the corresponding Neel temperature wouldeventually vanish. However, we have shown that the Neelordering would survive at reasonably large values of ∆.This is remarkable, because previous analytical argumentof mapping to the spin-1 model[18] is valid for ∆ → m and T Neel are tunable by controlling ∆, which has obviousimportant experimental implications.
Enhancement of super-exchange interaction .- The in-crease of J [cf. Eqs. (3) and (4)] in a double-well latticecomes from two sources. First, as seen from Eq. (4), inaddition to t s , t px and t py also enter the expression forthe super-exchange interaction J . The large value of t px then enhances the amplitude of J , similar to the antifer-romagnetism arising from p bands alone [9]. Second andmost importantly, t sx itself is significantly enhanced ina double-well optical lattice. It has been shown that t sx can be increased by one order of magnitude at a given lat-tice depth for some realistic experimental parameters[14].Thanks to the potential barrier in the center of each lat-tice site of a double-well lattice, the Wannier wave func-tion of the lowest band spreads its weight toward theedge of the corresponding unit cell, which consequentlyenhances the overlap between Wannier wave functionson adjacent sites, leading to an increase in the tunnel-ing amplitude. As a result, the Neel temperature canbe strongly enhanced, easily by one order of magnitude.The larger energy scale associated with the tunneling andsuper-exchange interaction will also help to reach equilib-rim faster in the strongly interacting region. We empha-size that this spin-1 antiferromagntism originates fromthe unique feature of the double-well optical lattice: The s and p bands can be tuned close to each other, and theresulting magnetic ordering incorporates both bands. Itis this feature that distinguishes our theory from previousproposals regarding p bands alone[8, 9]. Conclusion .- Using non-perturbative ‘DMFT withcontinuous-time quantum impurity solver’ direct numer-ical techniques, we study two-component fermions in adouble-well square optical lattice, with two interactingorbitals per site. The Mott insulator at filling two is con-stituted either by triplet ( n s , n p ) = (1 ,
1) or an admixture u (2 , − v (0 , t p contributes to J and t sx is significantly enlarged in double-well lattices, theNeel temperature can be one order of magnitude largerthan that of the one-band system in ordinary optical lat-tices, thus perhaps enabling the direct experimental ob-servation of the elusive Neel antiferromagnetism in coldatomic systems. Our work should facilitate the search ofmagnetic order in optical lattice systems. Acknowledgements:
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