Emergence of Topological Fermi Liquid from a Strongly Correlated Bosonic System in Optical Superlattices
aa r X i v : . [ c ond - m a t . o t h e r] F e b Emergence of Topological Fermi Liquid from a Strongly Correlated Bosonic System inOptical Superlattices
Bo-lun Chen and Su-peng Kou ∗ Department of Physics, Beijing Normal University, Beijing 100875, China
Recent experiments on quantum degenerate gases give an opportunity for simulating strongly-correlated electronic systems in optical lattices. It may shed light on some long-standing puzzles incondensed-matter physics, like the nature of high-temperature superconductivity in cuprates thathad baffled people over two decades. It is believed that the two-dimensional fermionic Hubbardmodel, or t - J model, contains the key to this problem; but the difficulty of unveiling the mysteryof a strongly-interacting fermionic system is also generally acknowledged. Here, as a substitute,we systematically analyze the property of bosonic t - J model simulated in optical superlattices nearunit-filling. In particular, we show the emergence of a strange topological Fermi liquid with Fermisurfaces from a purely bosonic system. We also discuss the possibility of observing these phenomenain ultracold atom experiments. The result may provide some crucial insights into the origin of high- T c superconductivity. PACS numbers: 03.75.Hh, 03.75.Lm, 64.60.Cn, 74.20.-z, 74.20.Mn
In recent years, the physics community has witnesseda series of exciting discoveries and achievements. Amongthem, using ultracold atoms that form Bose-EinsteinCondensates (BEC) or Fermi degenerate gases to makeprecise measurements and simulations of quantum many-body systems, is quite impressive and has become arapidly-developing field[1, 2]. Since atoms are cooleddown to temperature near absolute zero and trapped inoptical lattices building from six orthogonal laser beams,they provide us a peaceful playground for manipulatingatoms with unprecedented accuracy. Some pioneeringworks[3] revealed the promising potential of applying ul-tracold atoms to make quantum computer and quantumsimulator: By changing the intensity, phase and polariza-tion of incident laser beams, one can tune the Hamilto-nian parameters including the dimension, the hoppingstrength and the particle interaction at will. Peoplehave successfully observed the Mott insulator–superfluidtransition in both bosonic[4] and fermionic[5, 6] degen-erate gases, and have demonstrated how to produce[7]and control effective spin interactions in a double-wellensemble[8]. All these evidences imply that an era inwhich atomic and optical physics unites with condensed-matter physics is within sight.Particularly, the two-dimensional fermionic Hubbardmodel (or t - J model) is one of the most interest-ing issues depicting the nature of high-temperaturesuperconductivity[9, 10, 11]. Ever since the discoveryof high T c cuprates, tremendous efforts had been con-tributed to investigations of this model. Derived fromHubbard model at half-filling (one electron per site), the t - J model describes the motion of doped holes in an an-tiferromagnetic (AF) background. It carries the essenceof a strongly correlated electronic system with intrinsiccompetition between superexchange interaction of spins ∗ Corresponding author; Electronic address: [email protected] and hopping processes of charge-carriers (holes). Overthe past two decades, people have employed many meth-ods and developed different schemes, hoping to fully un-derstand this model. Although a lot of consensus havebeen accomplished, there are still some ambiguities tobe clarified. For example, can superconductivity evolvefrom a purely fermionic repulsive many-body system?Are there exist some new phases of matter in high- T c superconductors?Therefore, some people suggest to use its bosonic coun-terpart, the bosonic t - J model[12, 13, 14], as a trial modelto investigate, for bosons are much more easier to dealwith in both analytic and numerical approaches. Afterobtaining some experiences and conclusions, we may usethem as reminders and analogies to original fermionicmodel. Besides, considering the present situation of ex-periments in ultracold atoms, where bosons are moreaccessible to be cooled and controlled, we believe it isworthwhile to explore the bosonic t - J model in opticallattices which can be formally written into two parts:ˆ H = ˆ H t + ˆ H J = − t X h ij i σ (ˆ a † iσ ˆ a jσ + H.c. )+ J X h ij i ˆS i · ˆS j , (1)where Hilbert space is restricted by the no-double-occupancy constraint P σ ˆ a † iσ ˆ a iσ ≤
1, ˆ a iσ annihilatesa two-component boson ( σ ≡↑ , ↓ denoting two internalstates) and ˆS i is the (pseudo)spin operator at site i , ˆS i = P αβ ˆ a † iα σ αβ ˆ a iβ with Pauli matrix σ = ( σ x , σ y , σ z ), h ij i denotes the nearest-neighboring counting. ˆ H t and ˆ H J describes hopping ( t >
0) and an effective AF superex-change (
J >
0) interactions respectively.In a seminal experiment[8], Bloch et al. loadeda two-component Rb condensate, | F = 1 , m F = 1 i ( ↑ )and | F = 1 , m F = − i ( ↓ ), into arrays of isolated doublewells. By changing the relative phase of the two laserstanding waves, the potential difference ∆ in one double-well can be raised or ramped down. The oscillation ofthe condensate after such manipulation contained infor- U− ∆ U+ ∆∆ (a)V FIG. 1: Illustrations of optical superlattices. (a) Constructionof an optical superlattice. Combining two sets of optical lat-tices (wavelength λ and 2 λ ) with a relative phase difference π to produce a superlattice. V is the depth of the poten-tial well. The superexchange processes are realized throughvirtual hoppings, resulting perturbations U ± ∆ in the Hamil-tonian. (b) A visualization of a 2D superlattice, blue atomsdenote one spin species ( |↑i ) and red atoms denote the otherspecies ( |↓i ). mation of the strength and sign of the effective superex-change energy J . With proper bias ∆, they could convertferromagnetic interaction ( J <
0) into antiferromagneticone (
J > π , we can build a superlattice (with shallowsites and deep sites), where a certain type of bosons be-ing trapped in a certain type of sites, for instance, |↑i inshallow sites and |↓i in its neighboring deep sites.By changing the shape of the superlattice, one can liftthe potential difference ∆ to a fixed value to realize theeffective AF superexchange interaction ( J > t iσ of different inner states of atoms in different sites to be identical. Therefore, we can intro-duce the hopping effect while keeping a(n) (inhomoge-neous) Heisenberg-type interaction in an optical super-lattice. (see Methods for details.)In addition, according to ref. [15], one can introducevacancies in a BEC trapped in one-dimensional (1D) op-tical lattice by pointing an electron beam at specific sitesto remove atoms. Due to the small diameter (100 ∼ t - J Hamiltonian near unit-filling, we analyze the dopingeffect in a topological perspective. We show the existenceof several exotic phases as vacancies being gradually in-troduced. Then we discuss how to detect these featuresin currently-available techniques in ultracold atom exper-iments.
I. HOLONS: BOSONS OR FERMIONS?
At unit-filling, the bosonic t - J model is simple and canbe reduced to AF Heisenberg model, ˆ H = J P h ij i ˆS i · ˆS j .A variational wave-function based on Resonating Va-lence Bond (RVB) picture by using Schwinger-boson de-scription can produce[17, 18, 19] an unrivaled accurateground-state energy; a generalized version[18] can furtherprovide the staggered magnetization and spin excitationspectrum precisely. So the bosonic RVB picture is a nat-ural choice for describing AF Heisenberg model. Explic-itly, we can introduced two flavors of bosons on each site,created by a canonical operator ˆ b † iσ acting on the vacuum | i without bosons, satisfying ˆ b † iσ ˆ b iσ = 1. This represen-tation is equivalent to the following operator identity be-tween the spin and boson operators ˆS i = ˆb † i σ ˆb i . Here ˆb i = (ˆ b i ↑ , ˆ b i ↓ ) T is a bosonic spinon annihilation operator.The mean-field value is characterized by a bosonic RVBorder parameter ∆ sij = h ˆ b iσ ˆ b j, − σ i 6 = 0 for the nearest-neighbor sites, which depicts the short-range AF corre-lation as h ˆS i · ˆS j i = − | ∆ sij | . At zero temperature,spinon ˆb becomes massless and Bose-condensation takesplace with h ˆb i 6 = 0, corresponding to the long-range N´eelorder in x - y plane.To learn the property of an AF order with vacan-cies, we should generalize the idea of spin-charge sep- aration , which has become a very basic concept in un-derstanding the doped Mott insulator related to high- T c cuprate[10, 20]. Unlike a usual quasi-particle that carriesboth spin and charge quantum numbers in conventionalmetals, it states that the system has two independent el-ementary excitations, the neutral spinon and the spinlessholon. It is assumed that usual quasi-particle excitationsmay no longer be stable against the spin-charge separa-tion mechanism once being created, e.g., by injecting abare vacancy into the system; and it has to decay intomore elementary spinons and holons. In other words,to theoretically describe the introduction of a single va-cancy, one has to first annihilate a particle state | Ψ i to-gether with a bosonic spinon ˆ b iσ , then generate a spinlessoperator ˆ h † i denotes a holon (a vacancy) asˆ a iσ | Ψ i = ˆ h † i ˆ b iσ | Ψ i , | Ψ i = Y iσ ˆ b † iσ | i . (2)In addition, ˆ h † i and ˆ b iσ should satisfy the no-double-occupancy constraint, ˆ h † i ˆ h i + P σ ˆ b † iσ ˆ b iσ = 1. In contractto the case of high- T c cuprates, holons in bosonic t - J model are neutral for they are actually the absence ofcold atoms in a certain site.However, the situation changes when the hole moves.According to Marshall[21], the ground-state wave func-tion of the Heisenberg Hamiltonian for a bipartite lat-tice satisfies a sign rule. It requires that flips of twoantiparallel spins at nearest-neighbor sites are alwaysaccompanied by a sign change in the wave function: |· · · ↑↓ · · · i 7→ ( − × |· · · ↓↑ · · · i . To show the sign effectin detail, we divide a bipartite lattice into odd ( A ) andeven ( B ) sublattices and assign an extra sign ( −
1) to ev-ery down spin at A site. When the hole initially locatingat site i hops onto a nearest-neighbor site j , the Marshallsign rule is violated, resulting in a string of mismatchedsigns on the vacancy’s course[22]. The spin wave functionis changed into | Ψ i 7→ | ˜Ψ i = ( − P i ˆ n bi ∈ A, ↓ | Ψ i , where ˆ n bi ∈ A, ↓ = ˆ b † i ∈ A, ↓ ˆ b i ∈ A, ↓ is the number of down spinon each A -sublattice. In particular, if the hole movesthrough a closed path C on the lattice to return to itsoriginal position, it will get a Berry phase ( − N ↓ C , where N ↓ C is the total number of down spins “encountered” bythe hole on the closed path C .[23, 24, 25] This process isillustrated in Fig. (2a).To deal with this unavoidable Berry phase ( − N ↓ C in the ground-state wave function when there is mo-bile hole, we introduce a phase-string transformation[23] | ˜Ψ i 7→ e i ˆΘ | ˜Ψ i = | Ψ i , where ˆΘ = P ij θ ij ˆ n hi ˆ n bj ↓ , ˆ n hi andˆ n bj ↓ are occupation number operators of the hole anddown-spins, with a phase factor θ ij =Im[ln( z i − z j )]. Here z ≡ x + iy denotes position and the subscripts i and j denote lattice sites. Considering the single-occupancyconstraint, ˆΘ = − P ij ˆ n hi θ ij [1 − ˆ n hj − P σ ( − σ ˆ n bjσ ], where ( − ↑ ≡
1, ( − ↓ ≡ −
1. The phase-shift factor e i ˆΘ can also be regarded as a unitary transformation onan arbitrary operator: ˆ O e i ˆΘ ˆ Oe − i ˆΘ . For example, op-erators of holons ˆ h † i and spinons ˆ b iσ are transformed asfollow, e i ˆΘ ˆ h † i e − i ˆΘ = ˆ h † i e − i P j θ ij ˆ n hj + i P jσ ( − σ θ ij ˆ n bjσ − i P j θ ij ,e i ˆΘ ˆ b iσ e − i ˆΘ = ˆ b iσ e − i P j ( − σ θ ij ˆ n hj . Now we have restored the simple spin wave function | Ψ i as equation (2) originally defined, but have got a set ofnontrivial operators, and it is just the aim of the abovetransformation.Furthermore, by defining ˆ h ′† i = ˆ h † i e − i P j θ ij ˆ n hj , we cansee that except for a phase factor e i P j θ ij , holon ˆ h i andspinon ˆ b iσ are symmetric,ˆ h i ˆ h ′ i e − i P jσ ( − σ θ ij ˆ n bjσ × e i P j θ ij , ˆ b iσ ˆ b iσ e − i P j ( − σ θ ij ˆ n hj ; (3)namely, ˆ h i h → b, i → iσ ˆ b iσ and ˆ b iσ b → h, iσ → i ˆ h i . The phasefactor e i P j θ ij means an additional lattice π -flux-per-plaquette for holons. From equation (3), one can seethat there exists a mutual semionic statistics betweenholons and spinons, where holons perceive spinons as π -vortices and vice versa. In particular, after the trans-formation ˆ h ′† i = ˆ h † i e − i P j θ ij ˆ n hj , ˆ h ′† i obeys fermionic anti-commutation[26], { ˆ h ′† i , ˆ h ′ j } = δ ij . Finally, based on the RVB ground state by consideringthe phase string effect[23, 24, 25], the effective Hamilto-nian of holons in bosonic t - J model can be written asˆ H h = − t h X h ij i ( e i ˆ a ij − iφ ij ˆ h ′† i ˆ h ′ j + H.c. ) +∆2 X i ( − i ˆ h ′† i ˆ h ′ i + µ X i ˆ h ′† i ˆ h ′ i , (4)where t h ≈ | ∆ sij | t is the effective hopping amplitudeof holons, µ is the chemical potential. The gaugefield ˆ a ij satisfy the topological constraint P C ˆ a ij = ± π P l ∈ C (ˆ n bl ↑ − ˆ n bl ↓ ) for a closed loop C , φ ij describesa π flux per plaquette, P (cid:3) φ ij = ± π . It is obvious thatthe difference between fermionic t - J model and bosonic t - J model is the statistic of holons: in former case, holonsare bosons; and in latter case, they are fermions.Now we may be able to answer the question: what isthe fate of a holon in an AF order? As we have mentionedabove, the AF order lying in x - y plane can be seen asa Bose condensation of spinons, h ˆ b iσ i 6 = 0. Accordingto the above effective Hamiltonian (4), and the quantityˆ S † i = ( − i ˆ b † i ↑ ˆ b i ↓ e i P j θ ij ˆ n hj , a holon introduced here is atopological defect carrying spin twists and changing itsperipheral spin configuration, as a meron-like object[27,28, 29, 30], such a spin vortex (for a holon at the origin)may be characterized by a unit vector n i = ( − i h S i i /S , n i = r i | r i | , r i = x i + y i . This meron-like spin configuration is schematically shownin Fig. (2b).By now, one may imagine that a single hole wouldbecome a meron in a long-range AF order. However,the answer is not quite right. For a single meronconfiguration, the energy is logarithmically divergent, E ≈ J | ∆ sij | ln ( L/a ) with L the size of the system.In order to remove this infinite-energy cost, as in thepresent case, each holon-meron has to “nucleate” ananti-meron from the background spontaneously. De-fine n xi + in yi = e iφ + iφ i , with the unit vector n ≡ (cos φ , sin φ ) as the magnetization direction at infin-ity. In presence of a fermionic holon centered at − z k / z k /
2, we have[31, 32] φ ki = Im ln [( z i − z k / / ( z i + z k / z k ≡ e xk + ie yk .By using ˆe k = ( e xk , e yk ) to denote the spatial displacementof the holon and anti-meron centered, a dipolar spin con-figuration at a sufficiently large distance is obtained, φ i ≈ ( ˆz × ˆe k ) · r i | r i | , | r i | ≫ | ˆe k | . Thus, each pair of holons and anti-merons forms a com-posite, as shown in Fig. (2c). In contrast to the loga-rithmically divergent meron-energy, the energy a dipole E d becomes finite[31, 32], E d ≈ J | ∆ sij | ln [( | ˆe k | + a ) /a ], | ˆe k | & a .Physically, one may consider a bare hole (spinlessholon) created by an annihilation operator ˆ h † at point a , then it may jump to point b via some discrete steps,being connected by a phase string in between. Since theholon can reach b through different virtual paths orig-inated at a , this singular phase string is then replacedby, or relaxes to, a smooth dipole configuration. An anti-meron itself is a semi-vortex formed by condensed spinonsand it is immobile. Therefore, the hole-dipole as a wholemust remain localized (self-trapped) in space. The re-sulting spin configuration has a dipolar symmetry whilethe distortion of the direction of magnetization is longranged and decays as r − [32, 33]. II. TOPOLOGICAL FERMI LIQUID
In the lightly doped region, δ →
0, there exist local-ized holes that are self-trapped around anti-merons via alogarithmically confining potential V ( r ) ≈ q ln ( | r | /a ), q = 2 πJ | ∆ sij | , and their dipolar moments are randomlydistributed. Such kind of AF ordered state with randomdipoles has been studied in refs. [31, 36, 37, 38]. It isknown that when T = 0, the spin-correlation length is fi-nite for arbitrary doping δ . This implies the destructionof long-range AF order, as long as δ >
0. Consequently, -1-1-1 (a) e k -z k /2 ba z k /2 (c) FIG. 2: (a) A schematic demonstration of the phase stringeffect on a long-range AF order background. The point in A ( B )-sublattice is denoted by red(blue) arrows. The double-dot path is the course that a hole (black open circle) movesthrough. For clarity, spins on this path is plotted in lightcolors. Once the hole’s hopping results a down spin on A -sublattice, there is a sign change ( −
1) over the wave function | Ψ i , as described in the text. (b) An illustration of a meronconfiguration. Due to the mutual statistics, the vacancy hasbecome a fermionic particle. (c) An illustration of a holon-dipole configuration. It is a confined object composed of aholon-meron and an anti-meron at two poles: a ( − z k /
2) and b ( z k / ˆe k as a dipole moment. AF order is limited mainly in finite sizes, where the size ξ of a domain is determined by the hole’s concentration, ξ ≈ a/ √ δ . Accordingly, this state has been termed as acluster spin glass[39, 40]. The spin glass freezing temper-ature is then expected to vary as T g ∝ ξ ∝ a /δ , belowwhich holons tend to form a glass and their dynamicsstrongly slows down.When doping increases, more and more dipoles ap-pear. As a result, the “confining” potential V ( r ) be-tween these dipoles will be screened by polarizationsof pairs lying between them. When this screening ef-fect becomes so strong that even the largest pair has tobreak up, holons and anti-merons are liberated to moveindividually. This qualitative analysis suggests that alocalization-delocalization phase transition should occurat a critical point δ c , in a fashion of the Kosterlitz-Thouless (KT) transition. Using the renormalizationgroup method[31, 35], one can determine the critical holedensity δ c = δ c ( T ) or temperature T de = T de ( δ ) at whichdipoles collapse and holon-merons are “deconfined” fromthe bound state with anti-merons. At zero temperature,the critical hole density has been numerically determinedas δ c ( T = 0) ≈ . π = 0 . . Therefore, a zero-temperature quantum critical point ex-ists at δ c = 0 .
043 where hole-dipoles dissolve into holon-merons and anti-merons.When δ > . repulse each other, for theyare merons with same topological charges, then thesefermions cannot pair with each other and it makes theFermi surface stable. This is different from the case whereelectrons form Cooper pairs in superconductors, becausehere we have topological repulsions not electronic repul-sions.Thus, we display the global phase-diagram of bosonic t - J model in Fig. (3). There exists three different regionsglobally: region I, AF insulator, δ = 0; region II, insu-lating spin-glass (SG), δ ∈ (0 , . δ ∈ (0 . , . k B T ≈ c (1 − δ ) J, c is an integration quantity (seeMethods). Particularly, our results illustrate the emer-gence of an exotic fermionic state (TFL) in a pure bosonicsystem.In TFL state, the effective Hamiltonian of holons canbe written asˆ H h = − t h X h ij i e iφ ij ˆ h ′† i ˆ h ′ j + ∆2 X i ( − i ˆ h ′† i ˆ h ′ i + µ X i ˆ h ′† i ˆ h ′ i , and the dispersion relation is E k = µ ± q t h [cos ( k x ) + cos ( k y )] + ∆ , s = 0 c c III: TFLII:SG
Fermi Energy CurveIII: Topological Fermi LiquidI: AF Insulator II: Spin Glass T de DelocalizationLocalization k B T / J k B T / J / J FIG. 3: The global phase diagram. The solid black line indi-cates the phase boundary where | ∆ sij | = 0. Different regionsare demonstrated in the text. The inset depicts detailed phaseboundaries as well as the quantum critical point δ c in low dop-ing region ( δ < . µ ( δ ) implies theexistence of Fermi surface of deconfined holons. It is scaledby the superexchange energy J as shown in the inset, wherewe have chosen U/t = 12, namely, t/J = 3 and ∆ /J = 50. as Fig. (4) shows.In Fig. (5), we show the evolution of Fermi surfacesby plotting four doping cases, δ = 0 .
05, 0 .
1, 0 .
15 and 0 . t - J model of 15% hole concen-tration, of which highest Fermi energy with clear Fermisurfaces are predicted.Besides, when holon moves, the spin configurationchanges simultaneously, thus there is no long-range AForder anymore. Instead, the magnetic ground-state be-comes a spin liquid state with invariant spin-rotation andtranslation symmetry. Without anti-merons matchingwith holons, holons exert an “effective magnetic field” onspinons (recall the mutual relation between the two exci-tations), B h = πρ h with ρ h the density of holons. Hence,a new length scale is introduced to the spinon system,which is the magnetic cyclotron length l c = B − / h . l c will later be connected to the remaining magnetic corre-lation length.As we mentioned above, doping creates a Landau-levelstructure in the spinon spectrum. Thus, low-lying spinfluctuations are expected to be sensitive to dopants. Af-ter calculations, we find χ ′′ ( q , ω ) ∼ χ ′′ ( Q , ω ) e −| q − Q | l c / . Here χ ′′ ( q , ω ) is defined as ( β = 1 /k B T ) χ ′′ ( q , ω ) = 12 (cid:0) − e − ωβ (cid:1) Z dtd r e i ( ωt − q · r ) h S ( r , t ) · S (0 , i (5) FIG. 4: The dispersion relation of the topological Fermi liq-uid. The relative energy E k − µ of the upper branch is shownin the unit of t with ∆ /t = 20.FIG. 5: Illustrations of Fermi surfaces. The doping concen-trations are shown as titles of individual plots. where h S ( r , t ) · S (0 , i is the spin-spin correlation func-tion. This is a Gaussian type around the AF wave-vector Q = ( ± π, ± π ) /a with a width l − c . Conse-quently, it determines the spin-spin correlation in thereal space as cos ( Q · r ) e −| r | /ξ , where the correlationlength ξ = √ l c = a p / ( πδ ) is in the same order ofthe average hole-hole distance[41]. Namely, doped holesbreak up the long-range AF correlation into short-range AF fragments within a length scale of ξ .In contrast, the ground-state of fermionic t - J modelis a superconducting state of condensed holons, inwhich spin liquid state is always masked by holons’condensation[42]. While in bosonic t - J model, it is thetopological Fermi liquid state with massless fermionic ex-citations, in which one can easily detect properties of thisspin liquid state. For these reasons, the observation ofthis exotic quantum state (TFL with short range spincorrelation) in bosonic t - J model may pave an alterna-tive approach to verify the microscopic theory of high T c superconductivity. III. DETECTIONS AND SUMMARY
In region I and II, the system is an insulator in thefirst place, so the momentum distribution is universal andisotropic in the Brillouin zone (without considering thetrapping and boundary conditions). This feature can bedetected by measuring density profile measurement[44],or via Bragg spectroscopy[43] to observe the momentumspectrum. Additionally, in the N´eel state (I), the con-densate exhibits long range AF order, while in the spinglass state (II), this long ranged magnetic order disap-pears. This difference can be distinguished through spa-tial spin-spin correlations, h ˆ S z ( r ) ˆ S z ( r ) i . As demon-strated in refs.[45, 46], using a probe laser beam whichgoes through the condensate, one can measure the phaseshift or change of polarization of the outgoing beam h ˆ X out i to obtain the magnetization h ˆ X out i ∝ h ˆ M z i ∝ R d r φ ( r ) h ˆ S z ( r ) i , where φ ( r ) is the spatial intensity pro-file of the laser beam. The quantum noise h ˆ X i ∝h ˆ M z i ∝ R d r d r φ ( r ) φ ( r ) h ˆ S z ( r ) ih ˆ S z ( r ) i revealsthe atomic correlations in the system. This method candirectly examine the existence of 2D AF correlations.In region III, we have demonstrated that one naturalcharacteristic of TFL is the existence of Fermi surfacesin the Brillouin zone. Esslinger et al.[47] had successfullyobserved the Fermi surface in a 3D optical lattice filledwith fermionic atoms. Similarly, one may also observethese Fermi levels in our bosonic system, as shown in Fig.(5). Besides, the short-range AF fluctuations may alsobe observed. As equation (5) indicates, the spin dynamicstructure factor as well as the dynamic spin susceptibilityfunction are linked with spin-spin correlations. The latterone reflects the effective short-range magnetic correlationlength ξ .To sum up, in this paper, we first demonstrate theimplementation of bosonic t - J model in optical super-lattices filled with a two-component BEC. And for thefirst time, we systematically discussed the possible quan-tum phases of this model upon doping. A key featureis the mutual semionic statistics between the two ele-mentary excitations of the system, holons and spinons.When there are only a few holes around, they tend toform hole-dipoles, locally changing the underlying spintexture into meron-anti-meron pairs. At low doping, thedeformed spin configuration loses the long-range order,and there emerges a spin glass state. When holes are pre-vailing, through a quantum phase transition ( δ c = 0 . IV. METHODSA. Perturbation Theory in Large U Limit
The two-component Bose-Hubbard model in a biasedsuperlattice can be generally written asˆ H = − t X h ij i σ [ˆ a † iσ ˆ a jσ + H.c. + ∆2 t σ (ˆ n iσ − ˆ n jσ )] + U X i ˆ n i ↑ ˆ n i ↓ + U ′ X i ˆ n iσ (ˆ n iσ − , (6)where t = p /πE r ( V /E r ) / e − √ V/E r [7] with atomicrecoil energy E r = ~ k / m , k is the wave vectorof laser, m is the atomic mass, V describes exter-nal potential. The inter-species repulsion is defined as U = p /πka s E r ( V /E r ) / with a s the s -wave scatter-ing length among species, and the intra-species repulsionis U ′ = p /πka ′ s E r ( V /E r ) / , a ′ s is the correspond-ing scattering length. This term vanishes at exact unit-filling, but can contribute in higher-order tunneling pro-cesses.In the large U limit ( U ≫ t ) and nearly unit-filling n .
1, which is actually a prerequisite for building theaforementioned superlattices and is naturally satisfied,the hopping term can be treated as a perturbation. Thus,we introduce two projection operator: P projects theinitial Hilbert space onto the single-occupancy subspace( | i i , |↑i i , |↓i i ), Q = 1 − P projects onto the double-occupancy subspace ( |↑↓i i ). We further divide equation(6) into ˆ H + ˆ T mix , where ˆ H describes processes in P subspace and ˆ T mix mixes the upper and lower band viavirtual tunnelings.Applying canonical transformation ˆ H eff = e − ˆ S ˆ He ˆ S and eliminating first order terms of ˆ T mix , we findˆ H eff = ˆ H + 12 h ˆ T mix , ˆ S i , ˆ S = X mn | m i h m | ˆ T mix E n − E m ! | n i h n | . The projector | ǫ i h ǫ | can be chosen as P and Q . Energydifferences between upper and lower band are E Q − E P ≈ U ± ∆ , U ′ ± ∆ thus ˆ S = ( ˜ U − +2 ˜ U ′− )( P ˆ T Q−Q ˆ T P ) with˜ U = U − ∆ /U , ˜ U ′ = U ′ − ∆ /U ′ . After calculating the detailed terms, we can obtain the effective Hamiltonianas ˆ H eff = − t X h ij i σ (ˆ a † iσ ˆ a jσ + H.c. ) + X h ij i [ J z ˆ S zi ˆ S zj − J ⊥ ( ˆ S xi ˆ S xj + ˆ S yi ˆ S yj )] , (7)where J z = 4 t ˜ U − t ˜ U ′ , J ⊥ = 4 t ˜ U .
When a s and a ′ s do not differ very much (this conditioncan somehow be realized in experiments), J z ≈ − t ˜ U + ǫ with ǫ denoting a small spatial inhomogeneity. In fact,this small inhomogeneity is necessary in our following dis-cussions, for it can be seen as an analogy for the effectiveinter-layer-coupling in high- T c cuprates.At the first glance, it seems that bosons (say, |↑i ) inshallow sites may hop to neighboring deep sites moreeasily due to lower energy barrier they feel, comparingtheir counterparts ( |↓i ) in the reverse process. However,thanks to spin-dependent controlling techniques, whichcan be achieved by adjusting parameters of incident lightbeams, one can regulate tunneling amplitudes of differ-ent species, making atoms in different sites feel a similarhopping matrix element, t ↑ = t ↓ ≡ t . This is the reasonwe straightforwardly set an identical hopping term t atthe beginning.Meanwhile, to achieve an effective AF superexchangeinteraction, we need to set ∆ = √ U ( ˜ U = − U ) tochange signs of coefficients: J z ≈ t /U + ǫ , J ⊥ = − t /U in equation (7). As a result, the global coefficient of spininteraction becomes positive, J ≈ t /U , as equation (1)asks.Under these circumstances, we finally derive thebosonic t - J model with a small inhomogeneity in z di-rection, which can be formally written as equation (1).Quite recently, two papers [49, 50] suggested a similarproposal and made detailed discussions. B. Mean field calculation from slave-particleapproach
This representation is equivalent to the following op-erator identity between the spin and boson operators.Defining ˆ a iσ = ˆ h † i e i ˆΘ string iσ ˆ b iσ and ˆS i = ˆb † i σ ˆb i , where ˆ h † i creates a fermionic holon, leaving a non-local phase stringand ˆ b iσ annihilates a bosonic (Schwinger-boson) spinonat site i , we rewrite equation (1) as ˆ H = ˆ H h + ˆ H s :ˆ H h = − t X h ij i σ e i ˆ A fij ˆ h † i ˆ h j e i ˆ A fji ˆ b † jσ ˆ b iσ , ˆ H s = − J X h ij i σσ ′ e i ˆ A hij ˆ b † iσ ˆ b † j, − σ e i ˆ A hji ˆ b j, − σ ′ ˆ b iσ ′ , Here ˆΘ string iσ = ( ˆΦ bi − σ ˆΦ hi ) is a topological phasewith the contribution from spinon number ˆ n bσ , ˆΦ bi = P l = i,σ ( − σ Im [ln ( z i − z l )] ˆ n blσ and the holon’s (ˆ n h )contribution, ˆΦ hi = P l = i Im [ln ( z i − z l )] ˆ n h . And ˆ A fij andˆ A hij describe quantized fluxes bounded to spinons andholons. We then introduce a mean-field bosonic RVB or-der parameter ∆ s = P σ h e − iσ ˆ A hij ˆ b iσ ˆ b j, − σ i , and take theconstraint on total spinon number and dilute effect intoaccount, we apply mean-field approximation on ˆ H andsolve two self-consistent equations: δ = 2 − N X k λ k E − k coth ( βE k / , ∆ s = 1 − δN X k ξ k J − s E − k coth ( βE k / , to obtain the phase boundary k B T = 2 J (1 − δ ) (2 − δ ) (cid:18) ln 3 − δ − δ (cid:19) − that separates ∆ s = 0 and ∆ s = 0. Here N is the totallattice number. Detailed definitions can be found in ref.[24]. Acknowledgments
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