Microscopic Study of the Coupled-Wire Construction and Plausible Realization in Spin-Dependent Optical Lattices
MMicroscopic Study of the Coupled-Wire Construction and Plausible Realization inSpin-Dependent Optical Lattices
Valentin Cr´epel , , Benoit Estienne , Nicolas Regnault , Center for Ultracold Atoms, Research Laboratory of Electronics, and Department of PhysicsMassachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, USA Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes Energies, LPTHE, F-75005 Paris, France Department of Physics, Princeton University, NJ 08544, USA and Laboratoire de Physique de l’ ´Ecole normale sup´erieure,ENS, Universit´e PSL, CNRS, Sorbonne Universit´e,Universit´e Paris-Diderot, Sorbonne Paris Cit´e, Paris, France
Coupled-wire constructions offer particularly simple and powerful models to capture the essenceof strongly correlated topological phases of matter. They often rely on effective theories valid in thelow-energy and strong coupling limits, which impose severe constraints on the physical systems wherethey could be realized. We investigate the microscopic relevance of a class of coupled-wire modelsand their possible experimental realization in cold-atom experiments. We connect with earlier resultsand prove the emergence of fractional quantum Hall states in the limit of strong inter-wire tunneling.Contrary to previous studies relying on renormalization group arguments, our microscopic approachexposes the connection between coupled-wire constructions and model wavefunctions in continuumLandau levels. Then, we use exact diagonalization methods to investigate the appearance of thesefractional quantum Hall states in more realistic settings. We examine the parameter regimes wherethese strongly correlated phases arise, and provide a way to detect their appearance in cold-atomexperiments through standard time-of-flight measurements. Motivated by this experimental probe,we finally propose a realization of our model with cold-atom in spin-dependent optical lattices. Ourestimates show that the previous fractional quantum Hall phases lie within experimentally accessibleparameter regimes, giving a viable route towards their experimental study.
I. INTRODUCTION
Our understanding of the fractional quantum Hall(FQH) effect has heavily relied on the use of model wave-functions (WFs) [1]. Their strength resides in their si-multaneous predictive power and microscopic relevance.For instance, the Laughlin states [2, 3] allow to suc-cessfully predict the existence of quasiparticle excitationswith fractional charge [4, 5] and fractional statistics [6].They also fully screen the short-range and largest pseu-dopotentials components of the Coulomb interaction pro-jected in the lowest Landau level (LLL) [7, 8]. Such afeature ensures that it faithfully capture the low energyphysics of a fractional quantum Hall system at filling fac-tor ν = 1 /m , while providing an excellent approximationof the system’s ground state (GS).Another powerful and elegant approach able to pre-dict the universal behaviors of FQH phases is known asthe coupled-wire (CW) construction [9, 10]. This ap-proach has found applications in widely different mod-els [11–15], sometimes far away from the conventionalFQH phases [16–18]. In the CW construction, a higher-dimensional system is decomposed into a collection ofone-dimensional subsystems, the quantum wires. Spe-cific targeted topological phases arise due to suitablecouplings between the wires and careful choices of in-teractions [19]. The effectiveness of the CW construc-tion relies on powerful bosonization and renormalizationtechniques [20, 21]. In the FQH case, this constructionprovides an intuitive understanding of the bulk-edge cor-respondence [22]: the edge effective theory is used for the wires to generate the bulk physics [23]. Often, the inter-acting CW models are not chosen for their experimentalrelevance, but rather to provide an intuitive picture fora given topological phase or to ease the renormalizationgroup analysis.CW models may however find a greater microscopicrelevance and a wider range of application in ultracoldquantum gases [24]. These physical systems have beenenvisioned for the realization of FQH phases for the lastdecade [25, 26], due to the many experimental advancesin the generation of artificial magnetic fields [27–32]. Inthis prospect, the implementation of CW models withcold-atoms in optical lattices displays several advantages.First, cooling of one-dimensional wires below the Dopplerlimit [33], and strong artificial magnetic fluxes [34] havealready been demonstrated in such setups. Then, ul-tracold collisions between neutral particle in s − or p -wave [35, 36] naturally realize the idealized interactions ofcertain CW models [9]. Finally, sub-wavelengths spacingof the wire should provide the necessary tunneling coeffi-cients to create flat Chern bands [24]. More recently, ex-perimental breakthroughs in ultracold gases with a syn-thetic dimension [37, 38] open another route towards ex-perimental realizations of CW models, as theoreticallyenvisioned and numerically evidenced in Refs. [39–43].However, the long-range interactions in the synthetic di-mension tend to stabilize crystal phases rather than FQHones in these setups [44, 45].To support these prospective realization of FQHphysics in optical lattices, a closer microscopic under-standing of CW models is required. In fact, the low- a r X i v : . [ c ond - m a t . s t r- e l ] J u l energy and strong coupling limits, used in analyticaltreatments of CWs models, blurs crucial microscopicproperties. The precise range of tunneling and interac-tion strengths required for the emergence of FQH phasesremains unknown in most of these models. We still lacka microscopic characterization of CW ground states, forinstance via model WFs, to connect with simple experi-mental probes such as density. The competing phases inthese models and their distinctive features have yet to beidentified. These problematics demand sustained effortsin the microscopic study of CW model, either analyti-cally or numerically, but without recurring to effectivelow-energy theories.This paper addresses some of these questions for a classof coupled-wire models. In particular, we show how andin which regimes the CW construction is microscopicallyrelated to the continuous description of the FQH effect.Moreover, we numerically study the phase diagram of aCW model in realistic experimental conditions.The paper is organized as follows. In Sec. II, we be-gin with a short review of continuum FQH system andof the pseudopotential approach for the Laughlin state.In Sec. III, we show the emergence of a Landau levelstructure for quantum wires strongly coupled by tunnel-ing in an external magnetic field. Thanks to this cor-respondence, we adapt the pseudopotential approach toour coupled-wire system, and bridge the CW construc-tion and model WFs (see Sec. IV). In Sec. V, we use exactdiagonalization methods to sketch the phase diagram ofan interacting CW model. We characterize the weakly-interacting phases and highlight an experimental probeto discriminate them from the Laughlin phase. Finally,we propose a plausible experimental realization of ourmodel with cold-atoms in spin-dependent optical lattices(see Sec. VI). Our estimates indicate that the FQH-likephases could be observed in optical lattices for experi-mentally realistic parameters, provided temperature canbe kept low enough. II. QUANTUM HALL EFFECT ON THECYLINDER GEOMETRY
In this section, we briefly review the quantized motionof charged particles on the cylinder geometry, i.e. as-suming periodic boundary condition along one direction.We introduce model interactions for which the Laughlinwavefunctions (WFs) are the exact densest ground stateof the many-body problem [2].
A. Landau Levels
We consider a two dimensional gas of N particles ofcharge e moving in the ( x, y ) plane and subject to aperpendicular magnetic field B = ∇ × A . We assumeperiodic boundary conditions along the x direction andidentify x = 0 with x = L , thus mapping the problem to a cylinder. In the Landau gauge A = (sign( e ) By, k x is a goodquantum number and periodic boundary conditions im-pose k x = γk , with γ = 2 πL and k ∈ Z . (1)The kinetic Hamiltonian H kin = ( p − e A ) m , (2)splits the Hilbert space into Landau levels evenly spacedin energy by (cid:126) ω c = (cid:126) | e | B/m . Most of the QHE physicsis already apparent in the Lowest Landau Level (LLL).Hence, ignoring spin degeneracies and assuming the tem-perature is low enough, we will focus on the LLL physicsfrom now on. This subspace is spanned by momentumeigenfunctions of the form φ k ( x, y ) = e iγkx (cid:112) π √ L exp (cid:18) − ( y − y k ) (cid:96) B (cid:19) , (3)where (cid:96) B = (cid:115) (cid:126) | e | B (4)denotes the magnetic length. In this geometry, the mo-mentum label k also determines the center of the Gaus-sian wave-packet in the y direction, y k = γk(cid:96) B . We de-note by ˜ c † k the particle creation operator in orbital k ,thus φ k ( x, y ) = (cid:104) x, y | ˜ c † k | (cid:105) . For non-interacting fermions,the Integer Quantum Hall Effect (IQHE) occurs when aLandau level is completely filled. Here, we focus on thecompletely filled LLL with filling factor ν = 1. We canwrite the many-body WF as a Slater determinant involv-ing orbitals of the form Eq. 3. Denoting by z j = x j − iy j the complex coordinate of the j th particle, this many-body WF can be rewritten (up to a global normalizationprefactor) as:Ψ ν =1 ( z , · · · , z N e ) = (cid:89) i The study of the FQHE is much more difficult for stan-dard perturbation methods are not available. Indeed, theHamiltonian projected to the LLL consists solely of aninteraction term projected to the flat and highly degen-erate LLL. The theoretical understanding of the FQHEhas thus heavily relied on trial WFs [46, 47]. The mostcelebrated example is the Laughlin WF at filling ν = 1 /q with q a positive integer. It reads [2]Ψ /q ( z , · · · , z N e ) = (cid:89) i In this section, we consider an array of one dimensionalwires of free particles coupled by tunneling under a mag- netic field. We prove that for large enough tunnelingstrengths, this system has all the features of the equiva-lent continuum Laudau problem studied in Sec. II. At the FIG. 1. Periodic array of one-dimensional wires of length L (in red). Periodic boundary conditions are assumed along x , the direction of the wires. They are equally spaced andcentered at positions y = jd with j ∈ Z . A magnetic flux φ = BLd is threaded between each pair of consecutive wires(grey area). single-particle level, eigenfunctions are labeled by theirmomentum along the direction of the wires which alsodetermines their center of mass position as in Eq. 3. Al-together, these states form bands with non-trivial topo-logical properties which are unveiled by adapting Laugh-lin’s charge-pumping argument [53]. The results can beintuitively regarded as the consequence of taking the con-tinuum limit in the x -direction of the Hofstadter model. A. Model We consider an array of equally spaced one-dimensional wires of free particles in the ( x, y ) plane.As sketched in Fig. 1, the wires are along the x directionfor which we assume periodic boundary conditions andidentify x = 0 to x = L . The distance between two con-secutive wires is d such that the j -th wire lies at y = jd .A magnetic field is applied perpendicular to the ( x, y )plane and we denote the non vanishing component of thevector potential as A x ( y ) = sign( e ) By . The Hamilto-nian consists of two terms, the kinetic energy from theunconfined x direction and the tunneling between wires.We denote as c † j ( x ) the creation operator of a particle atposition x ∈ [0 , L ) on wire j . In second quantized form,the Hamiltonian reads H = (cid:88) j ∈ Z (cid:90) d x (cid:32) c † j ( x ) ( p x − eA x ( jd )) m c j ( x ) − t (cid:0) c † j +1 ( x ) c j ( x ) + c † j ( x ) c j +1 ( x ) (cid:1)(cid:33) , (10)where p x = − i (cid:126) ∂ x denotes the momentum operator along x and where we have introduced the tunneling strength t between neighboring wires. As in Sec. II A, the mo-mentum p x commutes with the kinetic Hamiltonian H and is quantized in unit of h/L due to periodic boundaryconditions. Introducing the Fourier components c † j,k = 1 √ L (cid:90) d x e iπL kx c † j ( x ) , (11)with k ∈ Z an integer, we can split the Hamiltonian ofEq. 10 into different momentum sectors H = (cid:80) k ∈ Z H k with H k = (cid:88) j ∈ Z ( (cid:126) γk − | e | Bjd ) m c † j,k c j,k − t (cid:16) c † j +1 ,k c j,k + h.c. (cid:17) . To simplify the notations, we introduce the natural ki-netic energy scale E = ( eBd ) m = (cid:126) m (cid:16) d(cid:96) B (cid:17) , (12)with (cid:96) B the magnetic length, and we define the dimen-sionless tunneling parameter λ = tE . (13)We also call ν w the inverse number of flux quanta φ = h/e threading the x − y plane between two consecutivewires ν w = φ φ = 2 π(cid:96) B Ld , (14)where the flux φ = BLd is depicted in Fig. 1. For a finitesize system with N w wires, it is equal to the total fillingfactor of wires in the system ν w = N w /N φ with N φ thenumber of flux quanta threading the system. Using thesenotations, the Hamiltonian in the momentum sector k becomes H k E = (cid:88) j ∈ Z ( j − ν w k ) c † j,k c j,k − λ ( c † j,k c j +1 ,k + c † j +1 ,k c j,k ) . (15)Its spectrum is depicted in Fig. 2 as a function of the mo-mentum k . We observe that the free parabolic branchesobtained for λ = 0 hybridize near crossing points whentunneling between wire increases. For λ > / 4, thegap opened by the new avoided crossing of width be-come larger than the initial position of the crossing lead-ing to a nearly flat low-energy spectrum. Higher in en-ergy, the unbounded kinetic energy dominates and tun-neling is negligible for the highly energetic bands. We can make these statements more precise by mapping theSchr¨odinger equation originating from H k on Mathieu’sdifferential equation, as detailed in App. A. This enablesto use the properties and asymptotics of the solutions ofthis differential equation [54, chap. 28].For small tunneling strengths λ (cid:28) 1, the perturba-tive picture of avoided crossing described above matchesthe exact solution and the eigenenergies of the systemcan be obtained as power series in λ near the uncou-pled wire point λ = 0. The first order correction aresimply those obtained within perturbation theory with agap between the first two bands (cid:39) λ , while higher orderterm can be obtained iteratively [55]. These power serieshowever have a finite radius of convergence ρ ( n ) in eachband n [54, chap. 28], which set the transition betweena perturbative regime λ < ρ ( n ) and the flat band regime λ > ρ ( n ) (as can be seen in Fig. 2). These radii havebeen numerically estimated in Ref. [56, chap. 2.4]. Theirresults ρ (0) (cid:39) . ρ (1) (cid:39) . λ > ρ (0) , we can no longeruse the previous perturbative expansion and must rely onuniform semiclassical approximations [57, 58] to obtainestimates of, for instance, the spread of the lowest band δ (0) (cid:39) λ (cid:29) (cid:114) π (cid:16) √ λ (cid:17) / e − √ λ . (16)Due to the exponentially small spread of the energybands, we shall refer to the limit λ (cid:29) λ ∼ 1. Finally, we come back on therole of kinetic energy in highly excited bands n (cid:29) ρ ( n ) ∝ n [59] explaining why the parabolic profile stilldominates for high energies in Fig. 2. B. Strong Tunneling: Emergence Of Landau Levels Let us define as d ( n ) † k the operator creating a particlein band n with momentum k , with energy (cid:15) ( n ) k . Thecorresponding eigenfunction ψ ( n ) k has the form ψ ( n ) k ( x, j ) = e iγkx g ( n ) k ( j − ν w k ) , (17)where the g ( n ) k is centered around zero and can beexpressed in terms of Mathieu special functions (seeEq. A7). We recover a structure analogous to the LLLon the cylinder studied in Sec. II where the momentumlabel k also determines their center y k along the cylinder: y k = ν w dk = γk(cid:96) B . (18) FIG. 2. Spectrum of Eq. 15 for several tunneling strength λ in a system of 25 wires with ν w = 1 and open boundary conditionsin j . The free parabolic branches obtained at λ = 0 hybridize near crossing points when tunneling between wire increases, untilthe lowest bands become flat. This happens to the lowest band (blue) for moderate tunneling strength λ ∼ . Beyond this point,the system becomes exactly analogous to the Landau levels studied in Sec. II. We now focus on the flat-band regime, and show thatthese eigenstates become analogous to the LLL orbitalsof Sec. II. This result can be rationalized if we interpretour model Eq. 10 as a variant of the Hofstadter Hamilto-nian in which the continuum limit has been taken alongthe wire direction x . In the strong tunneling limit, thesystem is asymptotically equivalent to an harmonic oscil-lator [60, 61] of characteristic frequency (cid:126) ω c = 2 √ λE .The convergence is extremely fast for low-lying bandssuch that the functions g ( n ) k for different momenta areexactly equal up to corrections exponentially small in √ λ . In particular, they all have the same expression g ( n ) and the same Taylor expansion up to arbitrarily largepowers of 1 / √ λ when λ (cid:29) 1. Similarly all eigenener-gies within a given band are equal up to exponentiallysmall corrections. For instance, we have the asymptoticbehavior [57, 62]: (cid:15) (0) k (cid:39) (cid:126) ω c + δ (0) sin(2 πν w k ) , (19)with δ (0) given in Eq. 16. As a consequence, we willapproximate the system in the flat-band regime by per-fectly flat bands of energy (cid:15) ( n ) and corresponding eigen-functions ψ ( n ) k ( x, j ) = e γkx g ( n ) ( j − ν w k ). As in Sec. II A,the single body wavefunctions of a given Landau level areshifted copies of the same envelope along y , and plane-waves along the compact dimension. Let us stress againthat this approximation is really well satisfied for the low-est band, where we only require λ > ρ (0) (cid:39) . λ = 1).Not only do the eingenvalues converge towards thoseof an harmonic oscillator, but the envelopes g ( n ) them-selves uniformly converge to hermite functions, as shownin Ref. [63]. In the lowest band, this reduces to: g (0) ( u ) −−−→ λ (cid:29) g (0) ( u ) = 1( π √ λ ) / exp (cid:16) − u √ λ (cid:17) . (20)For larger tunneling strength λ , a similar behavior is ob-served in the higher bands of the system. Although it is not necessary for the quantum Hall physics to arise [64–66], we can recover the initial isotropy of the Landauproblem on the cylinder (see Sec. II) by matching thewidth of the obtained Gaussian with the magnetic length.This is achieved by a fine tuning of either the tunnelingstrength or the inter-wire distance in order to get √ λd = (cid:96) B ⇐⇒ t = (cid:126) md . (21)We now summarize the results obtained in the flat-band regime, which for low-lying bands only requiresmoderate tunneling strengths λ > ∼ 1. First the spec-trum of our model Eq. 10 is, up to exponentially smallcorrections, made of highly degenerate flat bands cen-tered on n (cid:126) ω c + (cid:15) (0) with (cid:126) ω c = 2 √ λE and (cid:15) (0) = − λ + √ λ − / 16 [54]. The eigenfunctions with momen-tum k are centered around y k = γk(cid:96) B and their envelope g ( n ) does not depend on the momentum label k , recover-ing the Landau level structure of Sec. II. When λ (cid:29) C. Charge Pumping And Quantized HallConductance The momentum periodicity of the spectrum obtainedin Eq. 19 reflects another translational symmetry in ourmodel. Writing ν w = p w q w with p w and q w relatively prime,we observe that the Hamiltonian Eq. 10 is invariant underthe action of the magnetic translation operator sendingboth j → ( j + p w ) and k → ( k + q w ). This symmetryexplains why the spectrum at k and k + q w are equivalent.Similarly, it allows to relate the eigenfunctions of Eq. 17,derived in App. A, by the pseudo-periodic relation ψ ( n ) k + q w ( x, j + p w ) = e iγq w x ψ ( n ) k ( x, j ) . (22)From now on, we focus on the lowest band of the systembut the discussion applies equally well to more energeticones. The first consequence of Eq. 22 is that there areonly q w distinct functions g (0) k and we rewrite ψ (0) k ( x, j ) = e iγkx g r ( k ) ( j − ν w k ) , (23)with r ( k ) the remainder after division of k by q w .With this results at hand, we would like to repeatLaughlin’s charge-pumping argument [53]. In otherwords, we want to show the quantized Hall conductiv-ity of our model at filling ν = 1 and thus unveil thenon-trivial topology of our model. This will make theanalogy with a continuum LLL complete. The particlefilling of the lowest band is obtained as ν = n ν w with n = NN w (24)the number of particles per wire. Here, we implicitly con-sider a finite size system of N w wires with open boundaryconditions along j . However, we still use Eq. 22 whichremains extremely well satisfied in the bulk of the sys-tem where we can neglect the edge effects (see lower-rightpanel of Fig. 2 where N w = 25 and q w = 1).Laughlin’s charge-pumping argument starts by intro-ducing a time-dependent flux Φ = θ ( t )Φ , with Φ = h/ | e | the quantum of flux, threading the surface enclosedby the wires in a system at ν = 1[53]. This situa-tion can be described by the modified gauge potential A x ( y ) = sign( e ) By + Φ /L . The flux is increased from θ (0) = 0 to θ ( t f ) = q w adiabatically, i.e. , ω c ( ∂ t θ ) (cid:28) 1, inorder to adiabatically follow the eigenstates of the lowestband. We thus have ψ (0) k ( x, j ; θ ) = e ikx f θ ( k, j ) where thefunction f θ takes simple forms at specific values of θf ( k, j ) = g r ( k ) ( j − ν w k ) ,f ( k, j ) = g r ( k +1) ( j − ν w ( k − , ... f q w ( k, j ) = g r ( k ) ( j + p w − ν w k ) = f ( k, j + p w ) . (25)We emphasized a few well chosen intermediate stateswhich can be exactly described for any value of the tun-neling strength λ > 0. In the flat band limit, the dis-cussion simplifies since all g r are equivalent. After a fullramp-up of the flux Φ, all orbitals recover their origi-nal expressions, with a shift of their center of mass by∆ y = ν w p w d in the y -direction (see Eq. 18). This dis-placement leads to a current whose response is deter-mined by the transverse conductance [67, 68] σ xy = L ∆ P y θ ( t f )Φ , (26)with ∆ P y the polarization induced by threading opera-tion [69–71]. It can be computed as the density of dis-placed charged ∆ P y = | e | p w N φ LN w , where we have used thateach of the N φ was filled and the previously computed displacement ∆ y . Combining the different pieces, thisgives the quantized Hall conductance σ xy = e h ν = e h , (27)and reveals the non-trivial topology of the lowest bandof the system. IV. INTERACTIONS IN THE FLAT LOWESTBAND: FRACTIONAL QUANTUM HALLSTATES In this section, we want to extend our discussion toinclude interactions between particles originally on thesame wire. For simplicity, we consider the flat-bandregime and require the temperature of the system to sat-isfy δ (0) (cid:28) k B T (cid:28) (cid:126) ω c in order to project the wholedynamics onto the lowest band of the system. Our dis-cussion will closely follow that of Sec. II B, and we willborrow the exact results known in continuum FQH sys-tems to show that similar physics arise in the coupledwire model of Eq. 10. A. Projection Onto The Lowest Band Consider on wire density-density interactions depend-ing on the arbitrary potential V : H int = (cid:88) j ∈ Z (cid:90) d x d x (cid:48) V ( x − x (cid:48) ) : ρ j ( x ) ρ j ( x (cid:48) ) : , (28)where ρ j ( x ) = c † j ( x ) c j ( x ) is the density operator forthe wire j . Using the Fourier components V q = (cid:82) dxV ( x ) exp( − iπqx/L ), we can rewrite it as H int = (cid:88) u,k,l ∈ Z V k − l (cid:88) j ∈ Z c † j,u + k c † j,u − k c j,u + l c j,u − l . (29)Momentum conservation follows from translation invari-ance in the x -direction. Projecting this interaction ontothe occupied band ( n = 0 in Eq. 17) yields H int = (cid:88) u,k,l ∈ Z V k − l Γ uk,l d (0) † u + k d (0) † u − k d (0) u + l d (0) u − l . (30)The form factor Γ uk,l involved has the formΓ uk,l = (cid:88) j ∈ Z g (0) u + l [ j − ν w ( u + l )] g (0) u − l [ j − ν w ( u − l )] (31) × (cid:104) g (0) u + k [ j − ν w ( u + k )] g (0) u − k [ j − ν w ( u − k )] (cid:105) ∗ , which can be simplified in the flat-band limit, as detailedin Sec. III B. Using Eq. 20 within the flat band approxi-mation, we can evaluate the form factor explicitly:Γ uk,l = K α ( u ) ( λ ) e − ( rγ(cid:96) B ) ( k + l ) , r = (cid:20) (cid:126) mtd (cid:21) , (32) FIG. 3. Function K α ( λ ) of Eq. 34 appearing in the expres-sion of the form factors Γ uk,l . In the flat band regime ofSec. III B characterized by λ > , its is very well approxi-mated by (2 π √ λ ) − / . where we have defined α ( u ) = ν w u . This function onlytakes integer values when there is an equal number ofwires and fluxes (as in Fig. 2). It can take half-integervalues when there are twice as many fluxes than wires,and so on. Here, the tunneling-dependent function K satisfies K α ( λ ) = K − α ( λ ) = K α +1 ( λ ) (33)and reads: K α ( λ ) = 1 π √ λ (cid:88) j ∈ Z exp (cid:18) − j + α ) √ λ (cid:19) . (34)We have plotted this function in Fig. 3 as a function of λ for several values of α in the only relevant range [0 , / π √ λ ) − / when λ > H int = (cid:88) u,k,l ∈ Z (cid:104) V k − l e − ( rγ(cid:96) B ) ( k + l ) (cid:105) d (0) † u + k d (0) † u − k d (0) u + l d (0) u − l , (35)where we have included the multiplicative factor K α ( λ > 1) in the definition of the potential V . Due to the Gaus-sian form factors in Eq. 35, the correspondence with themodel interactions of Eq. 7 becomes more precise and wenow show that similar FQH physics can be stabilized. B. Laughlin States In The Continuum Limit In the flat band limit, the orbitals of the lowest bandEq. 17 can be recast in the more familiar form ψ (0) k ( x, j ) = e iγkx ( π √ λ ) / exp (cid:18) − r ( y − y k ) (cid:96) B (cid:19) , (36) thanks to Eq. 20. We have used the notation r = (cid:20) (cid:126) mtd (cid:21) / (37)and y = jd for more straightforward comparison withEq. 3. The anisotropy of the single particle state, mea-sured by how much r deviates from 1 (see Eq. 21), can beincorporated into a rescaling of the coordinates ˜ x = x/r and ˜ y = ry and a re-definition of the complex coordi-nate ˜ z = xr − iry . This also impacts the natural inverselength ˜ γ = rγ . In this stretched coordinate system, thesingle-particle WFs ψ (0) k exactly match their continuumcounterparts of Eq. 3.Furthermore, the projected interactions Eq. 35 also be-come equivalent to Eq. 7. Take for instance the one-dimensional version of the first pseudo-potential as on-wire interaction V (0) ( x ) = V δ ( x ). Eq. 35 gives its ex-pression after projection to the lowest flat band of thesystem H (0)int = V (cid:88) u,k,l ∈ Z e − (˜ γ(cid:96) B ) ( k + l ) d (0) † u + k d (0) † u − k d (0) u + l d (0) u − l , (38)which is exactly equivalent to Eq. 8 in a rescaled coor-dinate system. The pseudo-potential analysis [72] canbe repeated to obtain the densest zero-energy state of H (0)int [48] for a particle filling factor ν = n ν w = 1 / ν = 1 / 2. The only difference is the real-spacerepresentation of the many-body wavefunction Ψ whichdepends on the rescaled coordinatesΨ (˜ z , · · · , ˜ z N e ) = (cid:89) i We have just proved that FQH-like phases arise in ourmodel, when we consider the idealized limit λ (cid:29) λ and interaction strength. In thissection, we provide a full-fledged Exact-Diagonalization(ED) study of our interacting problem to precisely lo-cate the transition towards the Laughlin state Eq. 39.Moreover, our calculations show how to experimentallydiscriminate the strongly correlated Laughlin phase fromweakly interacting phases. A. Setup Let us first fix the scope of our ED study. Anticipatingthe ultracold alkali vapors considered in Sec. VI, we fo-cus on a bosonic system with on-wire contact interactions V (0) ( x ) = V δ ( x ). In all our finite size calculations, wework with N ≤ 12 particles distributed in N w = 5 wires .For simplicity, we assume that particles only occupy the N orb orbitals centered around k = 0 in the lowest energyband, as illustrated in Fig. 4. In other words, we keep allstates of the lowest band with momentum | k | ≤ N orb / While we focus on the case N w = 5 throughout Sec. V, we havealso performed ED simulations for finite size systems with 3 ≤ N w ≤ 7. All these systems display similar physics. FIG. 4. Lowest two bands for a finite system of N w = 5 wires with tunneling parameter λ = 0 . . The single particlestate of the lowest (resp. first excited) band, labeled by theirmomentum k , are represented with blue (resp. orange) dots.To simulate a system near total filling factor ν = 1 / , we tunethe magnetic field strength in order to keep precisely N orb =2 N + 3 orbitals below the single particle gap (darker blue).We only consider those orbitals in our ED calculations. Notethat N orb is larger than the natural number of orbitals for theLaughlin state. The truncated many-body Hamiltonian splits into a dis-persive and an interacting part H tot = H disp + H int with: H disp = (cid:88) | k |≤ N orb / ε (0) k d (0) † k d (0) k , (42a) H int = V (cid:88) p + p = q + q | p i | , | q i |≤ N orb / Γ q + q − ,p − d (0) † q d (0) † q d (0) p d (0) p , (42b)where we have used the short-hand notations p ± = ( p ± p ) / q ± = ( q ± q ) / 2. The Γ coefficients have beendefined in Eq. 31.To simulate a system near filling factor ν = 1 / 2, wherewe expect the bosonic Laughlin state Eq. 39, we tunethe magnetic field strength in order to have precisely N orb = 2 N + 3 orbitals below the cyclotron energy (cid:126) ω c ,as depicted in Fig. 4. While the densest Laughlin dropletonly needs 2 N − B. Laughlin States We first diagonalize the Hamiltonian 42 for λ = 0 . V . We observe fea-tures reminiscent of the Laughlin phase for strong inter-actions. The many-body ground state | Ψ GS (cid:105) is char-acterized by the relative contribution of its dispersiveand interacting energies, E disp = (cid:104) Ψ GS |H disp | Ψ GS (cid:105) and FIG. 5. a) Relative contribution of the dispersive energy E disp to the total ground state energy E GS = E disp + E int as a functionof the interaction strength V , for different particle numbers ≤ N ≤ . Beyond V (cid:39) . (cid:126) ω c , the system suddenly jumpsto a phase where the interaction is almost entirely screened (black). This behavior is reminiscent of the Laughlin state | Ψ / (cid:105) described in Sec. II B. b) Overlap of the many-body ground state | Ψ GS (cid:105) with the Laughlin state | Ψ / (cid:105) . The high overlaps, above95% for V = 0 . (cid:126) ω c (highlighted in white for each N ), allows us to identify the phase at large V as the asymptotic ν = 1 / Laughlin phase studied in Sec. IV B. c) Focus on ED results at N = 12 . The abrupt phase transition to the Laughlin state with E int (cid:28) E disp (respectively in orange and green) near V (cid:39) . (cid:126) ω c is accompanied with a closing of the many-body gap (bue).Other phases and phase transition can be observed, for instance near V (cid:39) . (cid:126) ω c . Their nature is investigated in more detailin Sec. V D. E int = (cid:104) Ψ GS |H int | Ψ GS (cid:105) . We denote the total energyby E GS = E disp + E int . Our numerical results are de-picted in Fig. 5a-c, where we observe an abrupt changeof behaviour near V (cid:39) . (cid:126) ω c . Beyond this point,the ground state almost entirely screens the interactionand all its energy comes from the dispersion of the lowestband. This feature is reminiscent of the Laughlin statewhich is an exact zero energy state of H int (see Sec. IV B).To confirm our intuition and to identify the sharp fea-ture of Fig. 5c as the transition towards the asymptoticLaughlin phase studied in Sec. IV B, we compute theoverlap of | Ψ GS (cid:105) with the Laughlin state | Ψ / (cid:105) at fill-ing factor ν = 1 / L cyl should be chosen to match the as-pect ratio of our anisotropic system. The overlaps pre-sented in Fig. 5b show a jump from zero to almost one atthe transition, and remain above 95% for V = 0 . (cid:126) ω c for all the system size considered. These results pro-vide clear evidence that the states at large interactions V > . (cid:126) ω c belong to the asymptotic Laughlin phasestudied in Sec. IV B. For large enough interactions, theLaughlin physics arise in our finite size system.Finally, we vary the tunneling parameter λ in order tolocate the boundaries of the Laughlin phase in the ( λ, V )parameter space. Our finite-size calculations suggest thatthe Laughlin physics can be stabilized near half-filling inour model provided λ > . V > . (cid:126) ω c . Asshown in Fig. 5a, we use E disp /E GS as a probe to charac-terize the strongly correlated FQH phase. This quantityis extracted for several values of ( λ, V ) and N = 12 par-ticles to obtain the phase diagram of Fig. 6. For λ > . V = 0 . (cid:126) ω c . This FIG. 6. Phase diagram corresponding to Eq. 42 obtained for N = 12 particles. As in Fig. 5a, we use E disp /E GS to dis-tinguish the weakly-interacting phases from the Laughlin one.The parameter range where the FQH physics can be realizedis surrounded by a dashed dark blue line. simply reflects the exponentially small dispersion of thelowest band in this tunneling regime (see Sec. III B). Forsmall λ , the width of the lowest band increases formingwell separated potential wells (see also Fig. 7a). Thisdispersion makes it harder for the system to build long-range correlation. There is however an intermediate re-gion 0 . ≤ λ ≤ . V . There, the density oscillations in-troduced by the dispersion of the lowest band are com-mensurate with those of the Laughlin state | Ψ / (cid:105) on a0cylinder with a small perimeter [73]. Our finite-size nu-merics tend to suggest that this commensuration effectmoves the transition towards lower interaction strengths.The wires eventually decouple, and we do not see any sig-natures of the Laughlin phase in our finite-size numericsfor λ < . C. Experimental Signatures: Momentum-SpaceDensity Distribution Having understood under which conditions the Laugh-lin physics arise in our model, we now highlight that themomentum-space density distribution provides clear sig-natures allowing to discriminate the FQH state from theweakly interacting phases of our model. This probe isaccessible in cold-atom experiments with standard time-of-flight (TOF) measurements, which further motivatesthe experimental proposal of Sec. VI.We have computed the mean occupation of all mo-mentum states (cid:104) N k (cid:105) = (cid:104) Ψ GS | d (0) † k d (0) k | Ψ GS (cid:105) , the quan-tity usually accessed through TOF measurements, for theground state of Eq. 42. Our numerical results are exem-plified in Fig. 7a-d for N = 12 particle and λ = 0 . V < . (cid:126) ω c , particles are gathered around theminima of the dispersion relation, which are highlightedin gray in Fig. 7a and b. On the contrary, in the Laughlinphase for V > . (cid:126) ω c , the momentum-space densitydistribution is almost flat and all orbitals in the bulk ap-proach (cid:104) N k (cid:105) (cid:39) ν = 1 / 2. For large interaction strengths,the ground state also presents the typical density fluc-tuations of the Laughlin state near the edges of the sys-tem [3].Fig. 7c shows the same quantity (cid:104) N k (cid:105) over a finer gridin the interaction parameter V . It highlights that thetransition between the two previous behaviors is abrupt,hence showing that the momentum-space density distri-bution can be used as a probe of the transition towardsthe FQH-like states identified in Sec. V B. As an illustra-tive example, we show in Fig. 7d that the mean deviationof (cid:104) N k (cid:105) from ν = 1 / λ, V ) and sys-tem sizes, we have witnessed the same signatures of theweakly interacting and Laughlin phases. This promotesthe momentum-space density distribution as a simpleprobe to appraise the appearance of the FQH-like statespredicted in Sec. IV B in our model. Let us insist on theexperimental realization of Fig. 7d in cold-atom experi-ments where the momentum space density distribution isaccessible with TOF measurements, and the interactionstrength V can be tuned by varying a static magneticfield near a Feschbach resonance. D. Nature Of The Intermediate Phase To complete our understanding of the phase diagram,we finally investigate the weakly-interacting phases of ourmodel in more details. They are well captured within amean-field approach that we briefly detail here, beforecomparing it to numerical results. 1. Very-Weakly Interacting Phase: Bogoliubov Theory For V = 0, the ground state is a Bose-Einstein Con-densate (BEC) with all particles occupying the lowestenergy orbital with momentum k = 0. Formally, it canbe written as | Ψ GS ( V = 0) (cid:105) = 1 √ N ! (cid:16) d (0) † (cid:17) N |∅(cid:105) (43)with |∅(cid:105) the state with no bosons. This situation is de-picted with blue lines in Fig. 7, where we indeed measure (cid:104) N k ( V = 0) (cid:105) = (cid:40) N if : k = 00 if : k (cid:54) = 0 . (44)Switching on an infinitesimally small interactionstrength V , the ground state can be obtained with astandard Bogoliubov analysis [74], as detailed in App. C.Here, we summarize the main ideas of this approach andadapt them to our finite-size systems.Within the Bogoliubov approximation, the weak inter-actions with the BEC slightly admix the original orbitalswith momentum ± k , with k > 0, together. Due to theweak interactions, excitations can still be described asquasi-particles of the mean-field Hamiltonian. To remainin the vacuum state for these new excitations, the BECis depleted by the creation of particle pairs with non-zeromomenta. The weakly-interacting ground-state is moreeasily described in the limit √ N (cid:29) | Ψ Bogo ( V ) (cid:105) = √ N (cid:29) exp (cid:32) − (cid:88) k> t k d (0) † k d (0) †− k (cid:33) | Ψ GS (0) (cid:105) , (45)derived in App. C, where we provide an explicit expres-sion for the coefficients t k .In order to compare the weakly-interacting theory toour ED results, we must adapt Eq. 45 to finite size sys-tems, where √ N is not much larger than one, and ensureparticle number conservation. We thus consider the fol-lowing ansatz | Ψ Bogo ( V ) (cid:105) = e − (cid:80) k> tkN d (0) † k d (0) †− k d (0)0 d (0)0 | Ψ GS (0) (cid:105) . (46)The coefficients { t k } k> are variationally optimizedaround their √ N (cid:29) FIG. 7. a) Dispersion relation of the lowest band for λ = 0 . and N w = 5 wires. There and below, we highlight the positionof the lowest energy orbitals with grey shades. b) Mean occupation number of momentum states (cid:104) N k (cid:105) = (cid:104) Ψ GS | d (0) † k d (0) k | Ψ GS (cid:105) asa function of the momentum k for various values of the interaction parameter V . We show the numerical results for a systemof N = 12 particles. The weakly interacting phases, investigated further in Sec. V D, show strong peaks near the minima of thedispersion relation. On the contrary deep in the Laughlin phase (red line), the momentum-space density distribution is almostflat and equal to ν = 1 / except near the edges where we observe density fluctuations characteristics of this FQH state [3]. c)Same as b) over a finer grid in V , in order to highlight the transition between the weakly interacting and Laughlin phases near V = 0 . (cid:126) ω c . d) The mean deviation of (cid:104) N k (cid:105) from ν = 1 / can be used to probe the transition towards FQH-like states. Thenon-zero contributions at large V are mostly due to the empty orbitals near the edge of the system (here | k | ≥ ). small finite-size effects . The depletion of the BEC is ex-plicitly accounted for by gluing the operator d (0)0 d (0)0 /N to the pair creation operator in the exponential of Eq. 46.This formally implements the required particle numberconservation.The finite size ansatz Eq. 46 almost perfectly cap-tures the ED ground states for very-weak interactionstrengths V ≤ . × − (cid:126) ω c , as shown by their over-laps in Fig. 8a. However, it quickly fails to capture thenature of the other intermediate phases, characterizedby more than one peak in their momentum-space densitydistribution (see orange and green lines of Fig. 7). 2. Extension To Other Mean-Field Phases Because of the multiple local minima of the lowestband (see Fig. 7a) and the finite range of the interac-tion (see Eq. 32), even weak interactions tend to favorthe creation of multiple condensates with very differ-ent momenta. Indeed, particles have a similar disper-sion energy in all the potential wells of the lowest band ε (0) k (cid:39) ε (0)0 . However, particles in distant minima of thedispersion relation hardly interact due to the finite range The number of variational parameters, ( N orb − / N =12 particles, remains much smaller than the many-body Hilbertspace dimension, equal to 33 427 622 for the same parameters. FIG. 8. a) For very weak interactions, the standard Bogoli-ubov approach | Ψ Bogo ( V ) (cid:105) well captures the physics of ourmodel. As explained in Sec. V D 2, it fails to describe theother weakly interacting phases which display multiple peaksin Fig. 7. For V > . × − (cid:126) ω c , we have to rely on anextended ansatz | Ψ ExtBogo ( V ) (cid:105) adapted to the multiple peaks inthe momentum-space density distribution of the weakly inter-acting phases depicted in Fig. 7 (see App. C). b) Overlap ofthe ED ground state | Ψ GS ( V ) (cid:105) with the extended Bogoliubovansatz | Ψ ExtBogo ( V ) (cid:105) (orange) and the bosonic Laughlin state | Ψ / (cid:105) at filling factor ν = 1 / (red). These two states ac-curately describe the physics of our model on each side of thetransition. All the results presented in this figure were ob-tained for N = 12 particles and λ = 0 . . N/ N particles at k = 0. This simple argument is confirmedby Fig. 8a, where we observe that Eq. 46 correctly cap-tures our ED ground state for V (cid:39) . × − (cid:126) ω c whichshould be compared to the energy difference between thetwo local minima N ( ε (0) k − ε (0)0 ) (cid:39) . × − (cid:126) ω c (thelast factor of N accounts for the different scalings of thedispersive and interacting Hamiltonian with respect todensity). The same reasoning shows that the BEC at k = 0 is destabilized towards intermediate phases withmultiple macroscopically populated momentum states foran interaction strength V ∝ /N . Therefore, we only ex-pect to see these latest in the thermodynamic limit (seeorange and green lines in Fig. 7).The previous Bogoliubov approach can be extended tothe case of multiple smaller condensates, and allows tocapture the other intermediate phases observed in Fig. 5cand Fig. 7c. The idea is to conserve the variational pa-rameters of Eq. 46, which measure the depletion of thecondensates by pair-creation in order to accommodatethe weak interactions of the system, while changing thestate they act on to describe the presence of multiplesmaller BECs. The explicit expression of this general-ized ansatz | Ψ ExtBogo ( V ) (cid:105) can be found in App. C. As canbe seen in Fig. 8a, this adjusted ansatz overcomes thelimits of Eq. 46 and perfectly captures the phases forweak interactions.In Fig. 8b, we compare our ED results with this gen-eralized Bogoliubov ansatz | Ψ ExtBogo ( V ) (cid:105) accross the tran-sition from the non-interacting regime to the Laughlinstate. We observe that all the mean-field phases are wellcaptured by this mean-field approach, which eventuallybreaks down when the Laughlin phase arise. VI. POSSIBLE REALIZATION INSPIN-DEPENDENT OPTICAL LATTICE In this section, we propose a plausible realization of thecoupled-wire model Eq. 10 where a one-dimensional spin-dependent optical lattice creates the initial wires [76].This configurations allows to cool the gas to sub-recoiltemperatures, mitigating the effects of inter-band mix-ing. Moreover, the sub-wavelength spacing of potentialwells in the spin-dependent potential allows to reach thestrong tunneling regime of Sec. III B. The spin degree offreedom of the potential is also used to selectively driveRaman-assisted hopping between wires [77], which cre-ates an artificial gauge field mimicking a uniform mag-netic field [30]. We now explain in more details how thesedifferent elements can be combined. A. Building The Wires: Spin-Dependent TrappingPotential The first ingredient of our experimental proposal is astrong optical lattice obtained by interfering a pair of red-detuned laser beams counter-propagating along y withintensity I and angle θ between their linear polarizationsas depicted in Fig. 9a. For simplicity, we consider al-kali atoms whose ground state manifold, characterizedby the hyperfine spin F , is immune to rank-2 tensorlight-shifts [78]. As a consequence, the spin-dependentpotentials created by the previous laser configuration,dubbed lin- θ -lin [79, 80], only linearly couples to thespin F . More precisely, it is diagonal in the spin basis F y | m y (cid:105) = m y | m y (cid:105) and reads [81–83]: U OL = U [cos θ cos(2 k L y ) + u sin θ sin(2 k L y ) F y ] , (47)with k L the lasers’ wavevector, U an overall multiplica-tion factor proportional to I and where the coefficient u gathers all the relevant informations on the excited hy-perfine structure [84]. As in Ref. [76], we focus on thecase F = 1 / u = 1 which constitutes the simplestrealization of the ideas put forward in this article. Ourconclusions can be extended to other atomic species. Forinstance, we refer the interested reader to Refs. [30, 85–87] for an extensive theoretical study and many experi-mental details on the F = 1 case of Rb.The spin eigenstates | m y (cid:105) diagonalizing Eq. 47 are fur-ther coupled by a magnetic field in the y − z plane givingthe additional energy: U ZM = (cid:126) ω Z ( F y + F z ) . (48)In absence of transverse magnetic field ω Z = 0, we canrewrite the spin-dependent potentials as U LO | ± / (cid:105) = ˜ U cos[2 k L ( y ± δy )] | ± / (cid:105) , (49)with δy = tan − (cid:18) tan θ (cid:19) , ˜ U = U (cid:112) θ + 1 . (50)The angle θ thus modulates the depth of the optical po-tential and shifts the | m y = ± / (cid:105) potential wells asshown with dotted lines in Fig. 9b. Upon applying themixing term U ZM , the degeneracies are lifted and we ob-tain two distinct branches( U OL + U ZM ) | χ ± ( z ) (cid:105) = U ± ( z ) | χ ± ( z ) (cid:105) (51)with U + ( z ) > U − ( z ). With an appropriate choice ofthe angle between the laser polarizations and magneticfield strength, we can design two decoupled lattices ofdouble-well potentials with an adjustable energy barrierbetween them. This situation is depicted in Fig. 9b,where the underlying color indicates the polarization (cid:104) χ ± ( z ) | F y | χ ± ( z ) (cid:105) of the spin eigenstates.We identify three almost equally separated potentialwells 1 (cid:13) , 2 (cid:13) and 3 (cid:13) which are depicted in Fig. 9b. We3 FIG. 9. a ) Sketch of the experimental setup. Two strong laser beams (blue arrows) counter-propagating along z in a lin- θ -lin configuration create a spin-dependent optical lattice. We identify three inequivalent Wannier centers within a unit cell ofthe lattice (see text and b ), here schematically depicted with three different tones of orange. The Raman beams (green arrows)counter-propagating along y generate complex tunneling coefficients between the wires (cid:13) and their neighbors, thus generating anartificial gauge field mimicking the Landau vector potential. b ) Spin-dependent adiabatic potential obtained by diagonalization of U OL + U ZM fr θ = 80 ◦ , U = 45 E R and (cid:126) ω Z = 5 E R . In absence of transverse magnetic field (dotted lines), the spin eigenvectors | χ ± ( z ) (cid:105) are eigenstates of F z . They are admixed close to the avoided crossing opened by the transversed magnetic field (solidlines), as can be seen from the background line color representing the spin polarization (cid:104) χ ± ( z ) | F z | χ ± ( z ) (cid:105) . Within one unitcell of size π/k L , we three potential wells denoted as (cid:13) - (cid:13) - (cid:13) give rise to localized Wannier states entering our tight-bindingdescription of the system. c ) While (cid:13) and (cid:13) are naturally coupled by tunneling matrix elements in the tight binding limit(see text), we need to engineer light-assisted hopping elements to couple them to (cid:13) . Due to the polarization of | χ ± ( z ) (cid:105) , thetransitions (cid:13) → (cid:13) and (cid:13) → (cid:13) can only de driven by a F − operator. d ) Close-up of Fig. 9b around the potential wells (cid:13) and (cid:13) . To estimate the tunneling matrix element between the Wannier states at energy E and E , we use a semiclassical analysisand see this reduced problem as simpler double-well potential [75]. The results of this semiclassical analysis involves the meanenergy E = E + E and the turning points z < and z > which are schematically pictured. would like to derive an effective tight binding model forthe later. Within an harmonic approximation, the smalloscillatory motion in the wells have natural frequenciesdetermined by Taylor expansion around the local minima (cid:126) ω = (cid:126) ω = (cid:113) U E R , (cid:126) ω = 2 ˜ U (cid:115) E R sin(2 k L δz ) (cid:126) ω Z , (52)where we have introduced the recoil energy E R =(2 (cid:126) k L ) / m . As stated earlier, the original unper-turbed optical lattice with ω Z = 0 enables the cool-ing of the atomic cloud to sub-doppler temperatures k B T < E R [33, 88]. In this temperature regime andassuming U (cid:29) E R , the previous energy scales (cid:126) ω i aremuch larger than the thermal energy of the atoms. Asa consequence, we can approximate the system’s dynam-ics by keeping only the lowest energy Wannier state ofeach potential wells 1 (cid:13) , 2 (cid:13) and 3 (cid:13) within a tight-bindingapproximation [89, 90]. B. Tunneling And Artificial Gauge Field To obtain the effective tight-binding model of our sys-tem, we first consider a single unit-cell of length π/k L along y . We denote the Wannier state centered aroundthe potential well i (cid:13) with the first quantized notation | i (cid:105) (see Fig. 9b), and its on-site energy as E i for i = 1 , , (cid:13) and 3 (cid:13) lie in the same band andcan overlap, hopping from these states to 2 (cid:13) is prohibitedby the orthogonality condition (cid:104) χ + | χ − (cid:105) = 0 [77]. Light-assisted tunneling may however be used to couple theseorbitals with two-photons Raman transitions. These twodifferent tunneling mechanisms are considered separatelythereafter. a. Real Tunneling Amplitude: To evaluate the tun-neling matrix element t between 1 (cid:13) and 3 (cid:13) , we followthe semiclassical treatment of a double well potential ofRef. [75] (see Fig. 9d). This approach does not accountfor the periodic structure of the lattice, usually underesti-mating the tunneling strengths of the original model [91].The investigation made in Ref. [75] gives the estimate t = (cid:126) ω π e − θ , θ = (cid:90) y > y < d y (cid:48) (cid:126) (cid:112) m [ U − ( y (cid:48) ) − E ] , (53)where the turning points y < and y > and the energy E areschematically depicted in Fig. 9d. For the parameters ofFig. 9, the numerical evaluation of these quantities give θ (cid:39) . t (cid:39) . E R . (54)This large value compared to other cold-atom experi-ments with optical lattices [92] is explained by the sub-wavelength separation of the potential wells 1 (cid:13) and 3 (cid:13) inthe spin-dependent optical lattice Eq. 47 [24].4 b. Raman Assisted Tunneling: We finally turn tothe engineering of light assisted hopping towards theWannier state 2 (cid:13) , which not only couples the latter tothe effective tight-binding model but also generates anartificial gauge field for our system. These coupling canbe achieved with two additional laser beams counter-propagating along the x -direction, very far detuned fromany atomic line. The first one is polarized along y and hasfrequency ω R while the second is polarized along z andpossesses two different tones ω R + ω and ω R + ω , with ω ij = ( E j − E i ) / (cid:126) (see Fig. 9a). They coherently couplespin states with | ∆ m y | = 1 through two-photon Ramantransitions and thus allow to drive the two transitions 1 (cid:13)→ (cid:13) and 2 (cid:13) → (cid:13) .Adiabatically eliminating the very off-resonantly cou-pled excited atomic states [93] and within the RotatingWave Approximation (RWA) [94], the Raman Hamilto-nian reads [95–97]: H Ram = − (cid:2) ( t R e ik L x + t (cid:48) R e − ik L x ) | (cid:105)(cid:104) | + h.c. (cid:3) − (cid:2) (˜ t R e ik L x + ˜ t (cid:48) R e − ik L x ) | (cid:105)(cid:104) | + h.c. (cid:3) . (55)All four Rabi frequencies t R , t (cid:48) R , ˜ t R and ˜ t (cid:48) R are propor-tional to the Raman lasers’ intensity I R . However, theycrucially involve different spinorial and spatial overlapsbetween Wannier states [77, 86], for instance t R ∝ (cid:90) d y (cid:104) χ + ( y ) | F − | χ − ( y ) (cid:105)(cid:104) | y (cid:105)(cid:104) y | (cid:105) , (56)while t (cid:48) R ∝ (cid:90) d y (cid:104) χ + ( y ) | F + | χ − ( y ) (cid:105)(cid:104) | y (cid:105)(cid:104) y | (cid:105) . (57)As observed in Fig. 9c, the strong polarization of the spinstates | χ ± ( y ) (cid:105) leads to t R (cid:29) t (cid:48) R . Similarly ˜ t R (cid:29) ˜ t (cid:48) R andfor simplicity we assume ˜ t R = t R such that the light-atominteraction with the Raman fields is described by H Ram = − t R (cid:2) e ik L x ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) + h.c. (cid:3) . (58)This formula obtained within the RWA is valid as longas t R (cid:28) min( E − E , E − E ). For the parameters usedin Fig. 9 and Eq. 54, this corresponds to t R (cid:28) E R .Under realistic experimental conditions, technical limi-tations will set in before the saturation of this inequality which is another nice feature of our spin dependent lat-tice. The intensity of the Raman lasers can be increasedor the detuning to the excited states decreased in orderto achieve t R (cid:39) t , which we will assume from now on.Eq. 58 shows that a particle moving along increasing y picks a phase proportional to x because of the momentumdifference between the two Raman beams, which is rem-iniscent to the effect of a gauge field. This analogy canbe made exact [85, 87, 96] (also see below) and was usedexperimentally to create artificial gauge field for neutralatoms [30, 37, 38]. The introduction of the Raman laserbeams thus creates an effective gauge field for the atomsand connects the Wannier state 2 (cid:13) to our tight-bindingmodel. C. Matching The Parameters Of The Two Models:Experimental Feasibility We have seen that the added pair of Raman laserbeams allows to create an artificial gauge field thanksto spin-selective two-photon transitions between the twobands of the optical lattice while the sub-wavelengthspacing of Wannier states provides significant tunnel-ing amplitudes t (cid:39) . E R . To make connection withSec. III, we call L the system length along the weaklyconfined x direction and introduce c † j ( x ) the creation op-erator at position x in the ( r + 1)-th Wannier orbital ofthe q -th unit cell with j = 3 q + r . the effective tight-binding Hamiltonian following from Secs. VI A and VI Bis H = (cid:88) j ∈ Z (cid:90) L d x c † j ( x ) p x m c j ( x ) − t (cid:104) e iφ (cid:48) ( j ) x c † j +1 ( x ) c j ( x ) + h.c. (cid:105) , (59)where the phases follow from φ (cid:48) ( j ) = (cid:40) k L if r = 0 , 10 if r = 2 . (60)The connection with Eq. 10 is cleaner after a gauge trans-formation c † j ( y ) → exp(4 ik L y/ c † j ( y ) under which H be-comes H = (cid:88) j ∈ Z (cid:90) L y d y c † j ( y ) ( p y − (cid:126) k L j ) m c j ( y ) − t (cid:104) e i ˜ φ (cid:48) ( j ) y c † j +1 ( y ) c j ( y ) + h.c. (cid:105) , (61)where the tunneling phases now sum to zero˜ φ (cid:48) ( j ) = (cid:40) k L if r = 0 , − k L if r = 2 , (62) showing that all the artificial gauge field have been trans-fered to the kinetic part of the Hamiltonian. Pushing the5analogy further, we can identify E = 169 (cid:126) k L m = 49 E R , (63)such that the typical tunneling parameters obtained inEq. 54 translate into λ = tE (cid:39) . > . (64)The ultracold atomic system reaches the flat-band limitstudied in Sec. III B for the experimentally relevant pa-rameters used in Fig. 9. This emphasizes the experimen-tal feasibility of our proposal, potentially leading to theobservation of quantum Hall physics in large ultracoldatoms ensembles. Sub-Doppler temperatures are how-ever necessary if one wants to only populate the lowestband of the system because of the rather small cyclotronfrequency (cid:126) ω (cid:39) E R /π in the cold atomic case.Finally, we comment on the possibility to reach thestrongly correlated states of Sec. IV. Since bosons at ul-tracold temperature only interact with s -wave scattering,corresponding to the zero-th Haldane pseudo-potentialEq. 7, it is natural to investigate the experimental condi-tions required to achieve a total filling fraction ν = 1 / n = N/N w (cid:39) ν w = 3 π k L L y (cid:39) . (65)This estimation leads to a length L y (cid:39) λ L along theweakly confined direction with λ L the Raman laser wave-length, which corresponds to L y ∼ µ m under typicalexperimental condition. This can be realized with a veryweak confinement along y . Most of the quantities com-puted here can be improved by numerical factors of orderone by either introducing an angle between the two Ra-man beams [98] or by choosing very different wavelengthfor the lattice and Raman beams. However, Eq. 64 andthe possible sub-Doppler temperatures already give veryfavorable estimates with regard to the experimental re-alization of quantum Hall physics in such systems. VII. CONCLUSION In this article, we have provided a microscopic char-acterization of a class of coupled-wire models. First, we have shown the emergence of Landau levels for stronginter-wire tunneling. This equivalence with a continuumsystem in the deep fractional quantum Hall regime allowsto adapt the pseudopotential approach to the coupled-wire system. In particular, we could exhibit model on-wire interactions stabilizing both bosonic and fermionicLaughlin phases in the thermodynamic limit. Turningour attention towards potential experimental implemen-tations of our model in cold-atom setups, we have usedexact diagonalization to observe and characterize the dif-ferent phases for realistic parameters. We provided ev-idence that time-of-flight measurements can distinguishbetween the Laughlin and the other weakly-interactingBogoliubov phases. Finally, we have proposed an ex-perimental realization of our model with cold-atoms in aspin-dependent optical lattice. The spin-dependent trap-ping potential leads to sub-wavelength spacing of the one-dimensional atomic wires, and allows to reach the strongtunneling regime. Our estimates indicate that the FQH-like phases could be observed in optical lattices for ex-perimentally realistic parameters if spontaneous heatingis mitigated. VIII. ACKNOWLEDGMENTS We are indebted to Ady Stern for discussions whichhave initiated the present work. We particularly thankChristophe Mora for sharing his knowledge and ideas onrelated topics. 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Appendix A: Mathieu Differential Equation AndAsymptotics In this appendix, we show how the stationarySch¨odinger corresponding to Eq. 15 maps onto Mathieu’sdifferential equation, we review its solutions and analyzefurther the weak and strong tunneling limits. 1. Relevance Of Mathieu’s Equation In this section, we fix the momentum sector k , and wewill omit the explicit dependence in momentum when itis unnecessary. We use the first quantized notations | j (cid:105) = c † j,k | (cid:105) for the state localized on wire j . The stationarySchr¨odinger equation for the eigenstate (cid:104) j | ψ (cid:105) = ψ ( j ) ofEq. 15 reads( j − j ) ψ ( j ) − λ [ ψ ( j + 1) + ψ ( j − µψ ( j ) (A1)with j = ν w k and where µ is the eigenenergy in unit of E . The coupled-wire system studied in Sec. III is thusequivalent to the well studied Cooper pair box Hamil-tonian [99, 100], which is usually solved introducing the8conjugate variable of j . Thus, we introduce the phase ϕ ∈ [ − π, π [, canonically conjugated to the wire position j ∈ Z : | ϕ (cid:105) = (cid:88) j ∈ Z e i ( j − j ) ϕ | j (cid:105) , | j (cid:105) = (cid:90) π − π d ϕ π e − i ( j − j ) ϕ | ϕ (cid:105) . (A2)It yields d ψ ( ϕ )d ϕ + [ µ + 2 λ cos ϕ ] ψ ( ϕ ) = 0 , (A3)where ψ ( ϕ ) = (cid:104) ϕ | ψ (cid:105) . Notice that the inclusion of j in the definition of the conjugate variable makes thefunction ψ ( ϕ ) pseudo-periodic ψ ( ϕ + 2 π ) = e iπj ψ ( ϕ ).This can also be seen as a gauge transformation in the ϕ representation. After the final change of variable x = ( ϕ + π ) / 2, this equation takes Mathieu’s equationstandard form [54, § w (cid:48)(cid:48) ( x ) + [ γ − q cos 2 x ] w ( x ) = 0 , (A4)with γ = 4 µ and q = 4 λ . 2. Formal Solutions Eq. A3 together with the pseudo-periodicity of ψ al-lows to identify ψ ( ϕ ) = me ν M (4 λ, ( ϕ + π ) / , (A5)where me ν M denotes is the Floquet solution of the Math-ieu’s equation with characteristic exponent ν M = 2( j + η ) , (A6)with η ∈ Z labeling the solutions of Eq. A3. To inter-pret η as a band index, we shall sort the correspondingeigenvalues µ = (1 / a ν M (4 λ ) in ascending order, wherewe used Mathieu’s characteristic function a ν M ( q ) [54,chap. 28]. For the n -th band, the sorting function η ( j , n )is a non-trivial function of j and of the band index whoseexplicit form is known [100]. Denoting as d ( n ) † k the cre-ation operator of a particle of momentum k in band n (see Sec. III B), and by | n (cid:105) = d ( n ) † k | (cid:105) the correspondingwavefunction, we finally arrive at the closed form expres-sion of the function g ( n ) k used in Eq. 17: (cid:104) j | n (cid:105) = g ( n ) k ( j − j ) = (cid:90) π − π d ϕ π e i ( j − j ) ϕ me [2( j + η ( j ,n ))] (4 λ, ( ϕ + π ) / . (A7)Our main interest in the formal mapping onto Math-ieu’s equation is to know whether the flat band approx-imation invoked in the main text is justified (see for in-stance Eq. 20 or Eq. 19). Let us define δ ( n ) ( λ ) as thespread of the n -th band of the spectrum of Eq. 15. Us-ing uniform semiclassical approximations [57, 58], it ispossible to obtain an expansion of the spread for largetunneling strength λ (cid:29) δ ( n ) ( λ ) (cid:39) n +3 n ! (cid:18) π (cid:19) / (cid:16) √ λ (cid:17) n +3 / e − √ λ × (cid:18) − n + 14 n + 764 √ λ + O (cid:18) λ (cid:19)(cid:19) , (A8)We have numerically diagonalized the HamiltonianEq. 15 for several tunneling and we compare in Tab. I theasymptotic behavior Eq. A8 to the numerically extractedvalues. In practice, the exponentially small spread makesthe flat band approximation very accurate, even for mod-erate tunneling strength, as evidenced by the large spreadto gap ratio gathered in Tab. I. TABLE I. We numerically diagonalized Hamiltonian Eq. 15for over 1000 wires. We compute the spread of the lowest band δ (0) and compare it to the asymptotic limit given of Eq. A8.We find that the flat band limit is extremely well satisfied evenfor very moderate tunneling strengths λ ∼ . We also extractthe flatness of the lowest band, defined as the ratio of its spreadto the gap separating it to the first excited band. We find thatthe gap is orders of magnitude greater for λ ≥ , justifyingour projection onto the lowest band in Sec. IV. λ δ (0) Asymptotic Eq. A8 Flatness0.10 1.649e-1 1.673e-1 8.266e-10.33 6.051e-2 6.333e-2 9.327e-20.66 1.618e-2 1.680e-2 1.366e-21.00 5.334e-3 5.394e-3 3.243e-33.33 1.894e-5 1.899e-5 5.605e-66.66 7.663e-8 7.673e-8 1.564e-810.0 1.009e-9 1.010e-9 1.665e-1033.3 1.42e-14 2.144e-18 1.259e-15 Appendix B: Laughlin State With Similar AspectRatio In the main text, we have compared our EDground states | Ψ GS (cid:105) with Laughlin states on a cylinder | Ψ / ( L cyl ) (cid:105) of perimeter L cyl . In this appendix, we pro-9 FIG. 10. Overlap between the ED ground state obtainedfor very large interaction strengths and the Laughlin state ona perimeter L cyl . Different colors indicate different tunnel-ing strength λ or a different number of extra orbitals (seeSec. V A), while dotted, dashed and solid lines are respec-tively used for N = 10, 11 and 12. The cylinder maximizingthe overlap is highlighted with gray vertical bars. For all theconsidered parameters ( λ, N φ , N, N orb ) , the optimal perimeterlies within ten percent of πν w λ / (cid:96) cyl B . This fact which is madeapparent by the normalization of the x -axis. vide more details on the choice of L cyl . To differenti-ate this cylinder from the wire system, we respectivelydenote as B cyl , (cid:96) cyl B , ω cyl c the magnetic field threadingthe cylinder, the corresponding magnetic length and cy-clotron energy.For an infinite system N w (cid:29) 1, we have derived arescaling of the coordinates x and y which allows to di-rectly compare the wire ground state with the cylinderLaughlin state (see Sec. IV B). In particular, equating theperimeter of the cylinder and the inter-orbital distanceleads to L cyl = L/r π ( (cid:96) cyl B ) L cyl = rdν w (cid:41) = ⇒ L cyl (cid:96) cyl B = N w (cid:29) πν w λ / . (B1)This formula can be understood as matching the aspectratio of the two models. However, it relies on the bulkproperties derived in Sec. IV assuming a very large num-ber of wires and cannot be used directly for the finite-sizesystems that we are interested with in Sec. V. We havethus relied on the following numerical approach.Consider the ground state | Ψ GS (cid:105) obtained with the pa-rameters ( λ, N φ , N, N orb ) and interaction strength V . Tofind L cyl , we first diagonalize the wire Hamiltonian Eq. 42for the same parameters ( λ, N φ , N, N orb ) but with V (cid:29) V = 5 (cid:126) ω c ). This leads to a new ground state | Ψ GS ( V (cid:29) (cid:126) ω c ) (cid:105) . Then, we generate several Laughlinstates | Ψ / ( L cyl ) (cid:105) with different aspect ratios r cyl = 2 πN orb (cid:32) (cid:96) cyl B L cyl (cid:33) , (B2) by varying te parameter L cyl /(cid:96) cyl B . Practically, we obtainthe states | Ψ / ( L cyl ) (cid:105) either by ED of the model interac-tion Eq. 8 or through exact matrix product states [101–103]. We finally choose the perimeter maximizing theoverlap |(cid:104) Ψ / ( L cyl ) | Ψ GS ( V (cid:29) (cid:126) ω c ) (cid:105)| .Our numerical results are presented in Fig. 10. Thebest overlaps obtained with this technique are never lowerthan 0.98, providing further evidence for the emergenceof the Laughlin physics in our model (see Sec. V B). Theyalso increase with greater tunneling strengths λ , as ex-pected from our analytical result of Sec. IV B in the limit λ > 1. Moreover, we observe that the perimeters maxi-mizing the overlaps lie close to L cyl (cid:96) cyl B (cid:39) π λ / ν w . (B3)Among all the considered parameters ( λ, N φ , N, N orb ),we have only observed fluctuations of less than 10% fromthis empirical formula. Therefore, we have used this em-pirical finding as an initial guess to limit our numericalburden.While we are not able to justifies Eq. B3 analytically,we can understand the scaling with λ as follows. Equat-ing r cyl of Eq. B2 to the aspect ratio r = N w d/L of thefinite wire system, we find: L cyl (cid:96) cyl B = 2 πν w λ / (cid:115) N orb N φ . (B4)For an infinite system, it reduces to Eq. B1 since N φ (cid:39) N orb . In our ED calculation however, the magnetic field B of the wire system is tuned away from B cyl such asto have precisely N orb orbitals below the single particlegap. This introduce a difference between N φ and N orb in the calculations of Sec. V and explains the differentscaling observed in Eq. B3. Considering the quadraticdispersion near the edge ε (0) k (cid:39) ( ν w k ) E , and requiringthat the orbital with momentum k = N orb / (cid:126) ω c (cid:39) √ λE (see Eq. V A), we get N orb N φ ∝ λ / ⇒ L cyl (cid:96) cyl B ∝ λ / . (B5) Appendix C: Bogoliubov Excitations In this appendix, we provide more details about theBogoliubov approach sketched in Sec. V D used to de-scribe the weakly-interacting phases in the model Eq. 42(see, for instance, Ref. [104] for a more thorough andcomprehensive review of the method). Due to both thefinite-size of our system and to the presence of other lo-cal minima in the dispersion relation (see Fig. 7), thestandard Bogoliubov analysis only holds for very smallinteraction strength. We thus provide a modified versionto take into account the finite size and the presence ofmultiple disconnected minima in the dispersion relation.0 1. Very-Weakly Interacting Phase:Thermodynamic Limit Let us first follow the standard derivation of the Bogoli-ubov quadratic Hamiltonian [104]. We assume that themacroscopically populated BEC of Eq. 43 is only slightlydepleted, such that we can replace N = d (0) † d (0)0 (cid:39) N and d (0)0 (cid:39) √ N by their expectation value. The differentterms of the Hamiltonian Eq. 42 can then be sorted indecreasing order of importance, according to their scal-ing with N . Keeping only terms at least proportionalto the particle number, we find the following Bogoliubovapproximation of the many-body Hamiltonian H Bogo = ε (0)0 N + V N v (0)+ (cid:88) k (cid:54) =0 (cid:104)(cid:16) ε (0) k − ε (0)0 (cid:17) + 2 V N (2 v ( k ) − v (0)) (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) D ( k ) d (0) † k d (0) k + (cid:88) k (cid:54) =0 V N v ( k ) (cid:16) d (0) † k d (0) †− k + d (0) k d (0) − k (cid:17) , (C1)where v ( k ) = Γ k/ k/ ,k/ = Γ ,k = Γ k, is real. Note thatthe inversion symmetry of the problem implies v ( k ) = v ( − k ) and ε (0) k = ε (0) − k . The first line of Eq. C1 cor-responds to the energy of the weakly interacting BEC E BEC ( N, V ) = ε (0)0 N + V N v (0). The last two linesshow the quadratic Hamiltonian which account for theentire many-body problem in the Bogoliubov approach.This quadratic Hamiltonian can be diagonalized as H Bogo = E BEC ( N, V ) + (cid:88) k (cid:54) =0 (cid:15) Bk B † k B k , (C2)where the operators B k are obtained by the squeezing (orBogoliubov) transformation d (0) k = u k B k − v k B †− k , d (0) †− k = u k B †− k − v k B k . (C3)The real coefficients u k and v k must satisfy u k − v k =1 for the new operators to obey bosonic commutationrelations [ B k , B † k ] = 1. They can thus be parametrizedby a hyperbolic angle u k = cosh θ k and v k = sinh θ k .Plugging Eq. C3 into Eq. C1, this angle is chosen tomake the the B k B − k and B † k B †− k term vanish:tanh(2 θ k ) = 2 V N v ( k ) D ( k ) . (C4)This allows to derive the following expressions for thequasiparticles eigen-energies and weights: (cid:15) Bk = (cid:112) D ( k ) − V N v ( k )] , (C5) u k = (cid:115) D ( k )2 (cid:15) Bk + 12 , v k = (cid:115) D ( k )2 (cid:15) Bk − . (C6) The ground state of the system is now defined asthe vacuum state for the quasi-particle operators, i.e. B k | Ψ Bogo ( V ) (cid:105) = 0. It can be expressed as: | Ψ Bogo ( V ) (cid:105) = exp (cid:32) − (cid:88) k> t k d (0) † k d (0) †− k (cid:33) | Ψ GS (0) (cid:105) , (C7)with t k = v k /u k . This equation makes clear that the sys-tem accommodates the weak interaction by the creationof particle pairs with non-zero momenta ± k , as stated inthe main text. 2. Very-Weakly Interacting Phase: Finite SizeSystems In order to compare the weakly-interacting theory toour ED results, we must adapt Eq. C7 to finite size sys-tems, where √ N is not much larger than one, and ensureparticle number conservation. We thus consider the fol-lowing ansatz, reproduced from Eq. 46: | ˜Ψ Bogo ( V ) (cid:105) = e − (cid:80) k> tkN d (0) † k d (0) †− k d (0)0 d (0)0 | Ψ GS (0) (cid:105) . (C8)The coefficients { t k } k> are variationally optimizedaround their √ N (cid:29) d (0)0 d (0)0 /N to the pair creation operator in the exponen-tial of Eq. C8. This formally implement the requiredparticle number conservation.As explained in the main text and numerically demon-strated in Fig. 8, Eq. C8 almost perfectly captures theED ground states for very-weak interactions V N (cid:28) 3. Other Weakly Interacting Phase As explained in Sec. V D, the multiple local minima ofthe lowest band (see Fig. 7a) and the finite range of theinteraction (see Eq. 32) tend to favor the creation of mul-tiple condensates with very different momenta. The pre-vious Bogoliubov approach can be adapted to this case aswell. The idea is to conserve the variational parametersof Eq. 46, which measure the depletion of the condensatesby pair-creation in order to accommodate the weak in-teractions of the system, while changing the initial statethey act on. This initial state is composed of P multipleBECs located near the minima of the dispersion relationwith momenta k , · · · , k P , and hosting N , · · · N P parti-cles. They are can be written as | ( k i , N i ) i =1 , ··· ,P (cid:105) = P (cid:89) i =1 √ N i (cid:16) d (0) † k i (cid:17) N i |∅(cid:105) . (C9)1The original state | Ψ GS (0) (cid:105) , corresponding to the bluelines in Fig. 7, is recovered by choosing P = 1, k = 0 and N = N . For all the other interaction strengths V in theweakly-interacting phases, we fix these new parameterssuch that | ( k i , N i ) i =1 , ··· ,P (cid:105) corresponds to the highest-weight number state in the many-body decomposition ofthe ED ground state | Ψ GS ( V ) (cid:105) . For instance, the casedepicted in orange in Fig. 7 has P = 3 peaks locatedat k = − k = 0 and k = 5, and hosting N = N = N = 4 particle each (before depletion by the pair-creation operators). Once the initial | ( k i , N i ) i =1 , ··· ,P (cid:105) has been determinedas detailed above, the Bogoliubov ansatz is obtained byappending the exponentiated particle-pair creation oper-ator as in Eq. 46. This results in the following ansatz: | Ψ Bogo ( V ) (cid:105) = exp (cid:32) − (cid:88) k> t k N d (0) † k d (0) †− k d (0) k i d (0) k j (cid:33) | Ψ GS (0) (cid:105) ..