Hidden order in bosonic gases confined in one dimensional optical lattices

We analyze the effective Hamiltonian arising from a suitable power series expansion of the overlap integrals of Wannier functions for confined bosonic atoms in a 1d optical lattice. For certain constraints between the coupling constants, we construct an explicit relation between such an effective bosonic Hamiltonian and the integrable spin-$S$ anisotropic Heisenberg model. Therefore the former results to be integrable by construction. The field theory is governed by an anisotropic non linear $\sigma$-model with singlet and triplet massive excitations; such a result holds also in the generic non-integrable cases. The criticality of the bosonic system is investigated. The schematic phase diagram is drawn. Our study is shedding light on the hidden symmetry of the Haldane type for one dimensional bosons.


I. INTRODUCTION
Systems of cold atoms trapped in optical lattices provide a remarkable tool to simulate quantum many body physics in engineered quantum systems [1,2]. In this context new perspectives are provided by the possibility to handle atoms or molecules with large dipole moment [3]. For these systems, the Bose-Hubbard models can be employed [4], provided that processes arising from the long range interaction are considered. The corresponding 'extended' Bose-Hubbard Hamiltonian can display a variety of quantum phase transitions between superfluid states to Mott Insulator (MI) ones (f.i. see [5]).
Recent investigations by Altman and coworkers indicate that a strongly interacting bosonic system can display a further gapped phase with 'exotic' order [6]. Based on numerical analysis of a Bose-Hubbard type model, such phase was demonstrated to be characterized by a non local hidden order of the Haldane type. Originally discovered in spin systems [7,8], Haldane phases are believed to play a crucial role in many different contexts including QCD [10] and High-T C superconductivity [11]. Recently the hidden order was noticed also in different types of electronic insulators [12,13].
The Haldane order in the bosonic systems was evidenced by rephrasing the physical concepts provided to study of the hidden order for spin systems. Although Altman and coworkers constructed a variational scenario in order to interpret their results, it would be desirable to trace how the bosonic hidden order emerges from the Haldane order in spin systems. This is a major purpose of the present work.
Based on exact methods we construct a precise relation between the bosonic and spin hidden orders. Relying on it the quantum criticality of the bosonic system is investigated. We construct the non-linear-σ model (NLσM) describing the system at long wave lengths. The phase diagram is investigated within the saddle point approximation of the NLσM. Our results are shedding light on the structure of the hidden order in bosonic systems, suggesting that the bosonic excitations in the Haldane Insulator (HI) are of triplet nature.
The paper is organized as follows. In Sec. II we derive the model Hamiltonian from the microscopic dynamics of a dilute system of interacting dipolar bosons. The emerging Hamiltonian defines a type of extended Bose-Hubbard model of interacting bosons in the lowest Bloch band. In section III, we establish the mapping between the bosonic model and the integrable higher spin-S XXZ Hamiltonian [18] (see Eqs.(4)-(10)). Because of the integrability of the spin model, the bosonic Hamiltonian is integrable by construction as long the microscopic parameters entering into bosonic Hamiltonian fulfill certain relations (see Eq. (11)). Relying on this result, the quantum criticality of the bosonic system is investigated in the Sec. IV where we construct the non-linear-σ model (NLσM) describing the system at long wave lengths. The phase diagram is constructed, beyond integrability, within the saddle point approximation of the NLσM (see Fig.(1); we note that the results are not restricted by Eq. (11)). In section IV we discuss and suggest experimental protocols to detect the Haldane insulating phase. Finally we draw our conclusions.

II. THE BOSONIC MODEL HAMILTONIAN
We start by analyzing the order of magnitude of the amplitudes involved in the 1d lattice Hamiltonian for trapped bosonic atoms; we include dipole-dipole interaction. We will obtain an effective model with densitydensity interaction and higher order hopping processes.
The general Hamiltonian for the atomic (of mass m) where V results from a combination of harmonic confinement with the optical lattice: V = V harm +V latt . We consider a 'cigar-shaped' configuration V harm = mω 2 (x 2 + γ 2 y 2 + γ 2 z 2 )/2 where the harmonic potential has a frequency ω along the x direction and is much more confined of an anisotropy factor γ ≫ 1 in the y, z directions. The 1d lattice, V latt = sE r sin 2 (πx/a), is arranged along x. E r = ( π) 2 /2a 2 m is the photon recoil energy, a is the lattice spacing and s measures the optical lattice depth in terms of E r . The interaction potential V int. = V sr + V dd contains two terms: an 'onsite' short range potential V sr ( r) = 4π 2 a BB δ( r)/m characterized by the s-wave scattering length a BB ; and a 'long range' anisotropic dipole-dipole , µ being the atomic magnetic dipole (µ 0 is the vacuum magnetic permeability), and θ being the angle of r − r ′ with the dipoles orientation.
The effective lattice (grand canonical) Hamiltonian emerging from the analysis above is where n i = b † i b i is the number operator. Terms in the ellipses involve higher powers in ε (see also [14]). The first two terms, with κ being the chemical potential and t the nearest neighbor hopping (in units of E r ), are the only low order terms from H 0 since the matrix elements t ij decrease as ε |i−j| 2 . U 0 = (2/π) 3/2 a BB as 1/4 /l 2 ⊥ is the onsite interaction neglecting the renormalization due to V dd (that is of order ε 2 ). Defining I dd = mµ 0 µ 2 /(2aπ 3 2 ) the contributions of V dd to the integral t ii+1;ii+1 , the coupling constants in (2) read We comment that the (form of the) second quantized Hamiltonian (2) is not affected by the gaussian approximation of the Wannier functions which might only modify the expression of the coupling constants in (3).
Besides the density-density interaction U 1 , H b provides for correlated t c and bosonic pair t p nearest-neighbors hopping processes. It is important to note that experimentally there are different ways of manipulating the relative strength of the coupling constants e.g. it is possible to change the optical lattice parameters [15], or use appropriate Feshbach resonances [16] to separately tune I dd and U 0 , independently from the expansion in terms of the parameter ε. This allows for the possibility of exploring a large portion of phase space.
Below we demonstrate that the 1d bosonic Hamiltonian (Eq. (2)) is obtained as large S limit of the integrable spin model generalizing the XXZ model to higher spin [17]. This will prove that the model (2) is integrable (see however the discussion below (10)).

III. THE INTEGRABLE HEISENBERG MODEL FOR HIGHER SPIN
Integrable XXZ models for higher spin [18,19] were intensively studied in the framework of the quantum inverse scattering method [20]. The Hamiltonian (E r is assumed as the energy's untit) is obtained as logarithmic derivative of the transfer matrix [21] where reducing to the ordinary gamma function for α → 0. The quantity J(α) is related to the (coproduct ∆ of) Casimir operator C S = S + S − + sinh αS z sinh α(S z + 1)/sinh 2 α of the quantum algebra su α (2) underlying the integrability of the theory: We remark that the anisotropy α enters into (4) deforming both the gamma function and the 'representation' J(α) of su α (2).
The limit of large S, with small αS is interesting for us. The first 'effect' of such a limit is that the ψ α (x) in (4) reduces to the ordinary digamma function ψ(x) = Γ ′ (x)/Γ(x). In such a limit the quantity J(α) can be obtained explicitely by resorting to the explicit expression of ∆C S By retaining the terms up to the second order in αS we obtain where λ = 1+α 2 k with k = S(S +1)+1/2. By exploiting its relation with ∆C S , J(α) can be written as 1/S perturbative expansion: where we used the Holstein-Primakoff realization of su(2) [22]. The bosonic model (2) is obtained first employing the Stirling formula for the asymptotics of the digamma function ψ(x) ≈ ln(x), then by expanding the equation resulting from (4) at second order in 1/S: where const. = N ln[S(2 + 3α 2 /4)] − N ψ(1). Translational invariance has been assumed. The spin S, and the anisotropy α are obtained by comparison of (10) with (2): We note that only one parameter results to be adjustable in (2) achieved from (4); we remark that the restrictions (11) can be achieved by tuning the relative strenght of contact versus dipole interactions. The resulting oneparameter-H b is integrable by construction (see also [23]); the exact solution will be studied elsewhere. In the isotropic case α = 0, H b is obtained from the Faddeev-Takhadjan-Tarasov-Babujian model [24]. In this case: The ground state of (4) is a singlet: S z tot = 0. For imaginary α the spectrum is gapless. For real α it was proved that the excitations are gapped [19]. Given the relation (10), it is intriguing to study the low energy spectrum of H XXZ in the limit of large S. This is what we are going to do by exploiting the NLσM.
The semiclassical continuous Hamiltonian arising from gradient expansion of (4) is H = Ω(x, τ )|∆C S |Ω(x, τ ) + . . . where the ellipses indicate terms with higher powers of a and 1/S (or combinations of thereof). H has λ − D form with D = α 2 ; the gradient expansion procedure spoils the integrability of the lattice model (10), since the t c and t p terms turn out of higher order in a [26]; nevertheless integrability manifests in the field theory in form of the restriction λ = 1 + kD, with k = S(S − 1/2), arising from the lattice theory. Generic values of λ, D can be considered through the 1/S expansion of the λ − D lattice model (that is not integrable), providing a different parametrization of the spin parameters in bosonic terms: The large scale behavior of the system is captured by the large S realization of the coherent state spin variables in terms of staggered magnetization n j (τ ) and quasihomogeneous l j (τ ) fluctuating fields through the Haldane mapping [7] Ω| S j (τ )|Ω = (−1) j n j (τ ) 1 − | l j (τ )| 2 where n j (τ ) 2 = 1 and l j (τ ) · n j (τ ) = 0. The NLσM description of (4) is obtained following a variation of the procedure originally adopted in [7,29,31] to deal with the S = 1, λ − D model. Due to the anisotropy of the λ − D model, we separate the staggered magnetization and its fluctuation in their perpendicular and transversal components: n j = ( n ⊥ , n z ) and l j = ( l ⊥ , l z ). In order to obtain the Lagrangian of the NLσM one needs to integrate out the l field in the Hamiltonian together with the terms coming from the Berry phase. After taking the continuum limit one finds a 2 2 S 2 λ (∂ x n z ) 2 + c z 2 (∂ τ n z ) 2 + M S 2 n 2 z , with M = 1 + S(S − 1/2)U 0 − U 1 , plus the topological phase associated with the Néel field n = (n ⊥ , n z ) T can assume only integer values for it is the winding number of the mapping n : Ê comp → Ë 2 which is classified by the second homothopy group π 2 (Ë 2 ) = . T contributes in the partition function as e 2iT πS , with T ∈ , so that it does not contribute for integer spins. In Eq. (14) the Néel field satisfies the constraint | n| 2 = 1. The coefficients are with µ = 1 + U 1 + kU 0 . The coefficients provide an additional interaction between the components of the Néel field. For M = 0, Eq. (14) is the sum of an O(2) NLσM in the field n ⊥ and a scalar model in n z . The former describes a free Gaussian model with a bosonic compactified field which is known to be a conformal field theory with central charge c = 1; the latter is also integrable. One of the most striking features of the Haldane phase is that the gap is provided by a triplet structure [7,27]. The masses of the particles ∆ ⊥ and ∆ z belong to the sectors S z = 0 and S z = ±1 respectively. For the case S = 1 ∆ z and ∆ ⊥ play the role of the excitations of a conformal field theory with c = 1/2 and c = 1 respectively [29]. Eventhough the Haldane order in the bosonic systems was evidenced through a non local 'order parameter' defined in analogy with the spin string order parameter [9], the phase diagram can be studied also by analysing the low energy properties of the system and specifically the particle-hole δE c and the neutral δE n energy gaps [6]. We shall see that ∆ z and ∆ ⊥ play the role of δE c and δE n respectively.
To obtain such quantities from the continous field theory we treat (14) by saddle point approximation (S = 1). The method assumes that < n 2 z >:= ζ. This means that (δn j ) 2 − δn j δn j+1 ≪ 1 with δn = n −n, indicating that only sparse charge-fluctuations around 'diluted DW states' (so called 'zero defects states' [30]) are considered. The constraint on the field n is taken into account by introducing a uniform Lagrangian multiplier η where There are two poles in the propagators of the effective action, providing the expressions in terms of ζ for the gaps in the z and in the perpendicular direction, respectively ∆ z and ∆ ⊥ , to be determined selfconsistently [31]. In order to derive the expression for the gaps we need the calculate the propagator. We substitute in the action the Fourier transform of the field n and of the Lagrange multiplier η, respectively ñ andη.
After performing a Gaussian integration on the fieldñ, the effective action becomes: where (K −1 ) jj are the diagonal entries of the inverse propagator with ∆q := q − q ′ , ∆n := n − n ′ and j = {1, 2, 3}. The notation v j , g j stands for v ⊥ g ⊥ if j = {1, 2} or v z and g z if j = 3. Ω n are the Matsubara-Bose frequencies, Ω n = 2πn/β. (K −1 ) 33 provides an expression for ζ.
With the ansatzη(q, n) = (βL)δ n0 δ q0 η, we can derive the saddle point equation (∂S/∂η = 0). For low temperatures (β → ∞) and in the thermodynamic limit (L → ∞) one finds where the ultraviolet cutoff Λ is a free parameter to be determined by fitting the gap to the known value for the isotropic point ∆ isotr = 0.41048 where ξ ⊥ = ξ z . The notation stands for ξ −2 γ := 2gγ η vγ with γ = {⊥, z}, ξ −2 z := ξ −2 ⊥ /λ and ν −2 vz . The expressions for the gaps are We can solve numerically the self-consistency equations (20) for different values of U 1 , U 0 and of the spin S (or equivalently t). From the numerical study of the gaps one can draw a (qualitative) phase diagram (see Fig.1). We see that ∆ z and ∆ ⊥ display a behavior that is very similar to δE c and δE n respectively (see Fig.1). In such a view, the phenomenology of the bosonic excitations in the HI would arise from the triplet nature of ∆ z and ∆ ⊥ . In particular, the line δE c = δE n that was evidenced numerically in Ref. [6] can be interpreted as the degeneracy of the triplet excitations of the field theory (14) (that includes the O(3) model λ = 0 [27]).
Further insights on the criticality of the bosonic system can be obtained by adapting the results for the spin S, λ − D model [32]. Specifically, the onset to the HI-density wave phase is suggested to be of the Ising type (second order), c = 1/2 and it is characterized by ∆ z = 0 with ∆ ⊥ = 0; the MI-HI phase transition is with c = 1 and it is caused by ∆ z = 0 with ∆ ⊥ = 0. Similar behavior of δE c and δE n was noticed in [6] (except that also δE c = 0 was displayed to vanish at the MI-HI transition). The density wave (DW) phase-MI transition is predicted to be of the first order; the c = 1/2 and c = 1 lines meet in a point that is governed by the continuos limit of the α = 0 integrable theory, with c = 3S/(1 + S), S = 1/(2t) [33]. The phase diagram of the system is summarized in Fig.1.

V. EXPERIMENTAL FEASIBILITY
Although a detailed analysis would go far beyond the scope of the present paper, we would like to sketch a possible method that could serve for detecting the HI experimentally (complementing the observations in [6]). The basic idea relies on the detection of the atomic current tracing back to the early experimental evidence of MI phase [34]: the MI conductivity is probed by applying a washboard potential to the lattice. A resonance in the atomic conductivity appears when the tilt between adjacent sites reaches the energy gap U 0 i.e. when it is resonant with the particle-hole pairs excitation energy. Because of the peculiar solitonic non local order [30] of the HI the resonance peak in the conductivity is narrowed, as already noticed with a different experimental situation (parametric resonance) by [6]. Placing the bosonic chain in a washboard potential, the distinct feature of the HI phase would be a dependence of the atomic current on the length of the chain (this is the analog of the diffusive spin transport evidenced in Haldane compounds [35]). We observe that, being an incompressible phase, the HI is expected to be robust to the parabolic confinement whenever the induced total energy offset (between center and the trap edges) is of the order of the energy gap. Following the numerical indications provided in [6] such a gap is estimated to be of the same order of magnitude of the MI one: ∆ ≃ t. Therefore we can estimate the magnitude of the allowed harmonic confinement to be mω 2 x 2 M < t = 2sεE r , x M being the size of the condensate along the optical lattice. It would be interesting to trace the analogous for the HI of the "wedding cake" structure that appears in MI in presence of strong confinement.
Finally, ring shaped (with circumference L) optical lattices with twisted boundary conditions [36] could be exploited to evidence the HI in an expansion experiment (we note that the energy offset due to harmonic confinement can be minimized in this case). In fact, adapting the results obtained for the spin diffusion in twisted Heisenberg integer spin rings [37], the atomic density current would display a characteristic parametric dependence on the boundary twist: a sawtooth like behaviour for correlation lenght ξ ≫ L, and it would be exponentially suppressed, with sinusoidal oscillations for ξ ≪ L. This could be a fingerprint of the Haldane gap for (finite) ultracold atomic systems.

VI. CONCLUSIONS
By exploiting a suitable expansion of the matrix elements in terms of the lattice attenuation parameter ε ≪ 1 we derived an effective model for bosonic atoms in a 1d lattice (2). Additional terms enter the Hamiltonian respect to the standard Bose-Hubbard model. For certain choice of the coupling constants, the model results to be integrable through a mapping with the spin-S-XXZ integrable model (see Eq. (10)). We note that the possibility of independently tuning the onsite-interaction relative to the density-density interaction, is available in the bosonic model Hamiltonian only in presence of dipoledipole interactions. This is precisely what allows degenerate quantum gases to explore different portions of the phase diagram beyond the MI region.
The direct relation between the spin and bosonic pictures are exploited to investigate the critical properties of the bosonic systems. From the present work, it is suggested that the correct context is provided by the spin S, λ-D model. The HI is investigated by studying the continuos field theory arising from gradient expansion of the lattice model: the Lagrangian has the form of a NLσM with further interactions between the components of the Neel field (see (14), (16)). We comment that the terms t p and t c entering the (2) do not contribute to the NLσM formulation indicating that such terms are irrelevant for the criticality we studied. Such an effect is a physical manifestation of the Affleck analysis [26]. The difference between integrable and non integrable lattice theories is reflected in a different parametrization of the coupling constants entering the NLσM. Interestingly enough the integrable parameterization of the continous model (14) we found should be tightly related to the so called pricipal chiral fields [38] indicating an emergent SU (2) × SU (2) symmetry of the bosonic Haldane phase. We notice however that the phase diagram of the system (14) is investigated, beyond the integrability, by means of the saddle point approximation; the integrable case might need separate discussion. The Haldane phase is found for any finite range of the interaction. This should be relevant for experiments where the scattering length can be tuned around zero thus evidencing the interaction between light-induced dipoles [16].