Polar Phase of 1D Bosons with Large Spin

Spinor ultracold gases in one dimension represent an interesting example of strongly correlated quantum fluids. They have a rich phase diagram and exhibit a variety of quantum phase transitions. We consider a one-dimensional spinor gas of bosons with a large spin $S$. A particular example is the gas of chromium atoms (S=3), where the dipolar collisions efficiently change the magnetization and make the system sensitive to the linear Zeeman effect. We argue that in one dimension the most interesting effects come from the pairing interaction. If this interaction is negative, it gives rise to a (quasi)condensate of singlet bosonic pairs with an algebraic order at zero temperature, and for $(2S+1)\gg 1$ the saddle point approximation leads to physically transparent results. Since in one dimension one needs a finite energy to destroy a pair, the spectrum of spin excitations has a gap. Hence, in the absence of magnetic field there is only one gapless mode corresponding to phase fluctuations of the pair quasicondensate. Once the magnetic field exceeds the gap another condensate emerges, namely the quasicondensate of unpaired bosons with spins aligned along the magnetic field. The spectrum then contains two gapless modes corresponding to the singlet-paired and spin-aligned unpaired bose-condensed particles, respectively. At T=0 the corresponding phase transition is of the commensurate-incommensurate type.


I. INTRODUCTION
Spinor Bose gases attracted a great deal of attention in the last decade as they exhibit a much richer variety of macroscopic quantum phenomena than spinless bosons (see [1] for review). The physics of three-dimensional spin-1 and spin-2 bosons is rather well investigated, both theoretically [2][3][4][5][6][7][8][9] and in experiments with Na and 87 Rb atoms [10][11][12][13][14]. The structure of the ground state strongly depends on the interactions, and in particular ferromagnetic, polar (singlet-paired), and cyclic phases have been analyzed on the mean field level and beyond the mean field [1]. The spinor physics of 3D spin-3 bosons is described in Ref. [15] and, after successful experiments with Bose-Einstein condensates of 52 Cr atoms (S = 3) [16], experimental studies of the spinor physics in this system are expected in the near future.
The observation of non-ferromagnetic states requires very low and stable magnetic fields (well below 1 mG) at which the interaction energy per particle exceeds the Zeeman energy.
Presently, the obtained stable field on the level of 0.1 mG is expected to reveal a transition between ferromagnetic and non-ferromagnetic states in chromium [17], and experiments using the magnetic field shielding and aiming at even lower fields are underway [18].
It is important to emphasize that a change of magnetization of an atomic spinor gas under variations of the magnetic field requires spin-dipolar collisions, since the short-range atomatom interaction does not change the total spin. In dilute gases of sodium and rubidium the spin-dipolar collisions are very weak, and the magnetization does not feel a change in the magnetic field on the time scale of the experiment. On the contrary, in a gas of chromium atoms which have a large magnetic moment of 6µ B , the spin-dipolar collisions efficiently change the magnetization and the gas becomes sensitive to the linear Zeeman effect [19].
Spinor Bose gases in one dimension (1D) are in many aspects quite different from their 2D and 3D counterpats and represent an interesting example of strongly correlated quantum fluids. In this paper, having in mind the gas of chromium atoms (S = 3), we assume that the system is sensitive to the linear Zeeman effect. We consider a 1D spinor gas of bosons where the dominant interactions are the density-density and the attractive pairing interactions. This choice is justified by the fact that in 1D only the latter interaction gives rise to a nontrivial quasi-long-range order. In contrast to 2D and 3D, in one dimension pairs with nonzero spinS do not condense. This is related to the fact that forS = 0 the symmetry of the condensate order parameter is non-Abelian. It is well known that strong quantum fluctuations in 1D dynamically generate spectral gaps for non-Abelian Goldstone modes which leads to exponential decay of the correlations (see, for example, [20]). As far as the polar phase (the condensate ofS = 0 pairs) is concerned, it can be formed because the symmetry of the order parameter is Abelian. However, in 1D its magnetic spectrum is quite different from that in 2D and 3D: in the absence of magnetic field the spin excitations have a gap. For a large spin S, the saddle point approximation gives a physically transparent description of the polar phase. A sufficiently large magnetic field closes the gap and leads to the transition from the singlet-paired (polar) phase to the ferromagnetic state.
The presence of the spin-gap strongly changes the physics of the 1D polar phase and the polar-ferromagnetic transition compared to higher dimensions discussed for spin-3 bosons in Ref. [15]. We investigate the 1D polar phase and this quantum transition and discuss prospects for their observation in chromium experiments.

II. THE MODEL
As the atom-atom short-range interaction conserves the total spin, the Hamiltonian of binary interactions for (1D) bosons with spin S can be written as a sum of projection operators on the states with different even spinsS of interacting pairs [1]: where x is the coordinate. For the 1D regime obtained by tightly confining the motion of particles in two directions, the interaction constants γS are related to the 3D scattering lengths a 3D (S) at a given spinS of the colliding pair. Omitting the confinement induced resonance [21] we have: where l 0 = ( /Mω 0 ) 1/2 is the confinement length, M is the atom mass, and ω 0 the confinement frequency.
We then use the relation SPS (x) =:n 2 (x) : wheren is the density operator and the symbol :: denotes the normal ordering, and reduce the interaction Hamiltonian to the form . For a positive value of (γ q − γ) the system is an ordinary Luttinger liquid, but for (γ q − γ) < 0 the situation may change. In 3D a negative value of (γ q − γ) would lead to a spontaneous symmetry breaking with a formation of the order parameter in the form of a condensate of pairs with total spin q. In one dimension only a quasi-long-range order is possible and only if q = 0 when the symmetry in question is an Abelian one [20]. Therefore, interactions with negative coupling constants, which have q different from zero or from 2S will not produce quasi-long-range-order. The case of q = 2S is exceptional because it corresponds to a ferromagnetic state where the order parameter (the total spin) commutes with the Hamiltonian. Therefore, at T = 0 this state can exist even in 1D. We do not discuss this interesting state, and the only possibility that remains is q = 0. So, in our model we have a (repulsive) density-density interaction and the pairing interaction that gives rise to the formation of singlet pairs.
In realistic systems the coupling constants γS are not equal to each other, although they are generally of the same order of magnitude. We thus have to single out the densitydensity interaction in a proper way and then deal with the rest. For example, the interaction Hamiltonian (1) can be represented as a sum of squares of certain local operators as is usually done in the theory of spinor Bose gases [1,15]: where In the case of 52 Cr we have c 0 = 0.65γ 6 , and the 3D scattering length is a 6 = 112a B [15], where a B is the Bohr radius. The exact value of the 3D scattering length a 3D (0) is not known and, hence, the constants γ 0 and c 2 are also unknown. In this paper, when discussing 52 Cr atoms we omit the :F 2 (x) : and :Ô 2 (x) : (renormalized) terms, treat c 2 as a free parameter and focus on the case of c 2 < 0.
We then write down the following Hamiltonian density in terms of the bosonic field operators Ψ j : where the spin projection j ranges from −S to S, the coupling constant g 0 is assumed to be positive, and we put = 1. The coupling constants g 0 and g are related to c 0 and c 2 . For example, in the case of 52 Cr we have g = 7c 0 = 4.55γ 6 > 0 and g 0 = −c 2 .

III. ZERO MAGNETIC FIELD. SADDLE POINT APPROXIMATION
We now consider the case of N = 2S + 1 >> 1 and apply the 1/N-approximation to the model described by the Hamiltonian density (4). First, we decouple the pairing from the density-density interaction by the Hubbard-Stratonovich transformation [22]: where ∆(τ, x) and λ(τ, x) are auxiliary dynamical fields. At large N the path integral is dominated by the field configurations in the vicinity of the saddle point ∆(τ, x) = ∆, λ(τ, x) = iλ 0 . The values of ∆ and λ 0 are determined self-consistently from the minimization of the free energy. The stability of the saddle point is guaranteed by the fact that the integration over the Ψ, Ψ + fields yields a term proportional to N and therefore the entire action is ∼ N.
The presence of large N in the exponent in the path integral suppresses fluctuations of the fields ∆ and λ, thus making the saddle point stable.
The bosonic action at the saddle point is where ǫ = k 2 /2M − µ and µ = µ 0 − λ 0 , with µ 0 being the bare chemical potential. From Eq. (6) we find the mean field spectrum of quasiparticles (we assume that µ < 0): The saddle point equations are: where n is the density of one of the bosonic species.
The quasiparticles (spin modes) constitute a (2S+1)-fold degenerate multiplet. As follows from Eq. (7), the quasiparticles have a nonzero spectral gap This result agrees with the one for N = 3 obtained in Ref. [24]. This is a special feature of one dimension. In 2D and 3D the integral in the saddle point equation (8) does not diverge at small ω, k for κ → 0, and such a gap is not formed. Therefore, one has a gapless spectrum of spin modes, which for S = 3, 2 and 1 has been obtained in the studies of spinor Bose gases (see, e.g. [1] and references therein). We would like to emphasize the fact that although Eqs. (8), (9), and (10) resemble the equations for a superconductor, due to the bosonic nature of the problem the order parameter amplitude ∆ is not equal to the spectral gap, and the latter is related to the parameter κ.
After the integration in Eqs. (8) and (9) we get the saddle point equations in the parametric form: where K(x), E(x) are elliptic functions [23]. From the form of these equations it is clear that the ratio κ/k 0 is a function of the parameter and k 0 can be written in the form:  Accordingly, Eq. (11) for the gap takes the form: so that the gap in units of ng 0 depends only on the parameter η. In the limit of weak interactions where η ≪ 1, we obtain: and Eq. (11) gives an exponentially small gap: For strong interactions, η ≫ 1, we have and the gap is given by The numerically obtained dependence of κ/k 0 is displayed in Fig. 1, and the function b(η) is shown in Fig. 2. The gap is presented in Fig. 3. The asymptotic formula (18) obtained in the limit of small η already works with 20% of accuracy for η = 0.05. With the same accuracy the large-η asymptotic formula (20) is already valid for η = 1.
In the limit of weak interactions, taking into account that |µ| ≈ ∆ and using Eqs. (7) and (15), we get µ ≈ −ng 0 /4M. Substituting this relation into Eq. (10) we obtain Hence the system is thermodynamically stable for g > g 0 .
The only gapless excitation of the system is the phase mode of the complex scalar field ∆. This excitation describes sound waves of the pair condensate. The effective Hamiltonian for the phase mode Φ is where Π is a canonically conjugate momentum. The velocity v and Luttinger parameter K s are extracted from the functional derivatives of the saddle point action and are given by the following equations: where G(ω, k) is the Green function of the field Ψ j , defined as Ψ j (ω, k)Ψ + j ′ (ω, k) = δ jj ′ G(ω, k), and The functions f 1 (k 0 /κ) and f 2 (k 0 /κ) can be expressed in terms of elliptic functions E( 1 − κ 2 /k 2 0 ) and K( 1 − κ 2 /k 2 0 ), but the expressions are combersome and we do not present them. In the limit of weak interactions we have: whereas for strong interactions these functions are given by The velocity v is given by the relation: so that for weak interactions using Eqs. (25), (15) and b ≃ 2 we have: For strong interactions equations (26), (15), and (19) The dependence of v on the parameter η is displayed in Fig. 4.
The Luttinger parameter K s follows from the relation: The scaling dimension of the ∆ field is d = K s /4π and it decreases very rapidly with η, which indicates that the mean field approximation works very well. In Fig. 5 we show the dependence of d on η for N = 7. In the regime of weak interactions the scaling dimension is exponentially small and it remains significantly smaller than unity even for η ≃ 2.
To conclude this part we give a brief summary of the properties of the paired phase.
With certain modifications, the properties for an arbitrary large spin S are similar to the ones for S = 1 described in Ref. [24]. Namely, all single particle correlation functions decay exponentially. This follows from the fact that the operator ψ + j always emits a gapped vector excitation (Bogolyubov quasiparticle) from the (2S + 1)-fold degenerate multiplet (for S = 1 it is a gapped triplet). Two-particle correlation functions of the operators ψ j ψ j and their Hermitean conjugates (no summation assumed) undergo a power law decay.

IV. MAGNETIC FIELD AND EXACT SOLUTION
In order to study the influence of the magnetic field on the properties of the singletpaired phase one can also use the saddle point approximation employed in the previous section. However, the saddle point equations become too involved. Therefore we resort to non-perturbative methods.  As was demonstrated in Ref. [25], the model described by the Hamiltonian density (4) possesses U(1)×O(2S+1) symmetry. Therefore it is reasonable to suggest that the low energy sector of this model is described by a combination of the U(1) Gaussian theory and the O(2S+1) nonlinear sigma (NLσ) model. For S=1 this was explicitly demonstrated in Ref. [24]. Both the U(1) theory and the sigma model are integrable and the exact solution gives access to the low energy sector of the model. At a special ratio of the coupling constants one can get even further, since it was demonstrated [25] that the entire model (4) is integrable at a particular ratio of g 0 /g. Below we restrict our consideration to the low energy sector where we are not constrained to this particular ratio. As is always the case for Lorentz invariant integrable models, all the information on the thermodynamics is contained in the two-body S-matrix which was found in Ref. [26]. Consider N = 2S + 1 (S integer) and physical particles which have a relativistic-like spectrum ǫ(θ) = m cosh θ, p(θ) =ṽ −1 m sinh θ, with mass (gap) m, velocityṽ, and the total energy For the particles confined in a box of length L with periodic boundary conditions, the Bethe Ansatz equations read: where S 0 is related in the standard way to the integral kernel K(θ): and K(θ) = e iωθ/π K(ω)dω/2π, The spectral gap m (the particle mass) and velocityṽ are related to the bare parameters of the model (4). For N ≫ 1 one can use Eq. (16) for the gap, and in the lowenergy limit the spectrum (7) of the spin modes becomes E(k) = √ m 2 +ṽ 2 k 2 withṽ = (b/2) ng 0 (1 + κ 2 /k 2 0 )/M, so that in the limit of weak interactions we haveṽ ≈ ng 0 /M. Equations (32) constitute a system of S + 1 coupled algebraic equations for the quantities θ 1 , ...θ n , λ and magnetization S z are determined by the following integral equations: where h = g L µ B B, with B being the magnetic field and g L the Landee factor. There is obviously one transition at h c = m/S. The magnetization is zero for h < h c and it gradually increases with the field for h > h c (there is another transition in high magnetic fields corresponding to the saturation of the magnetization, but the low energy theory cannot describe it). In order to find the magnetic field dependence of the magnetization near the transition, where B << 1, we approximate the kernel as and look for the solution of Eqs. (34) and (35) in the form: Substituting ǫ(θ) and ρ(θ) given by Eq. (38) into Eqs. (34) and (35)) we get: Keeping only the leading term in Eq. (39) and restoring the dimensions we have: with the critical magnetic field given by Equation (40) shows a typical field dependence of the magnetization for the quantum commensurate-incommensurate transition. This transition was first studied by Japaridze and Nersesyan [27] in the context of spin systems with a gap, where (as in our case) it is driven by the magnetic field. Later Pokrovsky and Talapov considered such a transition in the charge sector, where it is driven by a change in the chemical potential [28]. The magnetization is exactly zero below the critical field and increases as √ B − B c above the critical field near the transition. Note that this is quite different from the 3D case where the magnetization decreases continuously with the magnetic field when the latter goes below the critical value (see, e.g. [15]). fields corresponding to the saturation of the magnetization. However, it is not described by the low-energy theory and is beyond the scope of this paper. In the context of ultracold quantum gases, the commensurate-incommensurate transition has also been discussed for one-dimensional spin-3/2 fermions in the presence of the quadratic Zeeman effect [29].
The observation of the commensurate-incommensurate phase transition in a 1D gas of 52 Cr atoms would require (aside from a positive value of g 0 and, hence, a negative c 2 ) fairly strong interactions corresponding to the parameter Mg 0 /n ∼ 1, wheren = 7n is the total density, so that η ∼ 0.5. Then the spin gap in the polar phase is of the order of ng 0 and (assuming g 0 by a factor of 3 smaller than g) can be made on the level of 100 nK at 1D densitiesn ∼ 10 5 cm −1 . Then the transition occurs at the critical field of the order of 0.2 mG and can be observed at temperatures T ∼ 20 nK. This however is likely to require the 1D regime with a rather strong confinement in the transverse directions (with a frequency of the order of 100 kHz as in the ongoing chromium experiment in the 1D regime at Villetaneuse [17]).

Note added
After this work has been finished the Villetaneuse group has reported the observation of the demagnetization transition for 52 Cr atoms in the 1D regime, under a decrease of the magnetic field to below 0.5 mG. However, the experiment is done at temperatures ∼ 100 nK and the state which is reached by decreasing the magnetic field does not necessirily reveal the nature of the ground state due to thermal excitations (and due to diabaticity at the transition in the experiment).