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PHYSICAL REVIEW A 84, 051602(R) (2011)

Superfluid, solid, and supersolid phases of dipolar bosons in a quasi-one-dimensional optical lattice

Jonathan M. Fellows1 and Sam T. Carr2

1School of Physics, University of Birmingham, Birmingham B15 2TT, United Kingdom2Institut fur Theorie der Kondensierten Materie and DFG Center for Functional Nanostructures,

Karlsruher Institut fur Technologie, D-76128 Karlsruhe, Germany(Received 23 May 2011; revised manuscript received 18 August 2011; published 17 November 2011)

We discuss a model of dipolar bosons trapped in a weakly coupled planar array of one-dimensional tubes.We consider the situation where the dipolar moments are aligned by an external field, and we find a richphase diagram as a function of the angle of this field exhibiting quantum phase transitions between solid,superfluid, and supersolid phases. In the low energy limit, the model turns out to be identical to one describingquasi-one-dimensional superconductivity in condensed matter systems. This opens the possibility of using bosonsas a quantum analog simulator of electronic systems, a scenario arising from the intricate relation between statisticsand interactions in quasi-one-dimensional systems.

DOI: 10.1103/PhysRevA.84.051602 PACS number(s): 67.80.kb, 05.30.Jp, 05.30.Rt

Experimental developments in ultracold atomic systemscontinue to yield insights into exotic quantum phases that wereonce the dominion of theorists. Since the first Bose-Einsteincondensates were produced in the early 1990s [1,2], the steadymarch of experimental progress continues to allow us access toall manner of phases from Mott insulator to superfluids [3]; seeRef. [4] for a recent review. Solid theoretical proposals existfor many other unconventional collective states; for example,since early proposals to find the elusive supersolid in an opticallattice [5], there has been a spate of recent suggestions [6–10]of alternative setups in which a supersolid phase may beuncovered.

A promising possible application of these developmentsis that one might be able to use arrangements of ultracoldatoms in optical traps as analog computers for simulatingstrong correlations in electronic condensed matter systems[11]. As correlations are enhanced by low dimensions, itbecomes natural to consider systems involving weakly coupledquasi-one-dimensional tubes of particles, systems that sharetheir underlying geometry with that of some of the moreinteresting condensed matter systems. While it may at firstseem preferable to deal with fermions if one intends to compareback to electronic systems, in this paper we will demonstrate abosonic model with the same low-energy effective theory as awell-studied model of striped superconductivity in electronicsystems [12,13] that has been successfully applied to thecompound Sr2Ca12Cu24O41 [14] but may even have relevancefor high-Tc cuprate materials [15]. This equivalence betweenbosonic and fermionic models arises as a result of thequasi-one-dimensional nature of the setup, where the roles ofstatistics and interactions are intrinsically intertwined [16] andimplies that suitably chosen bosonic experiments can informus about the low-energy excitations of fermionic systems.

Experimental groups now routinely produce such tubes ofbosons. The obvious interaction to exploit within such systemsis the contact interaction as this is tunable through Feshbachresonance [17]. However, contact interactions are too shortranged to couple distinct tubes, and so it becomes preferableto deal with the dipole-dipole interaction, which itself is welltunable in these systems [18] and allows the manufactureof custom spatially anisotropic interactions in such systems

[19–22]. It is unimportant for the majority of our consider-ations whether the dipole moment is electric or magnetic;however, one may desire the larger energy scales offered by theelectric dipole to access some of the physics we will encounter.

In this manuscript we will discuss such a model in which agas of dipolar bosons is arranged in a two-dimensional planararray of one-dimensional homogeneous cigars; see Fig. 1. Thiscan be created by a strong two dimensional optical lattice inwhich the confinement in the z direction completely suppresseshopping in that direction, while that in the y direction allowssome weak hopping between the tubes. In principle, a weakoptical lattice may also be present in the longitudinal (x)direction without changing the physics, so long as the latticeis incommensurate with the density of bosons. We allowfor interactions between the tubes through a dipole-dipoleinteraction that can be tuned via the direction of the dipolemoment.

A similar experimental setup has been considered inRef. [23] in the limit of zero intertube hopping. The effectsof this hopping, but in the absence of the dipole-dipoleinteraction, was discussed in Ref. [24]. In this paper, wewill consider both intertube couplings to be present, payingparticular attention to the quantum critical region created bytheir competition. Given the anisotropy of the lattice and thefreedom of the dipole moment to point anywhere in 3-space,we have a great control over the interaction strength in thelongitudinal and transverse directions. If we choose the angleβ between the dipole moment and the normal to the plane tosatisfy sin β = 1/

√3 (β is referred to in the literature [21] as

the magic angle), then interactions in either direction will berepulsive, and it is this case that we will consider from hereon, fixing β and varying α.

The dipole-dipole interaction can be split into two con-tributions: intratube and intertube. While we keep explicitlythe full structure in the intratube part, we approximate theintertube part by a local nearest-neighbor interaction; thisapproximation is valid for the dipolar interaction when, as isthe case here, the anisotropy of the underlying lattice is large.For a fuller discussion see Ref. [21]; note here, though, thatthe dipole interaction in two dimensions (as opposed to threedimensions) is not truly long ranged as there is no divergence

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FIG. 1. (Color online) Proposed experimental setup. An opticallattice in the y direction separates the trapped dipolar bosons intoweakly coupled tubes. The dipole moments of the bosons arepolarized by an external field, B, which can be rotated in threedimensions to change the effective interactions.

in the integral of the interaction. This means that none ofthe essential physics is changed when considering nearest-neighbor interactions only as we are regardless operatingwithin the correct universality class. Now, if x measures alongtube n (at y = nλ) and α is the angle between the horizontalprojection of the dipole moment and the axis of the tubes(Fig. 1), then the contributions to the dipole-dipole interactiontake the form

V intradd = Cdd

4π

sin2 α

|x − x ′|3 δnn′ ,

(1)

V interdd = Cdd

4π

cos2 α

λ3δ(x − x ′)δn,n′±1,

where Cdd is the strength of the dipole-dipole interaction.The system is therefore described by a Hamiltonian H =∑n H

(n)tube + H

(n)inter, where the individual contributions are given

by

H(n)tube =

∫dx�†

n(x)

(− h2

2m∂2x

)�n(x)

+∫

dx dx ′ ρn(x)V intradd (x,x ′)ρn(x ′),

(2)H

(n)inter = −

∫dx J⊥[�†

n(x)�n+1(x) + �†n+1(x)�n(x)]

+∫

dx dx ′ ρn(x)wV interdd (x,x ′)ρn+1(x ′).

Here �†n(x) denotes the creation field operator in the nth

tube, the hopping parameter J⊥ can be found by integratingthe confining potential with respect to the appropriate Wannierfunctions of the longitudinal lattice, and w comes from theintegral off the Wannier functions in adjacent chains. We canalso include a contact interaction in the above model; however,for simplicity we consider the case where this is negligible, asin Ref. [25].

The above-described model has two clear strong couplinglimits; when the interaction is dominant, the ground state willbe a density wave (DW), while if interactions are weak, asuperfluid (SF) phase should emerge. We will now derive thephase diagram of this model as a function of experimentally

controllable parameters, paying particular attention to thequantum critical region between the DW and SF, where weshow that a supersolid phase can also manifest itself.

Firstly we bosonize the Hamiltonian according to thestandard procedure [26,27] via the prescription

�†n(x) ≈

√Aρ0e

−iθn(x),(3)

ρn(x) ≈ ρ0 − 1

π∂xϕn(x) + 2ρ0 cos[2πρ0x − 2ϕn(x)],

where ρ0 is the particle density (which we assume to bethe same for each tube), A is a nonuniversal dimensionlessconstant of order one, and ϕn and θn are the dual fields of thenth tube {[ϕ(x),∂x θ (x ′)] = iδ(x − x ′)}.

Now, neglecting higher harmonics, additive constants, andterms that vanish faster than 1/ρ0, we arrive at the followingbosonized description:

H(n)tube = hvs

2π

∫dx

[K(∂xθn)2 + 1

K(∂xϕn)2

],

(4)

H (n)pair = hvs

2π

∫dx

a2[gint cos 2(ϕn − ϕn+1)

− ghop cos(θn − θn+1)].

Here a ∼ ρ−10 is the short distance cutof,f and the dimension-

less coupling constants gint and ghop are given by

gint = a2ρ0wCdd cos2 α

λ3hvs

, ghop = 4πa2AJ⊥ρ0

hvs

. (5)

In the absence of intertube coupling (ghop = gint = 0), eachtube is governed by a Luttinger liquid-type Hamiltonian withLuttinger exponent K and sound speed vs . The relationshipbetween these parameters and the original model parametersis nonuniversal but has been determined numerically [28],where it is found that K varies smoothly as a function of ρ0r0,where r0 is the effective Bohr radius. In our case r0 is given interms of the dipole angle α by r0(α) = mCdd

2πh2 sin2 α, giving thedependence of K on α as shown in Fig. 2.

This bosonized Hamiltonian is now of a form similar to thatof a quasi-one-dimensional superconductor, so we shall adoptthe approach taken in Ref. [13]. If the interaction coefficientor the hopping coefficient is clearly dominant, then in eitherlimit we can consider the mean-field behavior and arrive ata sine-Gordon model that must be solved along with a self-consistency condition. Setting βint = 2K1/2 and βhop = K−1/2,we find a mean-field Hamiltonian of the form

H mf = hvs

2π

∫dx

[(∂xθ )2 + (∂xϕ)2 + gmf

a2cos(βϕ)

],

(6)gmf = 2g〈cos(βϕ)〉.

Though we explicitly write this for the interaction dominatedcase, due to the duality between θ and ϕ we can treat bothregimes simultaneously.

The expectation value 〈cos(βϕ)〉 is known within thesine-Gordon model [30], so we can solve the self-consistencyconditions to find that gmf ∼ 2g

12 (1+ 1

1−d ), where d = β2/4 isthe scaling dimension of the operator in question. The scalingdimensions of our two interchain perturbations are dint = K

and dhop = 1/(4K).

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FIG. 2. (Color online) A contour plot showing the variation ofthe Luttinger parameter K with the dipole angle α and the relativestrength of the dipole interaction. The most important contour forour purposes is that for K = 1

2 as we shall see this is where theDW and SF phases are maximally in competition. Inset: The barerelation between K and dipole strength based on the parametrizationin Ref. [28]. Although for cold atoms with a magnetic dipole moment,ρ0r0 1, cold molecules with an electric dipole can realistically bein the regime where ρ0r0 > 1 [29].

Inspecting the mean-field solutions we have acquired, wesee that the hopping dominated region is characterized bya nonzero expectation of 〈cos(θ)〉, which corresponds to〈�†

n(x)〉 = 0, and as such represents a SF phase. In the oppositecase when the interaction dominates, the phase is characterizedby a nonzero expectation of 〈cos(2ϕ)〉, which corresponds to〈ρn(x)〉 = ρ0 + C(−1)n cos(2πρ0x), and hence represents acheckerboard DW phase. Notice that if we were to abandonthe “magic angle” that is holding all interactions repulsiveand allow for attractive interactions, the DW phase can alsobe driven into a striped phase; a detailed study of the phaseboundary of these two DW phases is undertaken in Ref. [23].By changing β one could see either DW phase competingwith the SF phase, for concreteness we will hereon continueto assume that the relevant DW phase is the checkerboard.

In order to determine which phase is dominant for a givenparameter range, and hence find the phase diagram, we con-sider both the hopping and interaction terms as perturbationsover a Luttinger liquid and ask which has the higher criticaltemperature. We find the SF and DW susceptibilities (χsf andχdw, respectively) within the random phase approximationusing χ = χ (0)/(1 − gvs

a2 χ (0)). The LL susceptibility for eitherphase is (see, e.g., Ref. [27])

χ (0) ∼ 1

2

a2

vs

(πakBT

hvs

)2d−2

. (7)

The critical temperature Tc is estimated as the temperatureat which an instability arises from a divergence in theassociated susceptibility. As our system is two-dimensional,these transitions are in the Berezinskii-Kosterlitz-Thoulessuniversality class; however, this simple calculation still es-

0 4 2

Tc

Critical angleDensity waveSuperfluid

SF

DW

C

TDW

TSF

FIG. 3. (Color online) A plot of the critical temperature for theDW and SF phases and the implied phase diagram for an illustrativeset of parameters (gint = 0.1, ghop = 0.2, and R = 250) chosen suchthat K ∼ 1

2 at the critical point. Inset: Correction the phase diagramat the critical angle showing the onset of a supersolid (SS) phase.Based on realistic experimental parameters for the dipole momentof, e.g., KRb [33] and a lattice spacing λ ∼ 500 nm, we estimatethe maximum critical temperature of the DW phase to be of orderTDW ∼ 50 nK.

timates Tc correctly [13] in the weak coupling regime. Hencewe find Tc from the condition χ (0) ∼ a2

gvs, so we find that

the interactions dominate, and the system takes on a DW

phase when g1

2−2K

int � g2K

4K−1hop . In the opposing region the intersite

hopping dominates, and the system becomes a superfluid.The parameter we wish to focus upon is the angle α, which

comes from the geometry of our proposed setup. The explicitdependence of the critical temperature of the SF and DWphases on this geometric parameter is given by

kBT SFc = hvs

πa

(1

2ghop

) 2K(Rgmaxint sin2 α)

4K(Rgmaxint sin2 α)−1

,

(8)

kBT DWc = hvs

πa

(1

2gmax

int cos2 α

) 12−2K(Rgmax

int sin2 α)

,

where R = mλ3vs

2πhwa2ρ0and gmax

int = max[gint(α)] = gint(α = 0).Figure 3 shows the dependence of these critical temperatureson α for indicative values of R, ghop, and gmax

int .The above quasi-one-dimensional analysis is valid as long

as the parameters are sufficiently far from criticality. Infact, even though the ordered phases are necessarily two-dimensional, deep in either of them, the correlation functionsretain their one-dimensional character [31]. However, near thequantum-critical point (QCP) between the DW and SF, theentire quasi-one-dimensional formalism breaks down, and afull two-dimensional field theory is needed, the dimensionalanisotropy here being an irrelevant operator [34].

Exactly at the QCP, where DW can be smoothly rotatedinto SF, the order parameter acquires an enlarged (and non-Abelian) symmetry [13]. This is in analogy with SO(5)theories of the antiferromagnetic-superconductor transition[32], except that the enhanced symmetry at the SF-DWtransition is O(4) [34]. Following Jaefari et al. [34], in the

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vicinity of the QCP we can then write a 2 + 1 dimensionalnonlinear sigma model in terms of this enhanced O(4) orderparameter, which has a Lagrangian

Leff[N ] = 1

2g0(∂μN )2 − w∂μNaOab∂μNb − hNaOabNb.

(9)

Here N is the order parameter, g0 ∼ √gint + ghop is the

effective coupling constant, w ∼ K − 12 signifies the departure

from the point where the two interchain terms have the samescaling dimension, h ∼ gint − ghop measures the distance fromthe critical point, and O and O are two fixed matrices thatbreak the enhanced symmetry in the appropriate way. It turnsout [34] that the term proportional to w is marginally irrelevantin the renormalization group (RG) sense, meaning that thegeneric quantum critical behavior of our original model canbe described by the above nonlinear sigma model. Note that allwe are doing here is writing the most general Langrangian withall the correct symmetries. This model has been well studiedusing RG methods [34,35], where it is found that there is atetracritical point at g0 = gc ∼ 1 separating different types ofcritical behavior. In the case g0 < gc which is always satisfiedin our present weakly coupled chains scenario, there is a regionaround αc where there is coexistence of both the DW andSF order parameters, in other words, a supersolid phase. Thecorrected phase diagram in the vicinity of the critical point isshown schematically in the inset to Fig. 3.

Unfortunately precise predictions regarding the supersolidphase are not easily attainable because this phase is deep withinthe strongly correlated regime and not accessible from ourinterchain perturbation theories. Its existence must rather beinferred based on RG relating to the symmetries of the orderparameter at the quantum critical point, leading to a qualitativephase diagram. This said, as indicated in to Fig. 3, the relevantregion to look for the supersolid phase in any experimentalinvestigations of this system is set by the transition temperature

of the suppressed phase, which could be found by our mean-field calculations.

We can compare our results to those found in isotropic 2Danalogs of this system: In an isotropic lattice with nearest-neighbor interactions it has been shown that [5] the supersolidphase can arise at the interface of the SF and DW phases.The stability of the SS phase is considered in Ref. [36],in which the authors note that this phase is possible in thepresence of a lattice. This is as opposed to the free system inwhich, for a dipole-dipole interaction, it has been seen [37]that there is a depletion of the SF phase before the onset ofa dipole solid phase. Although our method is not applicableto the isotropic limit, the fact that the phase diagrams formodels on a 2D lattice are so similar suggests that therewill not be any additional transitions as we decrease theanisotropy.

Returning finally to the proposed experimental setup inwhich we expect this phase diagram to manifest itself, somegroups [38] have reported single-site resolution in flourescenceimaging of 87Rb in a two-dimensional lattice setup similar tothat we describe, so it may well be possible to probe the DWphase of this system in situ. A common way to probe theSF phase is releasing the bosons from their confinement, andlooking for high contrast in the interference pattern in theabsorbtion data [3]. The hallmark of the supersolid phase wehave described would be the presence of both a DW and SForder parameter.

In summary, we have shown how a popular model incondensed matter physics of striped superconductivity mayalso appear in a cold atom experiment with trapped dipolarbosons in a quasi-one-dimensional geometry. The rich phasediagram of the model can be easily explored by changing theangle of a polarizing field and shows crystalline, superfluid,and supersolid phases, on top of the high-temperature slidingLuttinger liquid phase.

We thank Joseph Betouras, Dima Gangardt, Chris Hooley,and Jorge Quintanilla for useful discussions.

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