Hydrodynamics of a superfluid smectic
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec Hydrodynamics of a superfluid smectic
Johannes Hofmann *, Wilhelm Zwerger * Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden Technische Universit¨at M¨unchen, Physik Department, James-Franck-Strasse, 85748Garching, GermanyE-mail: [email protected], [email protected]
Abstract.
We determine the hydrodynamic modes of the superfluid analog of asmectic A phase in liquid crystals, i.e., a state in which both gauge invariance andtranslational invariance along a single direction are spontaneously broken. Such asuperfluid smectic provides an idealized description of the incommensurate supersolidstate realized in Bose-Einstein condensates with strong dipolar interactions as well asof the stripe phase in Bose gases with spin-orbit coupling. We show that the presenceof a finite normal fluid density in the ground state of these systems gives rise to a well-defined second-sound type mode even at zero temperature. It replaces the diffusivepermeation mode of a normal smectic phase and is directly connected with the classicdescription of supersolids by Andreev and Lifshitz in terms of a propagating defectmode. An analytic expression is derived for the two sound velocities that appear inthe longitudinal excitation spectrum. It only depends on the low-energy parametersassociated with the two independent broken symmetries, which are the effective layercompression modulus and the superfluid fraction. ydrodynamics of a superfluid smectic
1. Introduction
The question whether superfluidity might persist even in a solid state has a long history.It was briefly discussed in the classic paper on off-diagonal long range order by Penroseand Onsager [1], concluding that no supersolid phase is possible because mobile defectsor interstitials would always be frozen out at zero temperature. The idea was takenup by Andreev and Lifshitz [2] who showed that supersolids are at least a theoreticalpossibility and are characterized by a sound-like rather than diffusive propagation ofdefects or interstitials. An upper bound on the associated superfluid fraction f s < f s . − was inferred from a reduced value of the rotational inertia in He below 250 mK by Kimand Chan [5, 6]. On the basis of a number of further experiments [7] and microscopic ab-initio calculations [8, 9], however, the likely conclusion is that the observed non-classicalrotational inertia in He is not caused by supersolidity (for a review, see Ref. [10]).In recent years, renewed interest in the subject has been triggered by a number ofexperiments with ultracold gases, in particular with Bose-Einstein condensates in drivensingle or double cavities [11, 12] or in the presence of spin-orbit coupling [13]. In bothcases, the period of the density profile is set externally, either by the wave vector of thecavity photons or the momentum transfer associated with the Raman coupling betweentwo internal states, which leads to a density modulation along a single direction. Morerecently, supersolid phases with an interaction-generated density modulation along theaxial direction of a cigar-shaped trap have been realized with dipolar gases in a regimewhere the dipolar length ℓ d is of the same order as the short-range scattering length a s [14, 15, 16]. These supersolids are generically incommensurate, with many atoms perunit cell. In particular, there is typically a large superfluid fraction, which allows toobserve signatures of supersolidity more easily.In the present work, we analyze the spectrum of hydrodynamic and Goldstonemodes for a general class of supersolids that exhibit a mass-density wave along a singledirection. They may be thought of as a superfluid version of a classical smectic A liquidcrystal [17]. As emphasized by Martin et al. [18], the structure of long-wavelength andlow-energy excitations in any thermodynamic phase is completely described in terms ofconserved variables and broken symmetries. The superfluid smectic, where two separatesymmetries — gauge invariance and translational invariance — are spontaneouslybroken, is thus expected to exhibit an excitation spectrum with two separate Goldstonemodes. This statement is not obvious, however, because Goldstone modes may be ydrodynamics of a superfluid smectic .For the superfluid smectic, broken translation invariance along a single directionimplies that the superfluid mass density tensor is anisotropic and there is a finite normalfluid density for longitudinal motion even at zero temperature. Besides the standardbulk sound mode, this gives rise to a separate second-sound type mode whose velocity isset by a combination of the layer compression modulus and the superfluid fraction. Thisis different from the individual cases of a smectic phase, where a secondary sound modevanishes for propagation both along or perpendicular to the layer [17], or a homogeneoussuperfluid, where second sound is an entropy wave that becomes ill-defined at lowtemperatures.This paper is structured as follows: In Sec. 2, we determine the spectrum oflongitudinal hydrodynamic and Goldstone modes of a superfluid smectic with bothGalilean and time-reversal invariance. The resulting first and second sound velocitiesturn out to depend only on three thermodynamic parameters, which are the bulk andlayer compression modulus together with the superfluid fraction. In particular, it isshown that the second sound mode derives from a combination of two diffusive modes inthe normal smectic, which are the heat diffusion mode and a characteristic permeationmode that describes defect diffusion. The physical nature of the second propagatingmode is discussed in detail in Sec. 3, where the connection to the classic Andreev-Lifshitz picture of supersolids in terms of a propagating defect density mode is made.Section 4 contains a discussion of our results and their relevance to recent measurementsof the excitation spectrum of supersolid phases realized with dipolar gases. There aretwo appendices, one on the Leggett bound for the superfluid fraction in superfluidswith an inhomogeneous density profile and a second one on the hydrodynamic modestransverse to the direction of spatial order. This allows to connect our results to earlierwork by Radzihovsky and Vishwanath on superfluid liquid crystal states in imbalancedFermi superfluids with either Larkin-Ovchinnikov or Fulde-Ferrell type order along asingle direction [23, 24]. T i = mj i that links thegenerator of translations to the particle current receives additional corrections that involve the spin-projection s z of the two-component Bose gas [21]. Since the smectic stripe phase is spin-balanced [22],these corrections do not enter. ydrodynamics of a superfluid smectic
2. Hydrodynamic and Goldstone modes in a superfluid smectic
In order to elucidate the similarities and differences between standard liquid crystals andthe superfluid version of the smectic phase considered in this paper, we start with thecase where no superfluidity is present. To simplify the discussion while still keeping theessential physics of superfluid smectic phases, we consider a two-dimensional situationwhere the smectic order shows up as a weak periodic modulation n eq ( r ) = ¯ n + ∞ X l =1 n l cos ( lq y ) ≈ ¯ n + n cos ( q y ) + . . . (1)of the density along the y -direction with a fundamental reciprocal lattice vector q . Fora non-vanishing Fourier component n = 0 in Eq. (1), translation invariance along y is broken. The associated new hydrodynamic variable is a scalar field u ( x, y ) that iscalled the layer phase [17]. It is defined by considering deviations from the equilibriumdensity (1) of the form n ( x, y ) = ¯ n + n cos [ q y − q u ( x, y )] . (2)At the level of a hydrodynamic description, there are four conserved quantities, whichare particle number, the two-dimensional momentum as well as energy. Combined withthe single symmetry-breaking variable u , there must be five hydrodynamic modes [18].Only one of them is a Goldstone mode, which counts twice in a hydrodynamic countbecause it is necessarily a propagating mode. As found by Martin et al. [18], theGoldstone mode of a smectic A liquid crystal is a transverse sound mode with a frequency ω t ( q ) ≃ p B/ρq q x q y ∼ sin ψ cos ψ that depends on the angle ψ between the wavevector q and the direction of density order. Here, ρ is the total equilibrium massdensity and B the layer compression modulus. It is defined by the elastic contribution f el = B ( u ′ ) / . . . to the free energy density associated with small longitudinaldistortions u ′ = ∂ y u of the smectic order [17]. The second propagating mode is abulk sound mode ω = ± c l q whose velocity c l ( ψ ) has only a weak dependence on theangle ψ [25]. In particular, for longitudinal propagation, its velocity c l ( ψ = 0) = K + Bρ (3)is determined by the sum of the (isentropic) bulk modulus K = ρ ∂p/∂ρ (cid:12)(cid:12) s,u ′ and the layercompression modulus B [18]. For weak modulations of the density n ≪ ¯ n , the bulkmodulus dominates and thus the sound velocity is essentially that of a fluid phase. Thelast remaining mode in addition to the Goldstone mode and the sound mode describesheat diffusion.Consider now the special case in which the wave vector q is directed either alongor perpendicular to the y -direction. Here, due to the peculiar angular dependence ydrodynamics of a superfluid smectic c t ( ψ ) ∼ sin ψ cos ψ of the transverse sound velocity, the Goldstone mode is absent. Bymode counting, there must then be three diffusive modes in addition to the propagatingbulk sound mode. The first one is the heat diffusion mode that is present at arbitraryvalues of the angle ψ . The second one is a transverse momentum diffusion modewith frequency ω = − iν q , where ν is a kinematic viscosity [17]. The third modewith frequency ω = − iD p q is special to smectic A liquid crystals and is called thepermeation mode [17]. It describes a diffusive process in which particles are exchangedbetween adjacent layers without changing the average periodic structure. The associateddiffusion constant D p = ζ B is determined by the layer compression modulus B anda dissipative coefficient ζ . The permeation mode may be viewed as an analog ofvacancy diffusion, a process that gives rise to an independent hydrodynamic mode in anycrystal [18]. As will be shown below, it is precisely the permeation mode in combinationwith the heat diffusion mode that turns into the Goldstone mode of the superfluidsmectic phase, where exchange between the layers occurs in a reversible manner bynon-dissipative, propagating mass currents. For a description of the low-energy excitations of a superfluid smectic phase, the presenceof superfluidity needs to be accounted for on a thermodynamic level by expressing thedifferential of the entropy density sT ds = dε − ( µ/m ) dρ − v n d g − h d ( ∇ u ) − j s d v s (4)as a function of the conserved variables energy density ε , mass density ρ , andmomentum density g = ρ n v n + ρ s v s together with the gradient ∇ u of the layer phaseand the superfluid velocity v s , which characterize the two broken symmetries. Thethermodynamic field h = ∂f el ∂ ( ∇ u ) (cid:12)(cid:12)(cid:12)(cid:12) T,A,N,v n ,v s = Bu ′ e y − K ∂ x u e x + . . . (5)conjugate to the gradient ∇ u of the layer phase determines the elastic free energy ofthe smectic. Here and in the remainder of the paper, we shall use the notation u ′ = ∂ y u for the derivative of the layer phase in the direction of the periodic modulation. Notethat for longitudinal modes, only the layer compression modulus B plays a role. Asecond Gaussian-curvature type elasticity appears in Eq. (5) for excitations with afinite component q x of the wave vector parallel to the layers. It involves the splayelastic constant K [17], which is relevant for the dispersion of Goldstone modes in aLarkin-Ovchinnikov phase of imbalanced Fermi superfluids as discussed by Radzihovskyand Vishwanath [23, 24], see the discussion in Appendix B.The conjugate variable to the momentum density g is the normal velocity v n ,which also appears in the superfluid mass current density j s = ρ s ( v s − v n ). Thelatter relation follows from the thermodynamic derivative j s = ∂f∂ v s (cid:12)(cid:12) T,A,N,v n , ∇ u , where the ydrodynamics of a superfluid smectic f ∼ − ρv n + ( v s − v n ) T ρ s ( v s − v n ) is dictated by Galileaninvariance [26, 27]. Quite generally, for superfluids with an underlying periodic structure,the normal velocity v n = ∂ t u is determined by the time derivative of the displacementfield u . This relation — which is valid at the linearized level around equilibrium andis thus sufficient for the derivation of the hydrodynamic modes — has been derived ingeneral form by Son [26] as a consequence of Galilean invariance. In particular, forhydrodynamic modes with wave vector q along the y -direction, the associated normalvelocity v n,y = ∂ t u is just the time derivative of the scalar layer-phase variable u . Similarto the elastic constants B and K , the superfluid and normal mass density tensors ρ s and ρ n , which are constrained by ρ s + ρ n = ρ
1, are thermodynamic variables definedvia j s as the conjugate field to v s . These relations are a straightforward tensor generali-zation of those from standard two-fluid hydrodynamics. For translationally-invariantfluids, the normal fluid density ρ n ∼ T d +1 ρ n ) yy ≃ ρ · ( n / ¯ n ) of the normal fluid density even atzero temperature. A strict lower bound for ( ρ n ) yy follows from a variational argumentdue to Leggett [3], which is discussed in Appendix A.The complete set of hydrodynamic modes in a superfluid smectic phase followsfrom the equations of motion for the conserved densities together with the two variablesthat describe the underlying broken symmetries. The latter are the superfluid velocity v s and the gradient of the layer phase ∇ u , which are both longitudinal vectors andthus effectively scalar quantities. Together with the dynamic equations for the particledensity ρ , the momentum density g and energy density ε , the six resulting equations ofmotion are given by ∂ t ρ + ∇ · g = 0 (6) ∂ t g i + ∂ j π ij = 0 (7) ∂ t ε + ∇ · j ε = 0 (8) ∂ t ( ∇ u ) − ∇ v n,y = 0 (9) ∂ t v s + ∇ µ/m = 0 . (10)The first three equations (6)-(8) are continuity equations that link the time derivativesof the densities to the divergences of the momentum density g , the stress tensor π ij ,and the energy current j ε , respectively. As already discussed above, Eq. (9) expressesthat a constant shift along the direction of smectic order changes the layer phase by aconstant . Finally, Eq. (10) is the Josephson equation (neglecting a quadratic term inthe velocities) that describes the dynamics of the superfluid phase.From the differential of the entropy (4) and the dynamic equations (6)-(10), we ζ ∇ h , where ζ is the dissipative coefficient that enters the diffusion constant D p = ζB of the permeation mode [17]. ydrodynamics of a superfluid smectic T ( ∂ t s + v n · ∇ s ) of the entropy densitythat depends on spatial gradients ∇ T , ∇ µ , ∂ i v n,j and ∇· j s of the thermodynamic forces.For the inviscid fluid considered here there is no entropy production, which implies aparticular series of constitutive relations that link the currents and the thermodynamicforces. To leading order in the velocities, these constitutive relations read: g = ρ v n + j s (11) π ji = pδ ij − ( h i δ j,y ) (12) j εi = ( ε + p ) v n,i + µj s,i /m. (13)Compared to a simple fluid, at this level the superfluid order modifies the particle andenergy current, while the smectic order modifies the stress tensor. In addition, as statedabove, the thermodynamic forces j s and h are linked to the velocities by j s = ρ s ( v s − v n )and h = B∂ y u e y .The linearized hydrodynamic equations of motion are obtained by substituting theconstitutive relations in the dynamic equations and expanding the thermodynamic forcesto leading order in the hydrodynamic variables around equilibrium. For motion alongthe direction of the smectic order (here, the y -direction), the resulting equations onlyinvolve the yy component of the superfluid mass density tensor, which we denote by ρ s = ρ − ρ n in the following. In this configuration, the transverse momentum degree offreedom decouples and gives rise to a diffusion mode. For the remaining five degrees offreedom, we obtain the characteristic equation − ω/q K/ρ − ω/q − B sT ρ s ρ n − ω/q − ρ ˜ sT ρ s ρ n K/ρ − ˜ s/ρc V − ω/q
00 1 /ρ n − ρ s /ρ n − ω/q δρg L δqv s u ′ = 0 . (14)Here, ω is the frequency of the mode and q the associated longitudinal momentum.Moreover, we introduce a heat current density variable δq = δε + ε + pρ δρ with˜ s = s/ρ the entropy per particle and mass, while c V = T ∂ ˜ s∂T (cid:12)(cid:12) ρ is the associatedspecific heat. Apart from the transverse momentum diffusion mode mentioned above,Eq. (14) contains another diffusive zero mode with eigenvector ( δρ, g L , δq, v s , u ′ ) =( − Bρ/K, , − Bc V / ˜ s, , ω + ω q (cid:20) − Kρ − Bρ n − ˜ s Tc V ρ s ρ n (cid:21) + q (cid:20) BKρ ρ s ρ n + ˜ s Tc V ρ s ρ n K − Bρ (cid:21) = 0 . (15) ydrodynamics of a superfluid smectic s T /c V at low temperatures, we obtain two undampedpropagating modes ω = ± c , q with velocities c , = K ρ + B ρ n ± (cid:20)(cid:18) Kρ + Bρ n (cid:19) − f s KBρρ n (cid:21) / . (16)The physics behind these two longitudinal modes and in particular also the associatedeigenvectors will be discussed in detail in the following section. We note that theresult (16) turns out to be equivalent to a result obtained by Yoo and Dorsey [27]for the longitudinal hydrodynamic modes of a crystalline supersolid. Indeed, Eq. (16)agrees with their Eqs. (48) and (49) if we identify the parameters 1 /χ → K withthe bulk compression modulus and λ → B with the layer compression modulus.Moreover, we do not include a strain-density coupling and thus γ = 0 in the notation ofRef. [27]. However, heat currents are neglected in Ref. [27] and therefore the contribution O (˜ s T /c V ) in Eq. (15) is absent. As discussed in Appendix B, this is of relevance if oneconsiders the hydrodynamic modes parallel to the layers, which involve a conventionalentropic second sound mode. In general, therefore, the hydrodynamic modes of asuperfluid smectic phase differ from those of supersolids in which translation invariance isbroken in all directions — it is only in the specific case of purely longitudinal propagationthat both systems behave in a similar manner. An example for this is provided bythe incommensurate supersolid phase of a two-dimensional Bose gas with interactionsdescribed by a soft disc potential, whose longitudinal modes have been determinednumerically [29, 30].
3. Defect density propagation and the limit of fourth sound
For a better understanding of the physics underlying the two propagating modes foundin Eq. (16) and in particular the connection to the classic picture of supersolids in termsof wave-like propagation of defects proposed by Andreev and Lifshitz [2], it is instructiveto rederive the longitudinal modes (16) and the associated eigenvectors with a slightlydifferent set of variables introduced by Yoo and Dorsey [27]. They decompose smallfluctuations of the mass density δρ = − ρ u ′ + δρ △ (17)into a contribution − ρ u ′ associated with deformations of the periodic structure andan additional defect density ρ △ . This separates the density variation of a defect freecrystal, for which a change in density is tied to the divergence of the deformation field,from the additional density change associated with the motion of defects or interstitials.The defect density obeys a continuity equation ∂ t δρ △ = − ∂ y ρ s ( v s − v n ) (18)whose conserved current g ∆ = ρ s ( v s − v n ) is just the Galilean-invariant superfluid mass-current density, determined by the counterflow between the superfluid and the normal ydrodynamics of a superfluid smectic v n = ∂ t u . The second time derivative of the defect density is coupled to thestrain field variable u ′ according to ∂ t δρ △ = ρ s ∂ y ( µ/m ) + ρ s ∂ t u ′ . (19)In a situation where the lattice is almost rigid, the contribution that involves the layerphase variable u ′ may be neglected. As a result, the defect density exhibits wave-likepropagation with a velocity given by c = f s ( K/ρ ). This is analogous to fourth sound ofsuperfluid He in narrow capillaries, where the normal fluid component is pinned by thewalls. It describes the oscillation of the superfluid with no motion of the lattice, a limitwhich is perfectly realized in the superfluid phase of bosons in an optical lattice [31],where the periodic modulation of the density is externally imposed and not caused byinteractions. For the superfluid smectic, this limit is reached when the layer compressionmodulus contribution
B/ρ n ≫ K/ρ in Eq. (16) dominates that of the bulk. In generalhowever, as Eq. (19) shows, the defect density and the longitudinal strain u ′ are coupled.An explicit result for the eigenmodes of the superfluid smectic phase thus requires tosimultaneously solve the equation for δρ △ and for u ′ , which reads ρ n ∂ t u ′ = ∂ y [ − p + Bu ′ + ρ s ( µ/m )] . (20)The solution of the coupled equations (19) and (20) does of course reproduce theresult (16) above. The associated dimensionless eigenvectors are δρ △ /ρu ′ ! = c / ( K/ρ )1 ! (21)for the first sound mode with speed c and δρ △ /ρu ′ ! = c / ( K/ρ )1 ! (22)for the second sound mode with speed c . Specifically, for an almost rigid lattice with B/ρ n ≫ K/ρ , the velocities reduce to c = B/ρ n + f n K/ρ and c = f s K/ρ with c ≫ c . In this limit, therefore, second sound is essentially a defect density modewith no involvement of the lattice. Recall that this mode derives from the diffusivepermeation mode of a smectic, which describes particle diffusion without a change inthe periodic structure. By contrast, the eigenvector ( δρ △ /ρ, u ′ ) = ( f s ,
1) for first soundin this limit involves the defect density with weight f s . In standard supersolids, where f s is expected to be small compared to one, this mode predominantly involves the strainfield, i.e., it describes the motion of the lattice.A rather different situation arises in the opposite limit of a small normal fraction f n ≪ f n ≃ ( n / ¯ n ) → B/ρ n inEq. (16) appears to diverge. This is not the case, however, since the elastic constant B ydrodynamics of a superfluid smectic B ∼ | n | vanishes like the square ofthe order parameter n . As a result, the ratio B/ρ n turns out to be finite in thelimit n → c right at the transition to the fluid phase. The layer compression modulus B ∼ | n | x must therefore vanish with an exponent x >
2. In fact, as was shown by Grinsteinand Pelcovits [32], the renormalized value B ren vanishes at long distances even in thesmectic phase with n = 0 due to anharmonic corrections to the linear elastic continuummodel used above. In the following, this complication will be ignored. The ratio B/ρ n is therefore a thermodynamic parameter which is finite in the superfluid smectic, butvanishes in the homogeneous superfluid where translation invariance is not broken. Inparticular, in the limit K/ρ ≫ B/ρ n of a weak density modulation, the velocities (16)approach c = ( K + B ) /ρ and c = B/ρ n . The velocity of the compression mode isthus unchanged compared to that in the normal phase. In terms of the variables δρ △ /ρ and u ′ , the eigenvector associated with first sound is dominated by the layer phasevariable with a negligible contribution from the defect density. Due to u ′ ≃ − δρ/ρ , theperiodic structure of the smectic therefore adiabatically follows the density fluctuations δρ in this mode, which describes oscillations of the lattice. The second sound mode,by contrast, whose velocity B/ρ n is determined by the ratio of the layer compressionmodulus B and the normal fluid density, involves both an oscillation in the longitudinalstrain field as well as the defect density with essentially equal magnitude. In physicalterms, it describes a wave-like propagation of particles in addition to that associatedwith variations in the smectic lattice structure, replacing the diffusive permeation modeof a normal smectic phase.
4. Summary and experimental implications of the results
We have derived the spectrum of hydrodynamic modes in supersolids that exhibit adensity modulation along a single direction only, with an arbitrary number of particles ina unit cell. This state may be viewed as a superfluid version of a classical smectic A liquidcrystal. It has a highly anisotropic spectrum of modes that is entirely determined bythe number of broken symmetries and a few thermodynamic parameters. In particular,the longitudinal excitations exhibit a second sound like mode that remains well-definedeven at zero temperature. An analytical result [Eq. (16)] has been derived for thesound velocities, which only contains the effective layer compression modulus and thesuperfluid fraction as the two low-energy parameters associated with broken translationand gauge invariance.As mentioned in the introduction, a variety of phases where superfluidity coexistswith periodic spatial order have been observed in the context of ultracold gases inrecent years. Among those, supersolids realized in driven cavities [11, 12] are special ydrodynamics of a superfluid smectic n may be tuned by thestrength Ω of the Raman coupling [35]. The role of the layer phase variable u is played bythe relative phase φ between the complex coefficients C and C in the Gross-Pitaevskiiansatz [35] (cid:18) ψ a ψ b (cid:19) = r NV (cid:20) C (cid:18) cos θ − sin θ (cid:19) e ik x + C (cid:18) sin θ − cos θ (cid:19) e − ik x (cid:21) (23)for the spinor wave function, where θ is a variational parameter and k sets the densitymodulation with wave vector q = 2 k . The spectrum of elementary excitations ofthe stripe phase has been determined by Li et al. [22] within a Bogoliubov approach.For wave vectors along the direction of ordering, there are two gapless modes whosedispersion ω ( q + q ) = ω ( q ) is periodic in the associated Brillouin zone. The upper oneis a density mode with a velocity c while the lower one is a spin excitation, again witha linear spectrum ω = c q at small values of the longitudinal wave vector q . It is anopen problem to extend our analytical result (16) for the two velocities in the supersolidphase of a single-component BEC to this two-component system and, in particular, todetermine the effective layer compression modulus B and the associated finite normalfluid density ρ n in terms of the microscopic parameters of spin-orbit coupled BECs.An example where our results are of direct experimental relevance are dipolar gasesin cigar-shaped traps. If the dipolar length ℓ d is larger than the tunable scatteringlength a s associated with short-range interactions, they exhibit a supersolid phase withan interaction-driven density modulation along the weakly-confined direction [14, 15, 16].In practice, these are highly inhomogeneous systems with only a few times 10 atoms.The modulation of the density in the supersolid phase, which has a typical lengthof the unit cell of around 0 . µ m, splits the BEC into a small number of coherentlycoupled droplets. Experimentally, a characteristic signature of the supersolid state in atrap compared to the standard BEC is the emergence of an additional collective modeat low energies [36, 37, 38]. Specifically, as observed by Tanzi et al. [37], the axialbreathing mode of a trapped BEC with frequency ω B = p / ω y shifts towards higherfrequencies beyond the transition to a state with finite density modulation. In addition,a new mode appears whose frequency goes down as the density contrast increases. Thisobservation can be understood on a qualitative level within our hydrodynamic approach ydrodynamics of a superfluid smectic q min ≃ /l y of the longitudinalwave vector in the trap is set by the inverse of the axial confinement length l y . Asdiscussed in Sec. 3, the supersolid phase is characterized by a compression mode tied toan oscillation in the lattice structure and a lower-energy Goldstone mode that describesdissipationless transport of defects, independent of a change in the lattice constant. Inthe presence of a trap, the splitting of the Bogoliubov-Anderson mode of a homogeneousBEC into two independent propagating modes in a supersolid phase shows up as abifurcation into a compressional mode at ω ≃ c /l y , the frequency of which is shiftedupwards in the supersolid phase because of the corresponding shift in the sound velocitythrough the additional contribution of the layer compression modulus B . In addition,a second mode appears at lower frequencies ω ≃ c /l y . It becomes increasingly softupon entering more deeply into the supersolid phase since the velocity c decreaseswith the superfluid fraction f s . In particular, in the limit where the periodic structureis essentially rigid, it approaches c = p f s K/ρ , which vanishes as the square root ofthe superfluid fraction f s → ω B = p / ω y , whichare completely independent of interactions, to the collective excitations studied in ourpresent work. A trap analog of the true Goldstone mode, which is properly definedonly in a homogeneous system, has also been seen in experiments by Guo et al. [36]in a small array of three droplets, where an in-phase dipole mode associated with theaxial motion of the whole cloud coexists with an out-of-phase mode at frequencies muchsmaller than that of the trap. In the latter, atoms are moving between the dropletsat a fixed center-of-mass position, corresponding to a counterflow between the periodiclattice and a superfluid of defects on top, analogous to the mode at ω ≃ c /l y discussedabove.A possible way to measure the sound velocities predicted in Eq. (16) — andthus to extract quantitative values for the layer compression modulus B and thesuperfluid fraction f s even with dipolar gases in cigar-shaped traps — is suggestedby the experiments of Petter et al. [39]. By an extrapolation to small wavevectors, theyallow to infer the presence of a sound-like excitation in the regime of a homogeneoussuperfluid with a linear dispersion ω = c q , which in fact persists for wave vectorsup to q = l − z ∼ . l − y [39]. Provided that the extrapolation into the linear regime ispossible in the supersolid phase, our analytical results for the mode velocities in Eq. (16)will allow to determine the three parameters involved in the thermodynamic descriptionof the superfluid smectic . Specifically, the bulk compression modulus K is fixed bythe velocity c = K/ρ of first sound in the homogeneous superfluid before the density ω q = c q into two independent ones in thesupersolid phase has been seen in numerical simulations based on a modified Gross-Pitaevskii equation,see Ref. [38]. ydrodynamics of a superfluid smectic B/ρ n and f s , using for examplethe relations c + c = K/ρ + B/ρ n and c c = f s · ( K/ρ ) (
B/ρ n ). Beyond a directmeasurement of the layer compression modulus B , this might allow to check the values ofthe superfluid fraction f s extracted from the contrast C = ( n max − n min ) / ( n max + n min ) ofthe density profiles in Ref. [40] via the Leggett bound. An open question in this contextis whether the superfluid-to-supersolid transition is continuous or first order with acorresponding jump in the contrast, as suggested by the discontinuity in the phononvelocities of the Goldstone modes obtained within a Bogoliubov description [30] anda recent generalization of the classical Hansen-Verlet criterion for freezing to quantumfluids [41].In a more general context, an interesting challenge for future studies is to investigatethe hydrodynamics of supersolid phases with a frozen, inhomogeneous density that is notperiodic. Such a superglass phase has been found in numerical simulations of Bose fluidswith strong short-range repulsion like He that are rapidly quenched to low temperatures[9]. They violate Galilean invariance and thus might realize a superthermal phaseconjectured some time ago by Liu [42], in which a finite temperature gradient canbe sustained in a static, non-dissipative situation.
Acknowledgments
It is a pleasure to acknowledge a number of helpful comments by A. T. Dorsey, A. J.Leggett, S. Moroz and L. Radzihovsky.
Appendix A. Leggett bound on the superfluid fraction
Leggett derived an upper bound for the superfluid fraction f s in a ground state withbroken gauge and translation invariance, which only involves the microscopic densityprofile [3]. In the special case of a purely one-dimensional configuration, Leggett’s resultstates that the superfluid fraction f s = ρ s ρ ≤ b ¯ n R b dy/n ( y ) (A.1)is bounded from above by an integral over a unit cell of the lattice (taken to be alongthe y -direction as in Eq. (1)) with lattice constant b = 2 π/q and average density¯ n . It is important to note that the bound does not rely on any commensurabilitycondition: it applies both to a commensurate situation, where the product ¯ n b = k of theaverage density and the lattice constant b is an integer k = 1 , , . . . , or the genericallyincommensurate case associated with a weak mass-density wave, which is of relevancefor the superfluid smectic. The bound becomes increasingly tight for densities thatare strongly suppressed at intermediate points within a unit cell, as expected in a realcrystal. In turn, superfluidity is favored if the density exhibits only small fluctuations ydrodynamics of a superfluid smectic n . Of course, the bound (A.1) does not provide a sufficient criterionfor superfluidity in a state with broken translation invariance: a finite value of the boundis still compatible with no superfluidity at all. What it shows, however, is that a groundstate of bosons with non-uniform density necessarily has a finite normal fluid fraction.Within a Gross-Pitaevskii description of the superfluid smectic phase (see,for example, Ref. [37]), it is assumed that the one-particle density operatorˆ ρ = P α λ (1) α | ψ α ih ψ α | is dominated by a single macroscopic eigenvalue λ (1)0 ≃ N . Inthe regime, where the ground state exhibits a weak density wave, the associatedeigenfunction h x | ψ i ∼ δ cos ( q y ) + . . . involves a small admixture of order | δ | ≪ y -direction (in Ref. [40], this is called thesine ansatz). The resulting equilibrium density n eq ( x ) = h x | ˆ ρ | x i → ¯ n (1 + δ /
2) [1 + δ cos ( q y )] (A.2)is then of the form assumed in Eq. (1) with n / ¯ n ≃ δ to linear order in δ . For | δ | ≥ | δ | = 1) or two different pointswithin the unit cell, which leads to a divergent integral in the denominator of Eq. (A.1).This is revealed by the special form f s ≤ (1 − δ ) / δ / → − δ for δ → | δ | >
1. As pointed out in the maintext, the Leggett bound implies that the normal fluid fraction f n ≥ δ + . . . in a stateof the form (A.2) is bounded from below by a finite value even at zero temperature.Unfortunately, the precise numerical factor connecting f n with the square of the densitymodulation is not determined by this variational argument.Finally, we emphasize that the use of the Leggett bound to extract a finite normalfluid density from a Gross-Pitaevskii ansatz for the supersolid ground state requiresto take into account excitations beyond the Gross-Pitaevskii description. Indeed, thebound (A.1) relies on applying a finite total twist Θ = R y ∂ y θ sf ( y ) of the local phase θ sf ( y ) of the eigenfunction h x | ψ i and allowing the system to adjust the functional formof θ sf ( y ) to minimize the energy. The minimum is achieved by putting in the imposedphase change Θ in regions of small density, which leads to the bound (A.1). In thecontext of a Gross-Pitaevskii description of supersolid phases, the problem of properlydefining a normal fluid fraction has been addressed by Josserand et al. [43, 44]. Theysuggest to decompose the amplitude and phase of the wave function into slowly- andrapidly-varying components and eliminate the latter. In practice, the elimination of thefast degrees of freedom, varying on the scale of the lattice constant, cannot be performedin explicit form. It is thus not clear whether this procedure is able to describe supersolidswith finite values of f n and eventually recover a normal solid phase with f n = 1. ydrodynamics of a superfluid smectic Appendix B. Transverse modes of a superfluid smectic and the relation tosuperfluids with unidirectional Fulde-Ferrell or Larkin-Ovchinnikov order
In this appendix, we provide results for the hydrodynamic modes of the smectic Asuperfluid beyond the purely longitudinal situation discussed in Sec. 2. Webegin by collecting the dynamical equations for the hydrodynamic variables( δρ, g L , g T , δε, v s , ∇ u ), Eqs. (6)-(10), written in Fourier space for arbitrary angles ψ between the direction of propagation and the direction of smectic order. For this, it ishelpful to decompose the current into a longitudinal and a transverse part defined by g L = q x q g x + q y q g y (B.1) g T = q x q g y − q y q g x . (B.2)In terms of the longitudinal current, the continuity equation (6) reads: − ωq δρ + g L = 0 . (B.3)To rewrite expression (7) for the longitudinal current, we introduce the longitudinal andtransverse part of the conjugate field h , which read with our choice (5) of the elasticfree energy h u,L = u ′ B cos ψ (B.4) h u,T = u ′ B cos ψ sin ψ. (B.5)Moreover, in expanding the pressure in terms of the hydrodynamic variables, we neglectits dependence on entropy, superfluid velocity, and layer phase gradient . We obtain − ωq g L + p − h u,L cos ψ = − ωq g L + (cid:2) Kδρ − B cos ψ u ′ (cid:3) = 0 . (B.6)Likewise, Eq. (7) for the transverse current becomes − ωq g T − h u,L sin ψ = − ωq g T − B cos ψ sin ψ u ′ = 0 . (B.7)In order to derive an expression for the energy density (8), we use the continuityequation (B.3), the constitutive relations for the current (11), as well as the expressionfor the pressure, − p = ε − sT − µρ − j · v n , (B.8) h and j s in combination with Maxwellrelations that link h and the pressure, which in turn can be derived from the thermodynamic differential d ˜ ε = T d ˜ s − pd (1 /ρ ) + v n d ( j /ρ ) + ( j s /ρ ) d v s + ( h u /ρ ) d v u for the energy per particle ˜ ε = ε/ρ . ydrodynamics of a superfluid smectic dp = sdT + ρdµ + j · d v n − j s · d v s − h · d ( ∇ u ) . (B.9)In addition, substitute the decomposition of the superfluid current j s,L = − g L (cid:20) ρ ys ρ yn cos ψ + ρ xs ρ xn sin ψ (cid:21) − g T cos ψ sin ψ (cid:20) ρ ys ρ yn − ρ xs ρ xn (cid:21) + ρv s (cid:20) ρ ys ρ yn cos ψ + ρ xs ρ xn sin ψ (cid:21) (B.10) j s,T = g L cos ψ sin ψ (cid:20) ρ xs ρ xn − ρ ys ρ yn (cid:21) − g T (cid:20) ρ ys ρ yn sin ψ + ρ xs ρ xn cos ψ (cid:21) + ρv s cos ψ sin ψ (cid:20) ρ ys ρ yn − ρ xs ρ xn (cid:21) (B.11)in the definition (13) of the energy current. Here, we abbreviate the xx and the yy components of the superfluid and normal tensor by a superscript x and y , respectively.Collecting all terms yields − ωq δε + ˆ q · j ε = − ωq δq − ˜ sT (cid:26) g T cos ψ sin ψ (cid:20) ρ xs ρ xn − ρ ys ρ yn (cid:21) − g L (cid:20) ρ ys ρ yn cos ψ + ρ xs ρ xn sin ψ (cid:21) + ρv s (cid:20) ρ ys ρ yn cos ψ + ρ xs ρ xn sin ψ (cid:21)(cid:27) = 0 . (B.12)In a similar way, the equation for the gradient of the layer phase becomes − ωq u + v n,y = − ωq u + cos ψρ ( g T − j s,T ) + sin ψρ ( g L − j s,L )= − ωq u + g L cos ψρ yn + g T sin ψρ yn − v s ρ ys ρ yn cos ψ = 0 . (B.13)To simplify the equation (10) for the superfluid velocity, use the differential (B.9) toleading order in the velocities in order to replace dµ : − ωq v s + 1 ρ ∂p∂ρ (cid:12)(cid:12)(cid:12)(cid:12) ˜ s,v s ,u ′ δρ − ˜ sρc V δq = 0 . (B.14)Collecting the equations (B.3), (B.6), (B.7), (B.13), and (B.14) gives the characteristicequation (abbreviating c = cos ψ and s = sin ψ ) − ωq Kρ − ωq − Bc − ωq − Bsc sT (cid:2) ρ ys ρ yn c + ρ xs ρ xn s (cid:3) ˜ sT sc (cid:2) ρ ys ρ yn − ρ xs ρ xn (cid:3) − ωq − ρ ˜ sT (cid:2) ρ ys ρ yn c + ρ xs ρ xn s (cid:3) Kρ − ˜ sρc V − ωq cρ yn sρ yn − ρ ys cρ yn − ωq δρg L g T δqv s u ′ = 0 . (B.15) ydrodynamics of a superfluid smectic ω + ω q (cid:20) − Kρ − Bρ yn c − ˜ s Tc V (cid:18) ρ ys ρ yn c + ρ xs ρ xn s (cid:19)(cid:21) + ω q (cid:20) BKρ c (cid:18) ρρ yn c + ρ ys ρ yn s (cid:19) + ˜ s Tρc V (cid:18) K (cid:18) ρ ys ρ yn c + ρ xs ρ xn s (cid:19) + Bc (cid:18) ρ ys ρ yn c + ρρ yn ρ xs ρ xn s (cid:19)(cid:19)(cid:21) + q (cid:20) − BKρρ xn ˜ s Tc V c s (cid:18) ρ ys ρ yn c + ρ xs ρ yn s (cid:19)(cid:21) = 0 . (B.16)In general, there are three distinct propagating modes. One of them is a generalizedtransverse sound mode ω t ( q ), which is the Goldstone mode associated with smecticorder found by Martin et al. [18]. As noted in Sec. 2.1, its velocity vanishes in thespecial case of purely parallel and perpendicular propagation. Formally, this is becausethe last term of the characteristic polynomial vanishes in this limit, giving rise to onlytwo sound modes plus two diffusive ones. For propagation in the longitudinal direction( c = 1 and s = 0), the matrix (B.15) reduces to the expression (14) in the main text(the transverse current component decouples and may be omitted). In the oppositelimit ψ → π/ ω + ω q (cid:20) − Kρ − ˜ s Tc V ρ xs ρ xn (cid:21) + q (cid:20) Kρ ˜ s Tc V ρ xs ρ xn (cid:21) = 0 . (B.17)Note that there is no dependence on B , ρ ys , and ρ yn . We obtain the standard first andsecond sound modes with speed c ( ψ = π/
2) = Kρ (B.18) c ( ψ = π/
2) = ˜ s Tc V ρ xs ρ xn , (B.19)which is as expected: in the special case of propagation along the smectic layers, thedensity wave structure does not affect the hydrodynamic modes, which are the same asfor a homogenous superfluid. Since the normal fraction ρ xn vanishes at low temperature,the second sound mode becomes ill-defined.Our results for the spectrum of hydrodynamic modes in a superfluid smectic phaseare also relevant for imbalanced Fermi superfluids in a situation where the spatialmodulation of the order parameter only appears along a single direction. As discussed inSec. 2, the superfluid smectic realizes two independent broken U (1) symmetries that areassociated with the thermodynamic variables v s and ∇ u in Eq. (4). The superfluidvelocity v s = ( ~ /m ) ∇ θ sf may be derived from an angular variable θ sf ∈ ( − π, π ].Moreover, since q u ( x, , y ) and q u ( x, y ) + 2 π give rise to identical distortions of thesmectic order [cf. Eq. (2)], the same type of symmetry breaking characterizes a smecticphase. Its fluctuations may thus be described by a different angle θ sm ∈ ( − π, π ]such that ∇ u = (1 /q ) ∇ θ sm5 . As pointed out by Radzihovsky and Vishwanath [23], θ sm and θ sf transform in an opposite manner under timereversal: θ sm is a true scalar while θ sf is a pseudoscalar. ydrodynamics of a superfluid smectic q = k F ↑ − k F ↓ of the two Fermisurfaces results in an uni-directional periodic modulation of the complex gap parameter∆ q . The associated low energy theory derived by these authors and discussed in muchmore detail in Ref. [24] is of the form H LO = K (cid:0) ∇ u (cid:1) + B (cid:16) ∂ k u −
12 ( ∇ u ) (cid:17) + ρ is ∇ i θ sf ) (B.20)where i = k or i = ⊥ refer to the directions parallel and transverse to the orderingvector. The two Goldstone modes associated with the Hamiltonian density (B.20) aredetermined by the elastic constants K and B together with the two different superfluiddensities ρ ⊥ s and ρ k s and a finite compressibility χ in the form ω sf ( q ) = q(cid:0) ρ ⊥ s q ⊥ + ρ k s q k (cid:1) /χ (B.21) ω sm ( q ) = q(cid:0) K q ⊥ + Bq k (cid:1) /χ . (B.22)The first mode is an anisotropic version of the Bogoliubov-Anderson mode of a neutralsuperfluid while the smectic phonon ω sm ( q ) is unique for the uni-directional LO state. Ithas a linear spectrum determined by the layer compression modulus B for wave vectorsalong the direction of ordering but turns into a mode with quadratic dispersion for q ⊥ q .The predicted mode structure differs from that of the superfluid smectic phase discussedabove, and indeed there are important differences between both phases. First of all, theuni-directional LO state only exists in the superfluid regime of the imbalanced Fermi gas.The elastic constants B and K therefore derive from a single complex order parameter∆ q . Moreover, in contrast to the superfluid smectic phase of dipolar BECs, it is assumedthat the spatial structure in ∆ q is not associated with a real density modulation andalso that the fermionic superfluid has no underlying zero-momentum condensate. Theseassumptions are valid for essentially incompressible Fermi systems in the BCS limit,where the condensate fraction is exponentially small. They imply that there is nocoupling of the symmetry breaking variables ∇ u and v s to the particle and momentumdensity. The energy density (B.20) associated with ∇ u and v s thus fully determinesthe spectrum of Goldstone modes . By contrast, the equations of motion (B.15) thatdetermine the hydrodynamic modes of a superfluid smectic phase depend crucially onthe coupling between density fluctuations δρ and both symmetry-breaking variables.The fact that the superfluid order parameter has a negligible coupling to the particledensity in the BCS-limit is well known. Specifically, it has been shown by Leggett [47]that even in the presence of strong Fermi liquid corrections in the normal state, the ρ s / (cid:0) ˆ q · v s − ( ~ q /m ) ∂ k u (cid:1) to the free energy density rather thantwo independent ones associated with superfluid flow and elastic distortions. The Fulde-Ferrell statetherefore realizes a broken relative gauge symmetry as in He A [46]. ydrodynamics of a superfluid smectic δn in particle density at thesuperfluid transition is related to the finite order parameter | ψ | by a linear relation δn = α ˜ κ | ψ | to lowest order. Here, ˜ κ = ∂n/∂µ is the compressibility and α is thecoupling constant between density and the order parameter, which is generically of theform − αδn | ψ | [49]. For weak-coupling BECs, one has α ˜ κ ≡
1. By contrast, for Fermisuperfluids, one finds that α ˜ κ ≃ ( T c /T F ) is exponentially small in the BCS limit andeven for a unitary gas the dimensionless coupling constant α ˜ κ turns out to be around0 .
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