Effective theory of chiral two-dimensional superfluids
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y TAUP-2965/13, NT@UW-13-18, EFI-13-9
Effective theory of chiral two-dimensional superfluids
Carlos Hoyos
Raymond and Beverly Sackler Faculty of Exact Sciences,School of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv 69978, Israel
Sergej Moroz
Department of Physics, University of Washington, Seattle, Washington 98195, USA
Dam Thanh Son
Enrico Fermi Institute, James Franck Institute, and Department of Physics,University of Chicago, Chicago, Illinois 60637, USA (Dated: May 2013)We construct, to leading orders in the momentum expansion, an effective theory of a chiral( p x + ip y ) two-dimensional fermionic superfluid at zero temperature that is consistent with non-relativistic general coordinate invariance. This theory naturally incorporates the parity and time-reversal violating effects such as the Hall viscosity and the edge current. The particle numbercurrent and stress tensor are computed and their linear response to electromagnetic and gravita-tional sources is calculated. We also consider an isolated vortex in a chiral superfluid and identifythe leading chirality effect in the density depletion profile. PACS numbers: 74.78.-w
I. INTRODUCTION
Chiral fermionic superfluids enjoy a long-standing andcontinuous interest in condensed matter physics. Thisgoes back to studies of thin films of He-A which is be-lieved to form a chiral condensate [1, 2]. More recently,low-dimensional chiral superfluids attracted some atten-tion due to the presence of Majorana zero-energy edgequasiparticles [3, 4] that might facilitate progress towardsa fault-tolerant quantum computer [5, 6].In this paper we construct and analyze the effectivetheory of a two-dimensional fermionic superfluid at zerotemperature that forms a chiral condensate which in mo-mentum space takes the form∆ p = ( p ± ip ) ˆ∆ , (1)where ˆ∆ is a real function of | p | and the sign defines thechirality of the condensate. In addition, p and p areprojections of the momentum vector on the orthonormalspatial vielbein which will be defined later. The orderparameter is the p-wave eigenstate of the orbital angularmomentum. We will assume that (1) is the energeticallyfavored ordering. The usual spontaneous breaking of theglobal particle number U (1) N symmetry is accompaniedby the breaking of the vielbein rotation SO (2) V sym-metry. The condensate (1) remains invariant under thediagonal combination of U (1) N and SO (2) V transforma-tions which leads to the symmetry breaking pattern U (1) N × SO (2) V → U (1) D . (2)This implies the presence of a single gapless Goldstonemode in the spectrum that governs the low-energy andlong-wavelength dynamics of the superfluid at zero tem-perature. The effective theory of this Goldstone field has an infinite number of terms and can be organized in aderivative expansion. This allows to include correctionsto the well-known Landau superfluid hydrodynamics ina systematic fashion and to study phenomena at lengthscales that are larger than microscopic length scales (e.g.the coherence length). In this paper we restrict our atten-tion only to the leading and next-to-leading order termsin the derivative expansion.Our guiding principle is the nonrelativistic version ofgeneral coordinate invariance developed in [7]. We putthe chiral superfluid into a curved space, switch on anelectromagnetic source and demand the invariance of theeffective theory with respect to nonrelativistic diffeomor-phisms and U (1) N gauge transformations. Since the gen-eral coordinate invariance can be viewed as a local versionof Galilean symmetry, the symmetry constraints on theeffective theory are (even in flat space) more restrictivethan just the ones imposed by the Galilean invariancealone. This approach proved to be useful before and ledto new predictions for unitary fermions [7, 8] and quan-tum Hall physics [9].While our effective theory might be viewed as a toymodel mimicking only some aspects of low-energy dy-namics in thin films of He-A and spin-triplet supercon-ductors, our predictions, in principle can be tested exper-imentally with ultracold spin-polarized fermions confinedto two spatial dimensions.
II. EFFECTIVE THEORY OF CHIRALSUPERFLUIDS
Our starting point is the effective theory of a conven-tional (s-wave) superfluid constructed in [7]. In curvedspace with a metric g ij and in the presence of an electro-magnetic U (1) N source A ν , the leading order action ofthe Goldstone field θ was found to be S [ θ ] = Z dtd x √ gP ( X ) , (3)where g ≡ det g ij , X = D t θ − g ij D i θ D j θ (4)and the covariant derivative D ν θ ≡ ∂ ν θ − A ν with ν = t, x, y . The superfluid ground state has a finite density,that we characterize by the chemical potential µ . In theeffective action it enters as a background value for theGoldstone field, that is decomposed as θ = µt − ϕ (5)with ϕ standing for a phonon fluctuation around theground state. This allows to identify the function P ( X )in Eq. (3) with the thermodynamic pressure as a func-tion of the chemical potential µ at zero temperature. Asdemonstrated in [7], the expression X transforms as ascalar under the infinitesimal nonrelativistic diffeomor-phism transformation x i → x i + ξ i provided δθ = − ξ k ∂ k θ,δA t = − ξ k ∂ k A t − A k ˙ ξ k ,δA i = − ξ k ∂ k A i − A k ∂ i ξ k + g ik ˙ ξ k ,δg ij = − ξ k ∂ k g ij − g ik ∂ j ξ k − g kj ∂ i ξ k . (6)In addition, X is invariant under U (1) N gauge trans-formations. These observations ensure that the leadingorder effective action (3) is invariant under general coor-dinate transformations.As becomes clear from the symmetry breaking pattern(2), in the case of a chiral superfluid we need a gauge po-tential for SO (2) V vielbein rotations. To this end we in-troduce an orthonormal spatial vielbein e ai with a = 1 , g ij = e ai e aj , ǫ ab e ai e bj = ε ij . (7)Such vielbein however is not unique and defined only upto a local SO (2) V rotation in the vielbein index a . Thisallows us to introduce a connection ω t ≡ (cid:16) ǫ ab e aj ∂ t e bj + B (cid:17) ,ω i ≡ ǫ ab e aj ∇ i e bj = 12 (cid:16) ǫ ab e aj ∂ i e bj − ε jk ∂ j g ik (cid:17) , (8) In this paper we follow the notation of [7] except that the massof the fermion is set to unity. Here we introduced the antisymmetric Levi-Civita symbol ǫ ij = ǫ ij , ǫ ≡ +1. The Levi-Civita tensor is then ε ij = √ g ǫ ij , ε ij = √ gǫ ij . where e aj ≡ e ai g ij and the magnetic field B ≡ ε ij ∂ i A j .Under a local (i.e., time- and position-dependent) in-finitesimal SO (2) V rotation e ai → e ai + φ ( t, x ) ǫ ab e bi , (9)the connection ω ν transforms as an Abelian gauge field ω ν → ω ν − ∂ ν φ . Using Eq. (8) together with the trans-formation law of the vielbein one-form δe ai = − ξ k ∂ k e ai − e ak ∂ i ξ k , (10)one can show that ω ν transforms as a one-form under thenonrelativistic diffeomorphisms, i.e., δω ν = − ξ k ∂ k ω ν − ω k ∂ ν ξ k . (11)In hindsight, this simple transformation of ω ν clarifiesthe appearance of the magnetic field B in the definitionof ω t .We are now in position to construct the leading ordereffective theory of a chiral superfluid. Quite naturally,the theory is still defined by the action (3) with the co-variant derivative now given by D ν θ ≡ ∂ ν θ − A ν − sω ν . (12)For the chiral p-wave superfluid s = ± / This en-sures that the Goldstone boson is coupled to the proper(broken) linear combination of U (1) N and SO (2) V gaugefields. One must set s = ± n/ n th partial wave.It is interesting to note that instead of the usual formal-ism, where the superfluid is described by a single Gold-stone boson θ , there is an alternative approach in whichthe Lagrangian depends on the phase of the condensate θ and the superfluid velocity v i . We present this alter-native description of a (chiral) superfluid in AppendixA and demonstrate there that by integrating out v i oneobtains the usual Goldstone effective action.We point out that Galilean invariance alone is notsufficient to fix the leading order action (3). Undera Galilean boost the magnetic field B transforms as ascalar, i.e., δB = − ξ k ∂ k B . For this reason the prefactormultiplying the magnetic field B in the first equation in(8) is not constrained by the Galilei invariance. Thusonly the general coordinate invariance fixes uniquely theleading order Lagrangian of a chiral superfluid.Time reversal and parity act nontrivially as T : t → − t, θ → − θ, A i → − A i , ω t → − ω t ; P : x ↔ x , A ↔ A , ω t → − ω t , ω ↔ − ω . (13) The effective action previously found in [10] is contained in Eq.(3). This becomes obvious after expanding P ( X ) around theground state X = µ . The infinitesimal Galilean boost with velocity v k is realized bya combination of the gauge transformation α = v k x k togetherwith the diffeomorphism ξ k = v k t . For a fixed s = 0, the effective theory (3) is not separatelyinvariant under neither time reversal T nor parity P . Onthe other hand, P T is a symmetry of the theory for anyvalue of s . Note however that the action (3) is separatelyinvariant under T and P if one transforms the chiralityof the ground state, i.e., s → − s .For s = 0 the action (3) is the leading order term ina derivative expansion that follows the counting in [7],where ∂ ν θ ∼ A ν ∼ g ij ∼ O (1) when expanded aroundthe ground state. In addition, in this power-counting aderivative acting on any field O increases the order byone, i.e., [ ∂ ν O ] = 1 + [ O ]. Non-linear effects of fieldswith [ O ] = 0 are included, since [ O n ] = n [ O ] = 0 for any n ≥
1. For this reason, we conclude from the scaling ofthe metric that [ e ai ] = 0 and [ ω ν ] = 1. As a result, D ν θ turns out to be of a mixed order: it contains terms bothof leading and next-to-leading order in the derivative ex-pansion. Since one can show that the only possible next-to-leading order contribution consistent with symmetriesis already contained in D ν θ , our theory is complete up toand including next-to-leading order. By introducing the superfluid density ρ ≡ dP/dX andthe superfluid velocity v j ≡ − D j θ the nonlinear equationof motion for the Goldstone field can be written in thegeneral covariant form1 √ g ∂ t ( √ gρ ) + ∇ i (cid:0) ρv i (cid:1) = 0 , (14)which is the continuity equation in curved space. Withrespect to nonrelativistic diffeomorphisms, ρ transformsas a scalar and v i transforms as a gauge potential, i.e., δρ = − ξ k ∂ k ρ,δv i = − ξ k ∂ k v i − v k ∂ i ξ k + g ik ˙ ξ k . (15)By linearizing the equation of motion (14) in the ab-sence of background gauge fields ( A ν = ω ν = 0) one findsthe low-momentum dispersion relation of the Goldstonefield to be ω = c s p , (16)where the speed of sound c s ≡ p ∂P/∂ρ is evaluated inthe ground state.From Eq. (12) the vorticity ω ≡ ǫ ij ∂ i v j can be ex-pressed as ω = √ g (cid:16) B + s R (cid:17) , (17)where we used that the Ricci scalar √ gR = ǫ ij ∂ i ω j in two dimensions. In the absence of a magnetic field The term in the Lagrangian of the form Q ( X )( ∂ t + v i ∂ i ) X isconsistent with continuous symmetries for an arbitrary function Q ( X ) and gives contributions to next-to-leading order. It doesnot however respect P T invariance and should not be included. in flat space, the superfluid velocity field is irrotational,i.e., ω = 0, except for singular quantum vortex defects.For an elementary vortex located at a position x v thevorticity is ω ( x ) = πδ ( x − x v ) /
2. Single-valuedness ofthe macroscopic condensate wave-function yields quanti-zation of the total magneto-gravitational flux on a com-pact manifold. For example, for a p-wave superfluid liv-ing on a sphere S with no magnetic field, the total fluxis R S ω = π , which is accommodated in two elementaryquantum vortices. III. CURRENT AND STRESS TENSOR
In this section, starting from the effective action (3), weconstruct and analyze the U (1) N current and the stresstensor. A. U (1) N current In curved space the temporal and spatial part of the U (1) N three-current are found to be J t ≡ − √ g δSδA t = ρJ i ≡ − √ g δSδA i = ρg ij v j | {z } convective + s ε ij ∂ j ρ | {z } edge . (18)In addition to the usual convective term, we find theparity-odd contribution to the current which is propor-tional and perpendicular to the gradient of the superfluiddensity. In a near-homogeneous finite system this part ofthe current flows along the edge of the sample, where thedensity changes rapidly. It is important to stress thatthe edge current is perpendicular to ∂ j ρ , but not to theelectric field E j ≡ ∂ t A j − ∂ j A t . Thus there is no staticHall conductivity in the chiral superfluid in agreementwith general arguments of [11]. The edge current wasfound in the study of superfluid He-A by Mermin andMuzikar [12].The current conservation equation is1 √ g ∂ t ( √ gJ t ) + ∇ i J i = 0 . (19)Since ∇ i J i edge = 0, this is consistent with the equation ofmotion (14).In a non-homogeneous chiral superfluid the edge cur-rent is flowing even in the ground state. This implies thatthe ground state has a nonvanishing angular momentum L GS . In flat space this is given by L GS = Z d xǫ kl x k J l = s Z d xρ | {z } N , (20)where N denotes the total number of fermions. Thisresult has a simple explanation in the limit of stronginteratomic attraction, where the many-body fermionicsystem can be viewed as a Bose-Einstein condensate oftightly bound molecules. In the chiral superfluid withpairing in the n th orbital wave every molecule has theintrinsic angular momentum l = ± n with the sign deter-mined by the chirality of the ground state. In the BEClimit interactions between molecules are weak and thusthe total angular momentum is just the sum of internalangular momenta of separate molecules. Note howeverthat Eq. (20) should be valid also away from the BEClimit as long as the density near the edge varies on lengthscale that is large compared to any microscopic lengthscale.The result (20) was derived for a finite droplet of thechiral superfluid, where the density decreases continu-ously to zero at the boundary. Let us consider now asuperfluid confined in a solid vessel. In this case, whenwe vary the action, together with the bulk current (18)we find an additional current that flows along the one-dimensional boundary. This current is localized at theboundary and is given by J i boundary = s ρε ij ˆ n j , (21)where ˆ n is a unit outer-pointing normal vector. Theboundary current flows in the opposite direction to theedge part of the bulk current (18). When the velocity iszero, the total current at the boundary is J i boundary + Z d x J i edge = 0 . (22)The value of the angular momentum is then the sameas in Eq. (20). However, the density is not an analyticfunction at the boundary in the topologically non-trivialphase. Therefore, the total angular momentum of thesuperfluid in the ground state confined in a solid vesselmay change. For a thorough investigation of this questionwe refer to [13].More general values of the parameter s are also possi-ble. Interesting examples are anyon superfluids of frac-tional statistical phase [14, 15] θ = π (cid:18) − n (cid:19) . (23)When n = 1 the anyon becomes a boson (scalar) andwhen n → ∞ it becomes a fermion. Similarly to thechiral superfluid, for any n > L = s ρc s A t B. (24)This was identified as responsible for the Landau-Halleffect in the anyon superfluid [14, 15]. The coefficient of this term was determined in [16] to be∆ L = e π (cid:18) n − n (cid:19) A t B. (25)For anyons, scale invariance fixes c s = ρ/ s = 12 (cid:18) n − n (cid:19) . (26)Therefore the orbital angular momentum is also frac-tional, as expected. B. Stress tensor
In curved space the invariance of the effective actionwith respect to an infinitesimal nonrelativistic diffeomor-phism ξ i implies S [ θ + δθ, A ν + δA ν , g ij + δg ij , e ai + δe ai ] = S [ θ, A ν , g ij , e ai ] , (27)and leads to the Euler equation1 √ g ∂ t ( √ gJ k ) + ∇ i T ik = E k J t + ε ik J i B. (28)where T ik ≡ T ij g jk and the contravariant stress tensor isdefined by T ij ≡ √ g δSδg ij . (29)Generically the Euler equation is the conservation equa-tion only if E k = 0 and B = 0.Now we wish to compute the stress tensor T ij for thechiral superfluid (3). In this case, the variation of thevielbein is related to the variation of the metric via e ai → e ai + 12 e aj δg ij + δλǫ ab e bi . (30)In this way the first relation in Eq. (7) is satisfied upto second order in δg ij . There is an ambiguity in thetransformation of the vielbein parametrized by δλ whichis related to the SO (2) V gauge freedom of the vielbein.In the following we set δλ = 0. For the contravariantcomponents of the metric, g ij g jk = δ ik → δg ij = − g il g jm δg lm , (31)so one cannot use the metric to raise the indices of themetric variation.First, consider the s-wave superfluid, i.e., set s = 0.Using δ √ g = √ gg ij δg ij we find T ijs =0 = 2 √ g δSδg ij = P g ij + ρv i v j (32)which is the stress tensor of the ideal fluid.For the chiral superfluid additional variation of theaction arise from the variation of the connection (8).Namely, we find δω t = − ε in g jk ∂ t g nk δg ij − Bg ij δg ij ,δω l = − ε in g jk ∂ l g nk δg ij − ε jk ∂ j δg lk + 14 ε mk ∂ m g lk g ij δg ij . (33) This leads to δS ch ≡ Z dtd x √ gP ′ ∂X∂ω µ δω µ (34)which in detail is given by δS ch = s Z dtd x √ g h P ′ ε in g jk ∂ t g nk − P ′ D l θε in g jk ∂ l g nk i δg ij + s ǫ kj Z dtd x ∂ k ( P ′ D i θ ) δg ij + s Z dtd x √ gP ′ g ij (cid:2) B + ε mk D l θ∂ m g lk (cid:3) δg ij . (35)After a tedious but straightforward calculation we find∆ T ij ch ≡ √ g δS ch δg ij = ( v i J j edge + v j J i edge ) + T ij Hall − s ρRg ij (36)with T ij Hall = − η H ( ε ik g jl + ε jk g il ) V kl . (37)Here the strain rate tensor V kl ≡ ( ∇ k v l + ∇ l v k + ∂ t g kl )and η H = − s ρ . With respect to general coordinatetransformations both V kl and T ij Hall transform as tensors. T ij Hall is known as the Hall viscosity part of the stresstensor and was discovered first in [17, 18]. It genericallyarises in two-dimensional many-body systems that breaktime reversal and parity [19, 20].In flat space with the metric g ij = δ ij the force densityarising from the Hall viscosity is given by f i Hall = − ∂ j T ij Hall = η H ǫ ij ∆ v j , (38)and thus the net work per unit of time produced by theHall viscosity in a region S surrounded by a boundary ∂ S is w = η H Z S v i ǫ ij ∆ v j = η H I ∂ S n k ǫ ij v i ∂ k v j , (39)where n k is the normal vector to the boundary ∂ S . Weconclude that the Hall viscosity is dissipationless in thebulk of the region S . Alternatively, its contribution tothe bulk entropy production is vanishing since T ij Hall V ij =0. IV. LINEAR RESPONSE
In linear response theory the induced current δ J islinear in the source δ A , i.e., δ J µ ( t, x ) = Z d x ′ Z t ′ 0. Inthis strict sense the Hall conductivity of a chiral super-fluid vanishes, which is in agreement with the argumentspresented in Sec III. B. Stress tensor response to gravitational source Here we consider a chiral superfluid in curved spacewith spatial metric g ij = δ ij + h ij where h ij is a smallperturbation around the flat background. The linearizedequation of motion for the phonon field is given by ∂ t ϕ − c s ∆ ϕ = − s∂ t ω t + c s ∂ t h/ sc s ∂ i ω i , (47)where h ≡ Tr h ij = h + h . This can be solved in theFourier space with the result ϕ = ic s ωh/ − isωω t − isc s p i ω i ω − c s p . (48)First, we find v i = ip i ϕ + sω i = s σ ( ω, p ) ρ GS E ωi + ic s σ ( ω, p ) ρ GS p i h i s ρ L ( ω, p ) ρ GS ǫ ij p j R, (49) δP | θ ( g ) ≡ P | θ ( g ) − P GS = c s δρ | θ ( g ) , (50)where the parity-odd “electric” field E ωi ≡ − i ( ωω i + p i ω t ). As explained in Appendix B, E ωi is a gradientof the vorticity of the displacement field. The variationof the pressure in Eq. (50) follows from the thermody-namic definition of the speed of sound c s = δP/δρ . Nowwe substitute Eqs. (49), (50) into the stress tensor (36).To linear order in sources the variation of the pure con-travariant stress tensor can be written in a matrix form δT | θ ( g ) ≡ T | θ ( g ) − T GS = (cid:0) δP | θ ( g ) − s ρ GS R (cid:1) σ + P GS δg − + η H h − iω h xx − h yy − c s σ ( ω, p ) ρ GS ( p x − p y ) h + is σ ( ω, p ) ρ GS ( p x E ωx − p y E ωy ) − s ρ L ( ω, p ) ρ GS p y p x R i σ + η H h iωh xy + c s σ ( ω, p ) ρ GS p x p y h − is σ ( ω, p ) ρ GS ( p x E ωy + p y E ωx ) + s ρ L ( ω, p ) ρ GS ( p y − p x ) R i σ , (51)where σ is the unity matrix and σ i are the Pauli ma-trices. The first two terms in the square brackets of Eq.(51) are linear in s and break parity, we will discuss themin more detail below. The last two terms are proportionalto s and do not break parity. They are of higher orderand might get corrections from next-to-next-to-leadingorder terms in the Lagrangian and thus are not predictedreliably by our theory.Now we are ready to extract the dynamic Hall viscosityfrom the linear response calculation. Indeed, as arguedin [24], the gravitational wave h ij ( t ) = h ij exp( − iωt ) (52)induces the following perturbation of the off-diagonalcomponent of the contravariant stress tensor δT xy = − P GS h xy + iωη ( ω ) h xy − iω η H ( ω )2 ( h xx − h yy ) , (53)where η ( ω ) ≡ η ( ω, p = 0) and η H ( ω ) ≡ η H ( ω, p = 0).Direct comparison with Eq. (51) gives us η ( ω ) = 0 , (54)i.e., there is no shear viscosity because at zero tempera-ture superfluid does not dissipate energy. In addition wefind η H ( ω ) = η H = − s ρ GS . (55)It is natural that in the leading-order theory (3) the Hallviscosity does not depend on frequency. C. Relation between Hall conductivity andviscosity It was demonstrated in [9] that for Galilean-invariantquantum Hall states the static electromagnetic Hall re-sponse at small momenta p receives a contribution fromthe Hall viscosity. Subsequently, the relation between theHall conductivity and stress response was generalized toother Galilean-invariant parity-violating systems [25]. Inaddition, it was shown in [25] that the relation can beextended to all frequencies. In particular, for a chiralsuperfluid one finds η H ( ω ) = ω ∂ ∂p x σ H ( ω, p ) (cid:12)(cid:12)(cid:12) p =0 . (56)Using Eqs. (46) and (55), we checked that this relationis satisfied within our leading-order effective theory. D. Parity breaking terms and Kubo formula The parity breaking contributions to the stress tensordue to gravitational sources can be grouped in the odd viscosity tensor, defined through the linear response for-mula T ij odd = − η ijkl odd ∂ t h kl . (57)The odd viscosity can be obtained from the Kubo for-mula involving the parity-odd part of the retarded two-point function of the stress tensor [25]. We define thefrequency and momentum-dependent odd viscosity η ijkl odd ( ω, p ) ≡ iω + G ijklR ( ω + , p ) odd , (58)where ω + ≡ ω + iǫ with ǫ → 0. In fact, the total viscos-ity tensor contains an additional contact term inverselyproportional to the compressibility [25], but it is parity-even and thus does not affect our discussion of the oddviscosity.The retarded two-point function is G ijklR ( ω, p ) = i Z ∞ dt Z d x e iωt − i p · x (cid:10)(cid:2) τ ij ( t, x ) , τ kl (0 , ) (cid:3)(cid:11) , (59)where τ ij is the stress tensor in the absence of gravita-tional sources.To leading order in the phonon fluctuation, the parityeven and odd contributions to the stress tensor are τ ij even = P GS δ ij − ρ GS δ ij ∂ t ϕ,τ ij odd = − η H ( ǫ ik δ jl + ǫ jk δ il ) ∂ k ∂ l ϕ. (60)The parity-odd contributions to the two-point functioncome from cross terms of even and odd contributions.Therefore, to leading order (cid:10) τ odd τ ij even (cid:11) = ρ GS η H δ ij (cid:2) σ ( ∂ y − ∂ x ) + 2 σ ∂ x ∂ y (cid:3) ∂ t h ϕϕ i , D τ even τ ij odd E = ρ GS η H σ ( ǫ ik δ jl + ǫ jk δ il ) ∂ k ∂ l ∂ t h ϕϕ i , (61)where we are using the same matrix notation as beforefor the first entry of the stress tensor in the two-pointfunction. In addition, we used h ϕ i = 0.Using the Fourier transform of the two-point functionof the phonon field G ϕϕ ( ω, p ) = 1 ρ GS c s ω − p , (62)one can recover the results we have derived before forthe linear response of the stress tensor to gravitationalsources. Specifically, the odd-even contribution producethe terms proportional to the trace of the metric per-turbation in the brackets in Eq. (51). These terms areespecially interesting because they are related to the factthat the superfluid is compressible. The metric perturba-tion changes the volume form by a term proportional toits trace δ √ g = h/ 2. This excites phonons that producethe stress we have computed above. The even-odd termsgive contributions to the variation of the pressure.The regular Hall viscosity term on the other hand doesnot involve the phonon propagator and therefore it is akinematic response that will appear in the correlationfunction as a contact term. Its origin is similar to thediamagnetic current, which is obtained from a term inthe action quadratic in sources. V. VORTEX SOLUTION Here we consider a vortex in a chiral superfluid in flatspace placed at the origin. We treat this problem inpolar coordinates ( r, φ ). Far away from the core, due tothe single valuedness of the condensate wave function v r = 0 , v φ = n r , n ∈ Z . (63)Here v r and v φ are the coefficients in the decomposition v = v r e r + v φ e φ with e r and e φ denoting the unit vectorsin the radial and angular directions.We will determine the asymptotic behavior of the su-perfluid density ρ ( r ) as r → ∞ . From Eq. (28) the staticEuler equation reads ρv j ∂ j v i = − ∂ i p + η H ǫ ij ∆ v j − ∂ j ( v i J j edge + v j J i edge ) (64)which after the projection onto the radial direction be-comes ρr v φ = c s ∂ r ρ − η H (cid:2) ∂ r v φ + 1 r ∂ r v φ − r v φ (cid:3) − f edge , (65)where f edge ≡ − δ ik e kr ∂ j ( v i J j edge + v j J i edge ) = − sn r ∂ r ρ .For the velocity given by Eq. (63) we have ∇ · v = 0 andthus ∆ v = 0. This simplifies the previous equation tothe form ρ n r = h c s + sn r i ∂ r ρ. (66) To leading order in the large-distance expansion h c s + sn r i → c s ∞ ≡ ∂P∂ρ (cid:12)(cid:12)(cid:12) r →∞ (67)and the differential equation (66) simplifies to dρρ = n dr c s ∞ r . (68)It is easily integrated giving the density profile ρ ( r ) = ρ ∞ (cid:2) − n c s ∞ r + O ( r − ) (cid:3) , (69)where ρ ∞ is the asymptotic value of the superfluid den-sity away from the vortex core. Since the chirality pa-rameter s does not appear in Eq. (68), the leading ordertail of the density profile is invariant under n → − n .Chirality effects arise first at next-to-leading order inthe large-distance expansion. By using c s = ∂P∂ρ = c s ∞ + ∂ P∂ρ (cid:12)(cid:12)(cid:12) r →∞ ( ρ − ρ ∞ )+ O (cid:2) ( ρ − ρ ∞ ) (cid:3) (70)together with Eq. (69), we find that to next-to-leadingorder we can replace h c s + sn r i → c s ∞ + (cid:16) sn − n ρ ∂ P∂ρ c s (cid:12)(cid:12)(cid:12) r →∞ (cid:17) r (71)in Eq. (66). The solution of Eq. (66) now gives thesuperfluid density profile up to next-to-leading order ρ ( r ) = ρ ∞ h − n c s ∞ r + n (cid:16) − ρ ∂ P∂ρ c s (cid:12)(cid:12)(cid:12) r →∞ (cid:17) c s ∞ r + sn c s ∞ r + O ( r − ) i . (72)Since in the chiral superfluid time reversal and parity arespontaneously broken, the density profile is not invari-ant under n → − n . The leading parity-violating correc-tion appears first in the 1 /r tail. The relative differencebetween the densities of a vortex (with n = 1) and anantivortex (with n = − 1) is asymptotically given by∆ ρρ ∞ = s c s ∞ r + O ( r − ) . (73) From the point of view of the microscopic theory,our hydrodynamic description becomes reliable only for r ≫ ξ , where ξ is the coherence length of the chiral su-perfluid. For weak interatomic attraction (BCS regime)the density depletion in the vortex core is small. Themaximal depletion away from the core is∆ ρρ ∞ (cid:12)(cid:12)(cid:12) BCS max ∼ s c s ∞ ξ ≪ c s ∞ ξ ≫ ρρ ∞ (cid:12)(cid:12)(cid:12) res max ∼ s c s ∞ ξ ∼ . (75)This happens because near the Feshbach resonance theonly relevant scale is given by the atomic density andthus c s ∞ ξ ∼ VI. CONCLUSION In this paper we constructed the leading-order low-energy and long-wavelength effective hydrodynamic the-ory of the simplest chiral fermionic superfluid in two spa-tial dimensions. The effective theory incorporates timereversal and parity violating effects and thus naturallygives rise to the edge particle current and Hall viscos-ity. In agreement with [25], we found a relation betweenthe Hall conductivity and Hall viscosity response func-tions. As an application of the formalism, we constructeda quantum vortex solution and discovered that the lead-ing chirality effect appears first in the 1 /r tail of thedensity depletion. Our predictions might be tested in ex-periments with spin-polarized two-dimensional ultracoldfermions.Since the chiral superfluid studied in this paper is afermionic topological liquid, it is known to posses a pro-tected gapless fermionic edge mode in the weakly-coupledregime [2, 11]. Although our theory contains only thebosonic Goldstone mode as a degree of freedom, we be-lieve that the fermionic gapless edge state is (at leastpartially) taken into account in our formalism. An im-portant input for the effective theory is the exact equa-tion of state P ( µ ) which defines the Lagrangian. Dueto the topological phase transition, the function P ( µ ) isnon-analytic at the critical value of the chemical poten-tial µ cr . In mean-field studies one finds µ cr = 0 [2, 11].Consider now the ground state of a finite droplet of thechiral superfluid in the weakly-coupled regime. While inthe bulk µ is almost constant and positive, it decreases Due to the marginal nature of interactions in two spatial dimen-sions, the effective range can not be set to zero at the Feshbachresonance. The dependence of physical observables on this lengthscale is logarithmic, i.e. weak, and can thus be neglected. towards zero near the boundary. This means that in thetopologically non-trivial weakly-coupled phase there is aclosed contour µ = µ cr which encircles the bulk, wherethe Lagrangian is non-analytic. This non-analyticity canappear only from integration of a massless mode. Thisone-dimensional closed contour is the domain where thegapless fermionic mode lives in the original fermionicmodel. On the other hand, in the topologically trivialstrongly-coupled regime there is no such a contour sincein this regime µ < µ cr already in the bulk. In summary,in the weakly-coupled regime the topologically protectedfermionic gapless mode was integrated out in Eq. (3) andmanifest itself via the non-analyticity of the equation ofstate. In future it would be useful to derive the effectivetheory from the microscopic fermionic model.It would be interesting to extend our theory to higherorders in the derivative expansion, where the nonrela-tivistic general coordinate invariance might put manymore additional constraints compared to those imposedby considering Galilean invariance alone. Generalizationof the theory to chiral superfluids with spin degrees offreedom might prove useful for better understanding ofthin films of He. Acknowledgments: The authors thank Tom´aˇs Brauner,Matthias Kaminski, Yusuke Nishida, Haruki Watanabeand Naoki Yamamoto for valuable discussions. Thiswork was supported by US DOE Grant No. DE-FG02-97ER41014, and NSF MRSEC Grant No. DMR-0820054and by the Israel Science Foundation (grant number1468/06). Appendix A: Alternative description of superfluid We postulate that v µ = (1 , v i ) transforms like a vectorunder spatial coordinate transformations, i.e., δv µ = − ξ λ ∂ λ v µ + v λ ∂ λ ξ µ , (A1)where ξ t = 0. This means δv i = − ξ k ∂ k v i + v k ∂ k ξ i + ˙ ξ i ,δv i ≡ g ij v j = − ξ k ∂ k v i − v k ∂ i ξ k + g ik ˙ ξ k . (A2)If such v i exists, then one can define the improved gaugepotentials ˜ A t ≡ A + 12 g ij v i v j , (A3)˜ A i ≡ A i − g ij v j , (A4)so that ˜ A µ transforms like a one-form δ ˜ A µ = − ξ k ∂ k ˜ A µ − ˜ A k ∂ µ ξ k . (A5)In this formalism the action of a conventional s-wavesuperfluid is S [ θ, v i ] = Z dtd x √ g h ρv µ ( ∂ µ θ − ˜ A µ ) − ǫ ( ρ ) i , (A6)0where ǫ ( ρ ) is the density of the internal energy that is notassociated with the macroscopic motion of the superfluid.After expanding ˜ A µ , this action becomes S = Z dtd x √ g (cid:20) ρ D t θ + ρv i D i θ + 12 ρg ij v i v j − ǫ ( ρ ) (cid:21) (A7)By integrating out v i , we find that v i = − g ij D j θ, (A8)i.e., v i is the superfluid velocity. Moreover, by integratingthe theory (A6) over v i and ρ we reproduce the effectivetheory (3).To describe chiral superfluids in this formalism we needto use the improved connection ω t ≡ (cid:16) ǫ ab e aj ∂ e bj + ε ij ∂ i v j (cid:17) , (A9) ω i ≡ ǫ ab e aj ∇ i e bj . (A10)It can be checked that ω µ transforms like a one-form.The action of a chiral superfluid becomes S [ θ, v i ] = Z dtd x √ g h ρv µ ( ∂ µ θ − ˜ A µ − sω µ ) − ǫ ( ρ ) i . 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