Odd Parity Transport In Non-Abelian Superfluids From Symmetry Locking

We consider relativistic non-Abelian superfluids, where the expectation value of the global symmetry currents relate space and internal indices, thus creating a"locked"phase. Locking a superfluid with SU(2) internal symmetry in 2+1 dimensions breaks parity spontaneously, and introduces parity-odd terms in the constitutive relations. We show that there are qualitatively different extensions of the rest frame locking to non-zero velocities. We construct the resulting superfluid hydrodynamics up to the first derivative order. Using an expansion close to the critical point, we estimate the ratio of the Hall viscosity and the angular momentum density. Our general hydrodynamic results are compatible with the holographic p-wave calculations in arXiv:1311.4882.


I. INTRODUCTION
Breaking parity introduces new transport effects in fluid hydrodynamics. The parity breaking can be explicit, spontaneous or as a consequence of quantum anomalies, each with its unique signature. An interesting parity-odd transport is the dissipationless Hall viscosity η H in 2 + 1 fluids [1,2]. It enters in the constitutive relations of the fluid as a term in the stress tensor Where v i is the (normal) velocity of the fluid. Latin indices i, j, k refer to space components.
Its effect is to repel or attract nearby flows due to a force perpendicular to the flow (for a recent review and references therein see [3]). In most systems the value of the transport coefficient η H can be obtained from the two-point function of the stress tensor using the Kubo formula where the brackets < ij > mean the traceless component and ω, k are the frequency and momentum of the Fourier transformed correlators. This can be taken as the definition of Hall viscosity even in the cases where there is no hydrodynamic description.
It has been shown that a large number of systems exhibit an interesting relation between the Hall viscosity and the angular momentum density ℓ, first derived in [6], η H ℓ = 1 2 . The aim of this paper is to study parity-odd transport in superfluid hydrodynamics with spontaneously broken parity, and in particular the general properties of the above relation between the Hall viscosity and the angular momentum density. Our general hydrodynamic results 1 The distinction with the ordinary Hall viscosity is a bit subtle, the torsional Hall viscosity enters in the canonical energy-momentum tensor, which is not necessarily symmetric, while the ordinary Hall viscosity is defined for the symmetrized energy-momentum tensor. Despite their similar structure they are independent quantities. are compatible with the holographic p-wave calculations in [13], where the relation between the Hall viscosity and the angular momentum density was argued to hold.
We will study the hydrodynamics of relativistic non-Abelian superfluids in a symmetry locked phase, that is where the expectation value of the global symmetry currents relate space and internal indices. In particular, we will be locking a superfluid with an SU (2) internal symmetry in 2 + 1 dimensions, which breaks parity spontaneously as a consequence of relating the SU(2) structure constants to space structure. We will define the locking when the normal component of the fluid is at rest, and we will show that there are various qualitatively different extensions to non-zero velocities. Locking will introduce parity-odd terms in the hydrodynamic constitutive relations. We will construct the resulting superfluid hydrodynamics up to the first derivative order. Incidentally, we find that in the locked phases a term of the form (1) appears in the stress tensor whose value is not determined by the Kubo formula (2), but by the two-point function between the stress tensor and the current Where J i a = O δ i a is the expectation value of the p-wave condensate and a = 1, 2 are SU(2) indices. We have labelled this transport coefficient η H to distinguish it from the usual definition (2). Using an expansion close to the critical point, we will estimate the ratio of the Hall viscosity and the angular momentum density and find that generically η H ℓ ∼ 1 for the standard Hall viscosity. On the other hand, we find that the ratio η H ℓ depends on the type of locking.
The outline of the paper is as follows: In §II we write down the constitutive relations for a general SU(2) superfluid to first dissipative order. The locking can be seen as expanding around a particular value of the superfluid velocities. Since eventually we are interested in the physics close to the critical point, we keep terms only up to second order in the superfluid velocities. In §III we define the locking when the normal component of the fluid is at rest and study possible extensions to non-zero velocities. In §IV we identify the contributions to Hall viscosity and angular momentum in the locked phase and derive Kubo relations for them.
In §V, we compare our predictions from the hydrodynamic analysis with the holographic p-wave model. We end by presenting our conclusions in §VI. Some technical results are collected in the Appendices.
Here we provide three tables to guide the readers. The ideal constitutive relations and notations are collected in Table I. Table II contains various projections of superfluid velocity discussed in §II. Some definitions used in Kubo formulas in §IV are also listed in Table III. µ, ν = 0, 1, 2 will denote space-time indices, i, j = 1, 2 space indices, A, B = 1, 2, 3 denote internal SU(2) indices, and a, b = 1, 2 the internal indices without the direction 3.
Internal symmetry currents

II. NON-ABELIAN SUPERFLUID HYDRODYNAMICS
In order to describe the hydrodynamic behaviour of a non-Abelian relativistic superfluid, we will make a generalization of the two-fluid model [19][20][21][22][23] 2 . The motion of the superfluid is determined by the conservation equations in the presence of a background metric g µν and gauge field A A µ , ∇ µ is the covariant derivative with respect to the background metric and The field strength is defined as usual The constitutive relations of the energy-momentum tensor and the current are 2 Non-Abelian relativistic normal fluids were analysed in [24].
Here u µ is the velocity of the normal component, P µν = η µν + u µ u ν is the projector in the transverse direction, q A is the normal charge density and ε n the normal energy density. p is the pressure and ξ µ A are the contributions to the currents due to the spontaneous breaking, usually identified with the superfluid velocity. In general we can expand the coefficients f AB as The terms π µν and ν µ A contain derivatives of the velocities and densities. The thermodynamic equations are We complement the hydrodynamic equations with the Josephson condition where H A depends on derivatives of the densities and superfluid velocities.
Using the equation One can show that there is a conserved entropy current when π µν = ν µ A = H A = 0, provided that 3 In the Abelian case this matches with the definition of the superfluid velocity as the covariant derivative of the Goldstone boson ξ µ = −∂ µ ϕ + A µ . In the non-Abelian case, there can be additional non-linear terms If we ignore gauge invariance, the gauge potential A A µ that determines the field strength would have as many independent components as ξ A µ , so the number of independent equations would be sufficient to fix the superfluid velocities. However, because of gauge invariance, the number of independent equations is smaller. In the absence of external sources, 3 One needs to use the first law in the following gauge-invariant form: if λ ξ = 0 there can be a gradient part ξ µ A = −∂ µ ϕ A that is not fixed by the equations of motion. If λ ξ = 0, then the part that is not fixed is of the form of a pure gauge SU (2) potential This justifies the addition of the Josephson condition (13) to the hydrodynamic equations.
When the dissipative terms are non-zero, the canonical entropy current is defined as The divergence of the entropy current obeys where we defined the electric field as E µ A = F µν A u ν . In order to impose the condition that the divergence of the entropy current is non-negative ∇ µ J µ s ≥ 0 we should be able to write the rhs of (19) as a sum of squares. This implies that the dissipative terms can only depend where the strain rate tensor is In principle it might be possible to modify the entropy current in such a way that there would be more allowed first order terms than the ones we present in (20). 4 However, as we discuss in Appendix A, even if such terms were present they would not affect to the analysis of the Hall viscosity and the angular momentum density. We will then keep the discussion with the canonical entropy current bearing in mind that more general non-Abelian superfluid hydrodynamics might be possible (if this were the case our analysis could be understood as a subclass of theories where some transport coefficients are zero).

A. Superfluid velocities in non-Abelian theories
We presented above a consistent set of hydrodynamic equations and thermodynamic relations, that reduces to a familiar form in the Abelian case with ξ µ being the superfluid velocity. In the non-Abelian case the properties of the superfluid should depend on the pattern of symmetry breaking. For a group G broken to a subgroup H, the Goldstone bosons parametrize a coset G/H. Let g ∈ G and h ∈ H be group elements, such that the coset is determined by the equivalence g ∼ gh −1 . We define the superfluid velocity as an element in the algebra 5 In order to describe a coset we have to demand that the hydrodynamics currents are invariant under a global transformation ξ µ → h −1 ξ µ h with h ∈ H. This means adding additional constraints on the superfluid velocities. We will not pursue this direction here but in the following we will study the case where the group is completely broken.
Another new characteristic compared to the Abelian case is that the symmetry can be broken if the currents acquire a non-zero expectation value. In the non-Abelian case the components of ξ µ A do not simply map to the gradient of the Goldstones, but they describe more generally the expectation value of the current. We will discuss this in more detail in the next sections. For now we will focus on finding an appropriate parametrization of the superfluid velocities.
There are two marked directions both in real and internal space. In real space the marked direction is determined by the velocity of the normal component u µ , while in the internal space it is determined by the chemical potential µ A . We will decompose the superfluid velocities in the directions parallel and transverse to both.
The completely parallel direction is determined by the Josephson condition (13). The dissipative term H A will not be important for us, since we will use the decomposition of the superfluid velocity in order to classify the first order terms. Then, at the ideal order we have We further decompose ζ µ A in the parallel and transverse directions to µ A : where the Abelian component of the superfluid velocity is andζ µ A µ A = 0. Note that for SU(2)ζ µ A in 2 + 1 dimensions has four independent components before using the equations of motion. In order to find a suitable parametrization we first define the 'spatial velocity' vectors v = (u 1 , u 2 ), m = (µ 1 , µ 2 ). (26) Then, the transverse components can be written in matrix form as (the first column corresponds to A = 3 and the first row to µ = 0): where λ = 0, 1, 2, 3 and theσ λ matrices are defined as the identity and the Pauli matrices: The four independent components ofζ µ A are parametrized by ζ λ .

B. Dissipative terms
We are interested in the behaviour of transport coefficients close to the critical point between the normal and the superconducting phase. 6 This implies that ζ µ A should be small, either because a large superfluid velocity will destroy the superconducting phase or because ζ µ a acts as an order parameter and it should vanish as the critical point is approached. We will perform an expansion for small ζ µ A ∼ ǫ ≪ 1, this means that both N µ andζ µ A are small ∼ ǫ. Within this expansion we will construct all possible terms to O(ǫ 2 ) that lead to a consistent hydrodynamic theory. Note, that the transport coefficient themselves can also be expanded in the scalars N 2 andζ α Aζ α A . We define the even and odd projectors in the directions transverse to the chemical po- To order O(ǫ) we can use the following two-index combinations of the transverse superfluid To order O(ǫ 2 ) we can use the following independent combinations Note, that ζ µν = ζ νµ andζ µν = −ζ νµ .
We will decompose the dissipative terms using the Abelian component of the velocity and the chemical potential All the possible first order terms to O(ǫ 2 ) can be found in the Appendix B 1.

III. SYMMETRY LOCKED PHASES
One of our goals is to understand the origin and how general is the relation between the Hall viscosity and angular momentum density found in [13] for the holographic p-wave model. In this model there is a nonzero chemical potential and charge density that we can choose to be µ 3 = 0, q 3 = 0. Lorentz and SU(2) symmetries are then reduced to spatial rotations SO(2) S and the U(1) 3 subgroup that leaves the chemical potential invariant. The parity breaking terms appear in a broken phase, where the currents acquire and expectation value J i a ∝ δ i a in such a way that space and flavour indices are related. We dub this as the 'locked' phase by analogy with the color-flavour locking phase of QCD [25]. In this phase the remaining symmetries are spontaneously broken to a diagonal U(1): Therefore, we expect this theory to have a single Goldstone mode. The origin of parity breaking is easy to understand, the SU(2) structure constants are epsilon tensors that break internal 'parity' transformations. After the locking, this breaking is transferred to the spatial directions as well. We can consider a transformation acting on the components of an object with one internal index V a as V 1 ↔ V 2 . The theory has also initially parity symmetry x 1 ↔ x 2 . When the locking is made, the components of the non-Abelian current become J i a ∼ δ i a , which is invariant only under a combination of the internal and parity transformations.
However, the internal transformation is not a symmetry because there are terms with epsilon tensors ǫ ABC that change sign. Then, the would-be parity symmetry allowed by the locking that is a combination of space and internal symmetries is broken by the epsilon terms. This means that there is no additional Z 2 symmetry in the superfluid phase.
When the normal fluid is at rest u µ = (1, 0), we can describe the locked phase in the hydrodynamic regime by setting Here ζ s is proportional to the p-wave condensate, more precisely If we demand that ζ s is constant in the absence of sources, this means that in the equation of motion for ξ µ A (16) the coefficient of the non-linear term should vanish λ ξ = 0. There are several possible extensions to non-zero velocities of the normal component that lead to qualitatively different results. We will distinguish between locking in the lab frame and locking in the rest frame of the fluid. We present them here and discuss in the next section how the Hall viscosity is affected by the locking.

A. Locking in the lab frame
We fix the locking to be (34) even at non-zero velocities. The transverse components of the superfluid velocity are in this case: We can also write it asζ µ A = µζ sP µ A . At the ideal order the currents are where we have used (36), and we identified the superfluid charge density as From (32), we are left with the following dissipative terms where V 3 = τ µ 3 = 0. After the locking, the basic building blocks allowed by the entropy equation (20) become (in the absence of external sources) We wrote explicitly all the first order terms that survive the locking to O(ǫ 2 ) in the Appendix B 2.

B. Locking in the rest frame
By 'locking in the rest frame' we mean that the normal component of the fluid and the chemical potential point in the same direction (taking 'time' to be the third direction). This can be achieved by setting , N µ = 0 and ζ λ = (ζ s , 0, 0, 0). Note that the normalization of the chemical potential is necessary in order to keep the condition µ A µ A = µ 2 .
For this type of locking the dissipative terms (32) are where The basic building blocks allowed by the entropy equation (20) become The allowed first order terms to O(ǫ 2 ) for this kind of locking can be found in the Appendix B 3.

IV. PARITY BREAKING EFFECTS
As we discussed, the locking will introduce parity breaking terms in the constitutive relations. We will first identify all the terms that can appear and at what order in ǫ. We will solve the hydrodynamic equations with external sources to identify the Hall viscosity and angular momentum density in the frame where there is no current J i 3 = 0, 7 which we identify as the ground state of the system. We will match the hydrodynamic solutions with linear response to derive Kubo formulas that determine the transport coefficients responsible for the parity breaking physics. This will be useful later to compare with the holographic p-wave model.
In the linear response analysis we set to zero the velocity of the normal component, so the results are valid for both the locking in the lab frame and in the rest frame. When the velocity is non-zero but small there will be a Hall viscosity term in the stress tensor of the It turns out that the coefficient of this term depends on the type of locking. As we will see it 7 Here we are referring to a physical frame and not to the ambiguity in the choice of hydrodynamic variables.
is the same as the linear response coefficient for a locking in the lab frame but parametrically larger (close to the critical point) for locking in the rest frame.

A. Terms in the constitutive relations
We list all the possible terms that appear in the locked phase in the Appendix B 2 and B 3.
For the locking in the lab frame, we list the terms that break parity and the order at which they appear in the expansion. We also give their approximate form for small velocities: • Tensor τ µν : This term introduces the Hall viscosity.
• Mixed tensor τ µ A : • Vector V µ : The second term introduces a Hall conductivity J i • Scalar Σ: no terms.
Note, that because s a ∼ ∂ a O , the associated terms are actually one order higher in ǫ than naïvely expected.
For the locking in the rest frame we have the same terms, plus a few additional more.
All the new terms are proportional to v µ A as given in (46), and they come from the term ∝ P µα ∂ α u A . We can decompose the transverse derivative of the velocity in shear, curl and scalar components: At small velocities we do the approximation where The extra terms are then • Tensor τ µν : The last term ∝ θ actually drops from the symmetric stress tensor, while the second term ∝ ω ij is a scalar contribution, as one can check by using ω ij = ǫ ij ω.
• Internal vector V A : no additional terms.
• Scalar Σ: There are two main observations we wish to make. The first is that locking does not necessarily introduce all possible parity breaking terms. For instance, terms depending on the vorticity are absent for the locking in the lab frame, and terms depending on the magnetic field are absent in both cases. For comparison, a complete list of parity breaking terms in normal fluids can be found in [26]. 8 The second is that terms depending on the strain rate 8 A similar study for non-relativistic fluids was made in [27].
σ ij (tensor) are parametrically larger when we do the locking in the rest frame. On the other hand, the terms depending on gradients of chemical potential and the expectation value O (vector) are the same for both lockings. The last are responsible for the angular momentum density, so we find that η lab However, the Hall viscosity as computed from linear response η H is not the same as η H for the locked phase in the rest frame. We will derive Kubo formulas for both Hall viscosities and for the angular momentum density in the following.

B. Response to external metric and viscosities
We introduce a background metric of the form g 00 = −1, g i0 = 0, g ij = δ ij + h ij (t). The only contributions to dissipative terms that are of linear order in the metric come from the strain rate tensor The non-zero components are space-like σ ij = ∂ t h ij . Using the results from the Appendix B 1 we get the following dissipative terms to O(ǫ 2 ) • Tensor τ µν (59) • Vector V µ 1,2 : none up to O(ǫ 2 ).
The stress tensor and the currents become where This implies that the correlation functions with the traceless component of the stress tensor The Kubo formulas for the parity-breaking coefficients are Similar Kubo formulas for the Hall viscosity were derived in [12,26,31].

C. Angular momentum density
The equilibrium solution in the locked phase has a finite normal density q 3 = q n and the following values for the superfluid velocities: We now allow the temperature, chemical potential and O to vary slowly over space, but keeping a static configuration and zero velocity for the normal component. In the absence of sources where we are expanding only up to first order terms. We are interested in configurations where the current vanishes J i 3 = 0, so that δξ i 3 = µ qs ν i 3 and The non-zero dissipative terms are proportional to Then, we have the following independent contributions to T 0i : All the contributions are O(ǫ 2 ). Let us assume the coefficients C n are approximately constant, then the total angular momentum is, for a smoothly changing condensate If we have a 'droplet' of superfluid of radius r 0 with constant density and condensate (so and similarly for µ), the angular momentum picks up a contribution from the boundary of the droplet: Therefore, we can define the average angular momentum density as Since the contributions are of order ℓ ∼ O 2 ∼ ǫ 2 , we found that generically

Kubo formulas
In order to find the Kubo formulas for the angular momentum density we will need to solve the hydrodynamic equations to leading order in derivatives and linear order in the sources. We set the velocity to constant u µ = (1, 0) and consider only static configurations.
As external source we will allow only a constant gauge potential A a 0 , and we will allow a fluctuation δξ 0 3 .
From the current conservation equation we get Then, The Josephson condition ξ 3 0 = µ 3 together with the equation for the superfluid velocity We will now use the conservation equation of the stress tensor ∂ k T ki = 0 where The derivatives are evaluated at constant . Combining everything, we find v i a = 0 and Then, the current becomes We obtain the following Kubo formulas, for the correlators evaluated at ω = 0, k = 0, We now introduce a non-zero space-dependent potential A 3 0 , so the electric field is nonzero E 3 i = ∂ i A 3 0 = 0 and allow δξ a 0 to fluctuate. The equations for current conservation imply that s A = 0. The Josephson condition δξ a 0 = δµ a together with the equation for the We will now use the conservation equation of the stress tensor ∂ k T ki = qE i The derivatives are evaluated at constant ζ 2 = 2 O 2 f 2 0 and µ 3 . Combining everything, we find We are left with the current Therefore, Note that C 3 is related to the Hall conductivity, we will comment more on this in the conclusions.

D. Hall viscosity term at non-zero velocity
The enhanced Hall viscosity that appears when we do the locking in the rest frame is generated by a term depending on v i a in π ij . Note that the equation of motion for the superfluid velocity and the Josephson condition imply Therefore, We introduce a source A a i which is time-dependent but independent of the spatial coordinates and satisfies the conditions The hydrodynamic equations are automatically satisfied to leading order in derivatives and the sources. The Fourier transform of the superfluid velocity is The stress tensor including the first order dissipative terms has the form Then, taking the variation with respect to the gauge field Note, that the conditions (91) for the gauge field are satisfied by the correlator The Hall viscosity coefficient is related to C η H as It is straightforward to derive the following Kubo formula

V. COMPARISON WITH HOLOGRAPHIC p-WAVE MODEL
In this section we will check the consistency of the general hydrodynamic analysis by comparing with the results obtained by Son and Wu [13] for the angular momentum density and Hall viscosity in the holographic p-wave model [28][29][30]. We also compute the rest frame Hall viscosity η H and find that the leading order contribution actually vanishes in this model.
In the following we present the basic features of the model and the results. We have collected the equations of motion and useful formulas in Appendix C.
The holographic p-wave model consists of Einstein gravity plus a cosmological constant coupled to a non-Abelian SU(2) gauge field in 3+1 dimensions.
The background metric and gauge field are charged black hole solutions of the form As z → ∞ the metric approaches asymptotically AdS 4 . Applying the holographic dictionary, this means that the dual field theory is a CFT with a SU(2) global symmetry at a finite density and finite temperature state. There can also be zero temperature black holes but we will not discuss them here.
The solutions for the gauge field are such that, as z → ∞, 9 From the dual field theory point of view this means that there is a non-zero chemical potential µ 3 = µ and an expectation value for the current J i a = O δ i a . Therefore, the SU(2) charged black holes presented above describe a locked phase like the ones we have analyzed using hydrodynamics. This model has a second order phase transition at a critical temperature T c from the locked phase to an unbroken phase. Close to the critical point and we can apply the same expansion that we used in the hydrodynamic model. The nearcritical expansion was used in [13] to compute the values of the Hall viscosity and angular momentum density, so we can make a direct comparison.

A. Correlators and Kubo formulas
Let us collect here the expected orders in ǫ from the hydrodynamic analysis: • Angular momentum density ℓ ∼ O(ǫ 2 ): Where Q is defined in (81) and α 0 in (86).
• Hall viscosity for locking in the rest frame η H ∼ O(ǫ): There are four correlators whose leading order in ǫ we need to estimate. The calculation of η H using the Kubo formula was made in [13] originally. One can use their result to show that it is O(ǫ 2 ), but in Appendix D we present a derivation that makes it more explicit.
The angular momentum density was also computed in [13], but using a different method.
It would be interesting to compare the exact (numerical) value obtained from the Kubo formula with their result, but here we will limit ourselves to an estimation of the order of magnitude.
The correlators of the energy-momentum tensor and global SU(2) current in the dual field theory can be computed by evaluating the (properly renormalized) on-shell action of small fluctuations around the background solution. The fluctuations of the metric and gauge field take the form whereḡ µν andĀ A µ are the background solutions. We perform an expansion of the equations of motion in A ∼ ǫ and solve the equations order by order. In most cases we will not need to find the explicit form of the solution to estimate the order of the transport coefficients.
Following [13], at zero momentum we can split the fluctuations according to their representation under the unbroken U(1) group mixing space and time components. For the metric and gauge field they group into tensor, vector and scalar. Both Hall viscosity coefficients appear from tensor fluctuations, while the angular momentum density has a contribution from the vector fluctuation and a contribution that originates from momentum-dependent fluctuations that mix scalar with vector fluctuations.
These are the fluctuations that we will turn on in order to compute each of the coefficients: • Hall viscosity η H : time-dependent h ij with δ ij h ij = 0.
• Angular momentum density: a constant vector contribution a 3 i and a a 0 , and a spacedependent mixed contribution a 3 i , a 3 0 .
• Hall viscosity in rest frame η H : time-dependent h ij with δ ij h ij = 0 and a a i with δ i a a a i = 0.
The expansion of the fluctuations close to the boundary z → ∞ is Where in the dual field theory H ij and A A µ are the sources for the energy-momentum tensor and global SU(2) current and T ij and J A µ are proportional to the expectation values, following the usual AdS/CF T dictionary. The correlators are found by taking variations of the on-shell action with respect to the sources The details about the renormalization of the on-shell action and the derivation of the equations of motion can be found in the original reference [13]. In the following we estimate the order of the transport coefficients.

B. Vector contribution to angular momentum density
For a vector fluctuation with h 0i = 0, a 3 i and a a 0 , the equations of motion are, to leading and next-to-leading order in A 10 To leading order in A we have the solutions To next order in A the solutions become The on-shell action is Since we are in the gauge where the radial components are zero, f A zµ = ∂ z a A µ . Then, the action becomes In particular, the cross term is The expansion close to the boundary is This gives Using that the expectation value of the dual current is we find that the two-point function to leading order in the vev and at zero frequency and momentum is Then, C. Mixed contribution to the angular momentum density We want to check what is the order of the J 3 0 J 3 i ∼ ǫ ij k j contribution. We will do it in the probe approximation, where the metric fluctuation is set to zero by hand. The equations of motion of the gauge fluctuation a A µ in the background A A µ are Assuming time derivatives are zero, this becomes We are working in the gauge a A z = 0 and impose the condition ∂ i a 3 i = 0. We can set a a i = 0, the A = 3, ν = z equation is automatically satisfied. The A = a, ν = z equation is The solution to this equation is The remaining ν = z equations are Where the field strengths are We can rewrite the equations as Then, the on-shell action will contain terms of the form The first two terms don't mix the a 3 i and a 3 0 fluctuations. The first two in the last term are roughly Therefore, and Together with (118), this confirms the hydrodynamic analysis:

D. Hall viscosity in the rest frame
To leading order in A, the equations for the tensor modes are 11 We are following the notation of [13], where h i = {h xy , h xx − h yy }, a i = {a 2 x + a 1 y , a 1 x − a 2 y }. The solution regular at the horizon for the metric is simply the constant solution h i = h b i . For the gauge field it takes the form where A (1) is the regular solution of the background equations of motion when they are linearized in A. It asymptotes the value α (1) 0 as z → ∞. To next-to-leading order, we have to solve the equations We can make δa i = 0. Using the equation of motion for A (1) , we can simplify the equation Then, the solution is Note that the solution is regular at the horizon. The expansion close to the boundary is To this order, there is only a mixed contribution to the on-shell action coming from the term From here one can derive the tensor-current correlator: When introduce this result in the Kubo formula (104) we find that η H vanishes to this order.

VI. CONCLUSIONS
Our original motivation for this work has been the question of whether parity breaking due to locking of internal and space symmetries in a superfluid phase leads generically to the relation between the Hall viscosity and the angular momentum density η H ℓ = 1 2 . For other sources of parity breaking it is known not to be true, as has been shown in several holographic models [26,32,33]. The reason to suspect that this could be the case is the possibility that locking may imply the same origin for the generation of both the Hall viscosity and the angular momentum density, thus linking their values.
In order to answer this question, we studied the first order hydrodynamics of relativistic non-Abelian superfluid in 2 + 1 dimensions, where we locked the SU(2) internal symmetry with the space symmetry. Note as a side remark, that the Goldstone bosons and the corresponding superfluid velocities pattern in a non-Abelian superfluid depends on the pattern of symmetry breaking, and the study of the general case is worth pursuing in the future.
The parity breaking due to the locking generated parity-odd terms in the constitutive relations, which we analysed in detail. In particular we studied the relation between the Hall viscosity and angular momentum density, which turned out to be generically of order one, but not necessarily one half. The holographic p-wave model studied in [13] falls within our class of locked superfluid hydrodynamics, and we showed that our general results are compatible with it. As part of our analysis we have derived a Kubo formula for the angular momentum density. We observe that it receives a contribution proportional to the Hall conductivity (88). This suggests that a similar formula exists for Abelian fluids, explaining the appearance of non-zero angular momentum density in holographic models with a Chern-Simons term for the dual gauge field [26,[32][33][34].
Finally, we demonstrated how locking corrects parity-even transport such as shear and bulk viscosities, and also found that there are qualitatively different extensions of transports and in particular Hall viscosity to non-zero velocities.
The canonical entropy current is The entropy current can have additional terms Where S µ should be such that In principle having S µ allows more independent first order terms than those allowed by the canonical entropy current alone, that for us were σ µν , θ, v µ A and s A . We make the following simplification. Both parity and time-reversal invariance are not broken explicitly, so there are no epsilon tensors appearing in the first order terms and Onsager's relations should be satisfied. This implies that there are no cancellations among first order terms in the divergence of the entropy current. Therefore, ∇ µ J µ s = (canonical quadratic terms) + (new quadratic terms) + (cross terms) .
The new quadratic terms can only come from the divergence of S µ .
In our case we expand where u µ V µ = N µ V µ = 0. For purely Abelian configurations, the analysis of [22] verified that there are no new independent terms when parity and time reversal invariance are unbroken and S µ could be set to zero. This implies that possible new terms should be proportional to the non-Abelian superfluid velocityζ µ A (terms depending on the chemical potential and external fields get an index but are otherwise the same). To the order we are working this means that the new quadratic terms must be quadratic inζ µ A , and it should be possible to write them as the square of new terms linear in the non-Abelian components of the superfluid velocity. If there are no such terms then the canonical entropy current will not be modified.
In the locked phase we set the Abelian velocity N µ = 0 and the sources to zero, so we will check whether, in this subclass of configurations, there can be additional terms in the entropy current. The first derivative terms we can have are Then, at O(ζ 2 ) in the scalar sector Σ u we have 6 terms In the vector sector V µ we have 13 terms Note that not all the terms are necessarily independent but they may be related by the ideal order equations of motion.
We write the O(ζ 2 ) contributions to the entropy as and From this expression we see that the possible quadratic terms are, from the scalar termŝ The last term comes from u µ ∂ µ S 6 . From the vector terms we havê The last two terms originate from the ∂ µ V µ i terms. All these terms can be written as the square of the following first-order termŝ The terms in second and fourth lines are tensors with a global index τ µν A , so they cannot appear in any of the dissipative contributions to the energy-momentum tensor or the current.
The last term in the second line could appear in H A , while the remaining terms could appear in ν µ A (in fact the terms proportional to θ are already there, so they do not introduce anything new).
The only terms that in principle could affect to our discussion of parity breaking in the locked phases are then the first two terms in the third line, that are proportional toζ µ A . However, their only effect is to add additional scalar contributions to the current, even after locking. Therefore, they do not affect to the Hall viscosity or the angular momentum.
In § § IV A the only effect is to add new terms in the scalar part below (51) and in (56) We collect in this appendix the dissipative terms that can appear to O(ǫ 2 ), in a general non-Abelian superfluid and in the different locked phases.

General non-Abelian superfluid
Using basic building blocks allowed by the entropy equation (20) we can construct the following terms 12 • Scalar Σ,

Locking in the lab frame
The terms that in principle survive after locking are

Locking in the rest frame
The terms that in principle survive after locking are In this appendix we collect some of the results of [13], in particular the equations of motion of the background and fluctuations and the form of the on-shell action. The background metric and gauge field are charged black hole solutions of the form ds 2 = −F (z)dt 2 + dz 2 F (z) + r 2 (z)(dx 2 + dy 2 ), As z → ∞ the metric approaches asymptotically AdS 4 .

Background equations of motion
We will denote derivatives with respect to z with primes: Linearized equation for the background A: Asymptotic expansion

On-shell action
Fluctuations around the background solution are denoted as δg µν = h µν and δA A µ = a A µ .
Indices are raised and lowered with the background metricḡ µν . The equations for fluctuations h i and a i are The renormalized quadratic on-shell action takes the simpler form: Appendix D: Calculation of Hall viscosity in the holographic p-wave We split the metric and gauge perturbations in leading order and corrections The equations of motion for the leading order part are We impose ingoing boundary conditions at the horizon z → z H And at the boundary z → ∞ the leading order terms are constant The equations for the next order corrections are In order to solve these equations we first write the corrections as Then, The 'source' terms are The solutions are easily found The choice of integration limits is necessary in order to preserve the condition that the solutions are ingoing at the horizon.
This fixes the solution to first order, but we will also need the solution to second order, given by the equations The 'source' terms are The solutions are then σ