Hydrodynamics on the lowest Landau level

Using the recently developed approach to quantum Hall physics based on Newton-Cartan geometry, we consider the hydrodynamics of an interacting system on the lowest Landau level. We rephrase the non-relativistic fluid equations of motion in a manner that manifests the spacetime diffeomorphism invariance of the underlying theory. In the massless (or lowest Landau level) limit, the fluid obeys a force-free constraint which fixes the charge current. An entropy current analysis further constrains the energy response, determining four transverse response functions in terms of only two: an energy magnetization and a thermal Hall conductivity. Kubo formulas are presented for all transport coefficients and constraints from Weyl invariance derived. We also present a number of Streda-type formulas for the equilibrium response to external electric, magnetic and gravitational fields.

in Ref. [22]. Our approach differs significantly from the latter in particular by the introduction of torsion and, in principle, can be applied generally to fluids of any type, although gapped quantum Hall states will be the focus of our attention here.
One of the most important features of the FQH problem is the presence of a large magnetic field separating the Landau levels, reducing the problem, in its most essential limit, to that of interacting particles confined to the lowest Landau level (LLL). The LLL limit can also be realized by taking the massless limit of a non-relativistic theory. From the point of view of the symmetries of a Newton-Cartan space, this massless limit is a regular limit. This important feature allows one to directly attack the LLL limit of the FQH problem. In this paper, we construct the hydrodynamic theory describing a finite-temperature FQH fluid in the LLL. (At finite temperature, the FQH plateaux will be smeared out, but for convenience we will continue to call any interacting systems of particles confined to the LLL a "FQH fluid.") Hydrodynamics is an effective theory describing the long distance physics of a system that is in local thermodynamic equilibrium. In the standard hydrodynamic theory we then have as variables a locally defined temperature T and chemical potential µ, as well as a fluid velocity v i that vary slowly in space and time. Their dynamics is given by conservation laws supplemented by constitutive relations expanded to some chosen order in derivatives. This construction of the hydrodynamic theory is simplest in relativistic physics, where covariance is manifest in the equations of motion and simple to implement in the constitutive relations, which are the most general expressions for j µ and T µν in terms of the fluid degrees of freedom T , µ, and u µ (the time-like four-velocity normalized so that u µ u µ = 1). One merely writes down all possible terms that have the correct index structure and may be obtained from T , µ, u µ and F µν only through contraction and differentiation; although the second law of thermodynamics puts extra constraints on this expansion. In non-relativistic physics, we have three conservation laws: those of particle number, momentum and energy (which are independent in this context) We would like the most general constitutive relations for j i nc , ε i nc , and T ij nc . The Galilean invariance of the equations is then imposed as an additional constraint.
Newton-Cartan geometry greatly simplifies the process of writing down the conservation laws and the constitutive relations. Currents that transform covariantly under diffeomorphism can be defined and covariant Ward identities derived Ref. [17]. (In contrast, the energy current ε µ nc and stress T ij nc are not spacetime covariant; the "nc"'s in Eqs. (2) are to distinguish the standard currents from the covariant ones we will be using throughout.) That a Newton-Cartan geometry naturally includes a source for the energy current has been noted in Ref. [23] and used to study energy transport in a recent paper [24]. Our paper is organized as follows. In section II we briefly recap the results of Ref. [17]. The derivative expansion and entropy current analysis then proceed entirely along the lines of the relativistic case. Section III obtains the most general constitutive relations and derives results of the massless limit. We find the FQH system is constrained to be force-free, which has powerful implications on the dynamics. In particular, all first order charge transport is determined by thermodynamics.
Section IV contains the entropy current analysis, completing the program outlined above. We find that what are in principle four independent parity odd response coefficients (on the basis of symmetries) are determined by only two: and energy magnetization M E and the Righi-Leduc (or thermal Hall) coefficient c RL . In all, on trivial Newton-Cartan backgrounds (i.e., in flat metric and zero field coupled to the energy density) we have Here T is the temperature, µ the chemical potential and E i , B the external electric and magnetic fields. p, n, ǫ, s and M are identified with the internal pressure, number density, energy density, entropy density and magnetization density while ζ, η andη are the usual bulk, shear and Hall viscosities and Σ T the thermal conductivity. All are arbitrary functions of the thermodynamic variables T , µ and B except for constraints from the usual thermodynamic identities and several positivity conditions The system is dissipationless if and only if all inequalities are saturated. Kubo formulas for all coefficients may be found in sections V (where they are presented in the Newton-Cartan formalism used throughout this paper) and VI D (where they are given in standard form). A recent analysis of 2+1 dimensional gapped phases derives the most general set of transport coefficients for zero temperature nondissipative systems [25]. Equations (3) generalize this to to an arbitrary hydrodynamic theory with nonzero temperature and chemical potential (though they are assumed to be slowly varying and far below the gap) giving us the dissipative viscosities and Righi-Leduc coefficient.
Finally, we present a set of generalized Středa formulas that characterize the equilibrium response to probing electric, magnetic and gravitational fields. A FQH fluid in thermody-namic equilibrium has nonzero electric and energy currents, where the first of which may be recognized as the usual Středa formula [26]. Here G i = ∂ i Φ is the external force exerted by a gravitational potential −Φ.
We give concluding remarks in section VII. The appendices contain additional constraints due to Weyl invariance and other materials of a technical character. In a companion paper [27] we present an alternative derivation of some of the results of this paper without the use of the Newton-Cartan formalism, compute the thermal Hall coefficient in the hightemperature regime and discuss the question of particle-hole symmetry of the hydrodynamic theory.

II. WARD IDENTITIES
We begin with a brief recap of recent work on the Ward identities of non-relativistic systems. For details we refer the reader to Ref. [17]. In this paper we derive covariant Ward identities using the Newton-Cartan structure of non-relativistic theories [18][19][20][21]. In considering response to a perturbing gravitational scalar potential we will need a torsionful version of this geometry (this has also been considered in Ref. [28]). This involves a degenerate metric g µν that measures spatial distances. It's degeneracy direction is spanned by a one-form n µ satisfying n ∧ dn = 0 that provides an absolute notion of space through it's integral submanifolds. It's convenient to also define an auxiliary "velocity" field v µ satisfying n µ v µ = 1 that allows one to invert the metric to a transverse projector The connection ∇ µ is then uniquely specified by and has torsion T λ µν = v λ (dn) µν . The velocity field is unphysical and may be chosen in whatever manner is convenient for a particular problem.
In Ref. [17] we demonstrate that for systems constrained to the LLL, the Ward identities following from gauge and diffeomorphism invariance in a nonrelativistic theory take the covariant form Ward identities for Newton-Cartan diffeomorphisms have also been considered in Refs. [29] and [23]. The above is a covariant generalization of these equations to arbitrary backgrounds, subjected to a LLL projection in the form of a massless limit. These identities also assume a spinful fluid of spin s = 1.
Here j µ and ε µ are the particle and energy currents and T µν a transverse symmetric stress The stress is conserved except for the action of external forces. The first of these is exerted by the familiar electromagnetic field strength F µν = (dA) µν , but there is also a torsional field strength G µν = (dn) µν that couples to the energy current. Before the LLL projection the equation for stress conservation contains terms involving the momentum current. These however drop out upon taking the massless limit m → 0 and stress conservation becomes the force balance (11). The first equation expresses conservation of charge current while the second is the workenergy equation. Here is the shear tensor. Although the Ward identities appear to depend on a choice of v µ , one can demonstrate that the implicit and explicit dependence cancel and they are in fact invariant under v µ redefinitions. Finally note in all cases the divergence operator takes the form ∇ µ − G µ where G µ = T ν νµ which is the correct form of the divergence on a torsionful manifold.
In writing these formulas, we have chosen g-factor g = 2 and spin s = 1 as we are always free to do. The former is necessary for a regular massless limit, the later is a matter of convenience. A given system may not satisfy these conditions, but in Ref. [17] we present a precise dictionary that allows one to translate our results to the general case.

A. Coordinate Expressions
To aid in the interpretation of Eqs. (9)- (11) and comparison to the usual treatment of non-relativistic fluid dynamics, we collect here a number of coordinate dependent expressions for the above structure. Because we demand n ∧ dn = 0, a Newton-Cartan geometry admits a convenient set of coordinates called global time coordinates (GTC) in which for some scalar potential Φ. It is instructive to have a few coordinate expressions for the structure outlined above in GTC. In these coordinates we may generally parameterize the metric and velocity vector as It's then a matter of calculation to show that ∇ i being the standard spatial connection. There is a unique volume element ε µνλ that is compatible with the connection, where we specialize to 2 + 1 dimensions from this point forward. If we define then ε µν plays the role of the spatial volume element. Again in GTC we have where ε ij is the antisymmetric tensor with ε 12 = 1 √ g . In GTC the Ward identities then read We see that G i = ∂ i Φ plays the role of an external gravitational field that couples to the energy density so we may think of −Φ as the non-relativistic gravitational potential.

III. CONSTITUTIVE RELATIONS
In 2 + 1 dimensions there are four independent one-point Ward identities: current conservation, the work-energy equation and Newton's second law. In the low energy, long wavelength limit, we expect that the system admits a fluid description, that is, the remaining degrees of freedom are also four-fold: two thermodynamic variables, which we take to be the temperature T and chemical potential µ, and the fluid velocity. The Ward identities then suffice to determine the evolution of the system and serve as equations of motion. However, in the massless limit we lose two of these degrees of freedom. The momentum current drops out of the final Ward identity which now contains no time derivatives. What is typically a dynamical equation for the momentum flow reduces to a force-free constraint: since the fluid is massless, it is obliged to flow in such a manner that the applied forces cancel. We will use this in what follows to solve for the charge flow. What remains is two equations of motion that will determine T and µ for all time given initial conditions. Of course for these to say anything we need to specify constitutive relations, that is T µν , ε µ and j µ in terms of the fluid degrees of freedom T , µ, and the external fields. In the long wavelength, low energy limit when the fluid description is assumed to hold, we can assume that only low powers in the derivatives of these variables are important. In this section, we present the most general constitutive relations consistent with non-relativistic diffeomorphism covariance to first order in a derivative expansion.
Our derivative counting scheme for the background fields is as follows. The FQH problem assumes a large, nonvanishing magnetic field, which we will take to vary slowly in space and time. The fluid is also assumed to be moving in a nearly flat geometry and to have only slightly departed from thermodynamic equilibrium; that is, F µν , g µν , T and µ are all O(0).
µν ∇ λ f , the torsion must already be at O(1). To organize the independent data appearing at each order we first note a few convenient facts. To begin, any vector w µ may be uniquely decomposed into a part parallel to v µ and perpendicular to n µ A similar decomposition may be carried out for tensors of all types. In particular for (2, 0) tensors we have for some spatial vectors b µ and c µ and a spatial tensor d µν . As a result, we need only consider scalars, transverse vectors, and transverse tensors in our classification. Since transverse 2-tensors may be further decomposed into a trace, a symmetric traceless part and an antisymmetric part (which we will not need), we are left in the end with scalars, transverse vectors and transverse symmetric tensors. We further subdivide this classification by evenness or oddness under parity.

A. Zeroth Order
Let's begin by analyzing the force-free constraint (21). To our order we have The charge current must be proportional to the unique zero eigenvector of F µν . We make a "choice of frame" so that v µ tracks this equilibrium charge current where n will be some function of the zeroth order data Here B = 1 2 ε µν F µν is of course the magnetic field. Note that in this frame we have In the reference frame comoving with the charge, we expect that the equilibrium state is invariant under spatial rotations. This implies that the energy current must coincide with the charge current and the stress must be pure trace. Hence where ǫ and p are again functions of T , µ and B. However, p, ǫ and n are not entirely arbitrary being constrained by thermodynamics. It can be shown from statistical considerations (see appendix A) that ǫ is the energy density and n number density of the fluid. The hydrodynamic pressure p is sometimes called "internal pressure" and is related to the grand potential density (sometimes called the "thermodynamic pressure") p thm = p thm (T, µ, B) by a Legendre transformation To simplify some of our formulae we prefer to work with the internal pressure p = p(T, µ, M) which is naturally a function of T , µ and the magnetization density M = ∂ B p thm . We will thus exchange B for M as the independent variable in what follows. The functions p, ǫ and n satisfy the thermodynamic identities Only one of these functions (say p) is independent. It is called the equation of state.

B. First Order
We now seek the most general corrections to T µν , ε µ and j µ to first order. Denote these as The complete set of first order data is Here Θ = ∇ µ v µ is the expansion and σ µν = τ µν − Θg µν the traceless shear. The "tilde" operation is defined for symmetric two tensors asÃ µν = 1 2 A µλ ε λ ν + A νλ ε λ µ . We do not include the material derivative of all three thermodynamic variables since one may always be eliminated by the constraint Not all of this data is independent on-shell and we may choose to eliminate some in favor of the others by solving the equations of motion. In our case, there are two scalar equations: the continuity equation and the work-energy equation. We use these to eliminate the material derivatives of T and µ, as indicated by parentheses.
Before we continue, a few comments on fluid frames are due. Since we will be considering small departures from thermal equilibrium there is an inherent ambiguity at first order in derivatives in how we define T and µ. This is a problem extensively discussed in the literature on nonequilibrium fluids [30,31]. We differ from the usual case only in that we do not have any independent definition of a fluid velocity that would require additional fixing. Hence we have a two parameter ambiguity which we choose to fix by going to the Landau frame Note that we have π µν n ν = 0 for free since the stress is a transverse tensor. The most general first order constitutive relations are then where a tilde denotes oddness under parity. We derive Kubo formulas for these coefficients in section V.

C. Force-Free Flows
As mentioned previously, we may use the force balance constraint to completely solve for the charge current All charge transport coefficients are thus determined by the equation of state. Also note that all longitudinal responses are zero. This is because the Lorentz force must cancel forces from pressure gradients and the magnetic field always produces a force perpendicular to the current; hence the current must be perpendicular to pressure gradients.

IV. ENTROPY CURRENT ANALYSIS
The constitutive relations (35) subject to the restrictions (36) are the most general possible that are consistent with the equations of motion and constraint. However, it is still possible to generate flows that violate the second law of thermodynamics. For example, it is well known that a negative shear viscosity allows one to remove entropy from an isolated system and so we should have η ≥ 0 [32]. To derive all such restrictions, we perform an entropy current analysis along the lines of Ref. [30]. Lacking a spacetime picture of non-relativistic physics, previous analyses were restricted to the Lorentzian case and in particular did not include an independent energy current. Our results reproduce theirs for those coefficients that we have in common as well as derive new results for energy transport.
The canonical entropy current is but out of equilibrium we should in principle once again expand in first order data where Now we impose the second law. For non-negative entropy production between all spatial slices, we must have Using the equations of motion in the form one may check that the divergence of the canonical entropy current is a quadratic form in first order data and so the only genuine second order data in (40) is where we have used the Newton-Cartan identities ε µν G µν = 0 and ε µν ∇ µ G ν = 0. Since each term may be independently varied to have either sign, all coefficients appearing in this equation must be zero. The remaining first order data is then Note that by ∂ µ we mean the partial derivative with respect to the chemical potential, not a spatial derivative. For clarity we will always use ∇ µ for the spatial derivative when there is the possibility of confusion. The ∇ µ T ∇ µ T , G µ ∇ µ T and G µ G µ terms need not be separately constrained. We obtain a less stringent condition by setting Σ T = − 1 T Σ G , in which case they arrange into a perfect square We note in passing that in thermal equilibrium there can be no entropy production. This implies or ∂ i T = T ∂ i Φ in coordinates. The physics of this clear: −Φ is the source that couples to the energy density and plays the role of a Newtonian gravitational potential. Heat will tend to flow from regions of higher −Φ to lower −Φ. Equilibrium is reached once the temperature profile is such that (46) is satisfied. This result is also follows from the treatment of equilibrium statistical mechanics in appendix A. In general relativity this is known as the Tolman-Ehrenfest effect which states that the redshifted temperature T ||ξ|| is constant in thermal equilibrium for ξ a timelike killing field [33]. In the non-relativistic case we have T n µ ξ µ = const. From (44) we immediately obtain the expected signs of the parity even viscosities and thermal conductivity The remaining terms place new restrictions on the energy and entropy coefficients We seek the most general solution to these constraints. Begin by eliminating the entropy coefficients by taking the curl of the third equation and plugging in the second Since the left hand side is the curl of a vector, the right hand side is divergenceless and it appears as if we might obtain another constraint. However one may check that this is automatically satisfied by virtue of the constraints (36) and the thermodynamic identities (31). We may simplify the partial differential equation (49) by a substitution that isolates the energy response's dependence on the equation of state andΣ G We see that since (g T ,g µ , g M ) is curl free, it must be the gradient of some functioñ

Summary
This completes the entropy current analysis. For convenience, we collect our results in this section. FQH fluids may be generally viewed as massless fluids in a Newton-Cartan geometry. For the special values g = 2, s = 1 of the parity breaking parameters we have the following constitutive relations: The charge-current response is purely transverse where all coefficients are determined in terms of thermodynamics Since v µ = e Φ (1, we have a pure Hall conductivity σ H = e Φ n B . The energy-current takes the form There is one longitudinal response, the thermal conductivity The remaining four coefficients are all transverse and depend only on the equation of state and two arbitrary functionsΣ G andg of T , µ and M Using (48) we find the entropy current is determined byg andζ G whereζ Finally, the stress is determined by the internal pressure and three viscosities The bulk and shear viscosities must be non-negative whereas the Hall viscosityη is unconstrained. In a Weyl invariant theory, the bulk viscosity must vanish. The complete set of restrictions imposed by Weyl on the coefficients considered above are given in appendix B.

V. KUBO FORMULAS
Fractional quantum Hall transport is determined by p, ζ, η,η, Σ T ,Σ G andg, some of which are subject to positivity constraints, but are otherwise arbitrary functions of T , µ and M. In this section we provide Kubo formula's for these functions. For concreteness, perturb around a flat background with wavevector k µ . The wavevector may be decomposed into temporal and transverse parts k µ = ωn µ + q µ . We will be considering response in both the "rapid" (q µ → 0) and "slow" (ω → 0) cases. The two-point functions of interest are Explicitly, these are The contact terms in these equations do not contribute to the imaginary parts of the respective Green's functions in momentum space, which will appear later in Kubo's formulas.
In this section we prefer to take all coefficients as functions of T , µ and B rather than T , µ and M as it is less awkward to deal with electromagnetic perturbations. It's a straightforward matter to translate the T , µ, M dependence of equations (52) and (54) to T , µ, B by plugging in M(T, µ, B) = ∂ B p thm (T, µ, B) and use of the chain rule.

A. Viscosities
The viscosities have already been discussed at length in the literature [14,30], but we rederive their Kubo formulas in our language for completeness. Our treatment is particularly close to that of Ref. [30]. Consider a rapid metric perturbation δh µν . Using the definition (13) of the shear, we find δT and δµ may of course be solved for using the linearized equations of motion but we will not need to do so here. Π µν λρ andΠ µν λρ are the even and odd symmetric traceless projectors Using these identities and the symmetry properties Π µνλρ = Π λρµν andΠ µνλρ = −Π λρµν , it is then straightforward to verify that Here and further, whenever we write lim ω→0 , we assume that spatial momentum is put to zero (q = 0) before the limit is taken. Vice versa, when we write lim q→0 we implicitly assume that the frequency has been put to zero (ω = 0) befor the limit is taken.
To get at the bulk viscosity, use δΘ = 1 2 iωg µν δh µν and take the trace and imaginary part of (64) Before deriving the remaining Kubo formulas, we would like to make some comments on the relation Σ T = − 1 T Σ G obtained from the entropy current analysis and rederive it from an alternative point of view that highlights the underlying physics. The Einstein relation identifies the conductivity and dissipation σ = −χ µ of any charged fluid where These seemingly unrelated coefficients are connected by the following physical consideration. Apply a static but spatially varying electric potential δA 0 . Charges will flow, but give the system time to relax and the current will again vanish. The chemical potential will adjust to match the profile of the electric potential δµ = δA 0 . Consistency then demands that σ = −χ µ . Σ T = − 1 T Σ G follows along similar lines. In the presence of the gravitational potential −Φ, energy will flow from regions of large potential to small potential until equilibrium is reached. From appendix A we have for a static background where c is the time circle passing through the fluid element under consideration. Now add a time independent perturbation δn µ = −δΦn µ and the temperature and chemical potential adjust by Altogether we have We now impose consistency with the linearized equation of motion which gives the gravitational Einstein relation Σ T = − 1 T Σ G . Luttinger first used the gravitational Einstein relation to obtain a Kubo formula for Σ T [34]. We perform a derivation in our language for completeness. Take a rapid transverse perturbation δn µ such that v µ δn µ = 0, where we have used ε µν k ν = 0. Upon application of projectors we have Finally, we derive a Kubo formula for the functiong. Under a slow perturbation δn µ = −δΦn µ we have The energy current varies as Plugging in the explicit form ofΣ T andΣ µ , we obtain This only determinesg up to a function f ( µ T ). We can fix this ambiguity by response to a slow electric potential perturbation δA µ = δa 0 n µ . We then have giving Recall here that derivatives are taken at constant B rather than at constant M. The Kubo formulas forΣ G andg completely determine the parity odd energy transport.

VI. PHYSICAL INTERPRETATION
To compare with physical results, we first need deal with two issues. First, the covariant currents ε µ and T µν have implicit dependence on v µ that must be removed. This can be done by instead considering the noncovariant currents defined by In Ref. [17] we demonstrate that these noncovariant currents are simply (these relations are greatly simplified by our use of the massless limit and selection of s = 1). The only change from the above is in the energy current, which we defer discussion of until later. Written out explicitly, we have The second issue is that to perform the LLL projection we have taken the g-factor to be g = 2 (and the spin to be s = 1 though this is not essential). To compare to standard expressions used in literature we need to transform back to the values commonly assumed, g = s = 0. The result turns out to be rather trivial, in the end giving us back (85) with shifted transport coefficients, but it is worth demonstrating how this comes about. In the process we find simple formulas that demonstrate how to recover the physical transport coefficients from those calculated in the massless limit. The general procedure for how to do this is explained in Ref. [17] and we merely outline the results here.
Note that to simplify the resulting formulas we assume that E i is O(1) in derivatives. In the above, the electric field was potentially large; however, since it's variations are assumed to be small, a frame where E i is small everywhere may always be obtained. In such a frame a large number of terms are higher order and neglected. Indeed we have already used this in (84) to neglect terms that involve the mass which we are otherwise restoring.
The g = s = 0 currents are then This has a simple interpretation and with a little physical insight we could have guessed the form given here. Recall from Ref. [17] that redefining g involves a shift to the electric potential This also shifts the ground state energy of the system and so the chemical potential changes µ g=2 is the chemical potential that appears in (88b). Similarly, setting g = 2 alters the intrinsic magnetic moment of the fluid: each particle carries an excess magnetic dipole moment of 1 2m . The g = 2 and physical magnetizations are then related by accounting for the final term in (88b) and the shift to the internal pressure in (88c) since p = p thm − MB. Finally, setting s = 1 overestimates the intrinsic angular momentum per particle by 1, giving the observed shift in the Hall viscosity. In the end, the constitutive relations simply revert to the form (85) where we are using the g = s = 0 values of µ, M, andη.
The non-covariant currents then satisfy equations of motion [17] 1 Now consider the current response to the electric field E, the gravitational field G = ∇Φ and gradients of T , µ and B We find a Hall conductance This equation is can be obtained trivially by going to the coordinate system moving with the velocity (E ×ẑ)/B, in which the electric field vanishes. The Hall diffusivity is which using Maxwell's relations can be written as From this equation it is easy to argue the existence of Hall plateaus when the chemical potential lies in a gap. In the T → 0 limit, small variations in the chemical potential cannot induce electron transport and so σ µ H = 0. Equation (96) then immediately implies where we have taken Φ = 0 and ν is some constant (which we of course know to be the filling fraction). (We are working in units where e = = 1.) Similarly we also find so in particular σ B H is simply the magnetic susceptibility.

B. The Energy Current
We now turn to energy transport. A redefinitioñ will aid in the physical interpretation of the formulas that follow. Including the T ij v j shift to the covariant energy current we have The g = 0 values of the energy density ǫ, energy magnetization M E , and Righi-Leduc coefficient c RL that are used in this formula are related to the g = 2 values by Defining thermal conductivities we have

C. Středa Formulas
One notable feature about the formulas (86) and (100) is the charge and energy currents that persist in thermal equilibrium. We now turn to these, deriving a set of Středa-like formulas for two dimensional fluids. First note from the definitions of the temperature and chemical potential in appendix A that in thermal equilibrium we have ∂ i µ = e Φ E i + µG i and ∂ i T = T G i where G i = ∂ i Φ is the gravitational field exerted by the potential −Φ. The equilibrium currents are then Expressing these in terms of the externally applied fields E i , B and G i we have

D. Noncovariant Kubo Formulas
For the reader's convenience, we restate here the Kubo formulas found above in terms of the energy magnetization M E and thermal Hall coefficient c RL without the use of the Newton-Cartan formalism. They are expressed in terms of two-point correlators of the noncovariant currents These are the correlators being evaluated on the trivial background g ij = δ ij , Φ = 0, E i = 0 and ∂ i B = 0.

VII. CONCLUSION
The proper coordinate invariant description of non-relativistic physics is that of a Newton-Cartan geometry, which naturally includes a source n µ for the energy current in addition to those present for the stress and charge current. As discussed in recent work, with some care, diffeomorphism covariant currents may then be defined and 1-point Ward identities follow naturally as in the nonrelativistic case.
In a fluid dynamical description, the Ward identities become equations of motion once constitutive relations have been supplied. We have given the most general constitutive relations consistent with diffeomorphism covariance and derived their Kubo formulas. We argue that a fractional quantum Hall fluid is distinguished as being a force-free fluid in 2 + 1 dimensions. The force-free condition immediately gives powerful constraints on fractional quantum Hall transport, determining all charge transport in terms of thermodynamics.
A straightforward entropy current analysis was then performed. The expected restrictions on the signs of parity even viscosities and thermal conductivity are obtained, in addition to new constraints on the transverse energy response. These four coefficients are not independent but are instead determined by two free functions of T , µ and M: the thermal Hall conductivity and energy magnetization. The derived constitutive relations imply a set of formulas for the equilibrium response that generalize the well-known Středa formula. These new formulas characterize the system's response to Newtonian gravitational fields and inhomogeneous magnetic backgrounds.
It is our hope that the approach outlined here to non-relativistic fluids finds further use. In this approach spacetime coordinate invariance is automatic, just as in the standard treatment of relativistic fluids and computations are streamlined. Here we brought our formalism to bear on FQH fluids, but it is sufficiently general to treat arbitrary fluids in any dimension.
is characterized entirely by a temperature and a chemical potential where c is the time circle passing through that element. Note that T and µ may depend on space.
To zeroth order in derivatives, we then have B being the only other covariant scalar that may be constructed at zeroth order. The detailed form of p thm will depend on the microscopic physics but will not be needed here. Had we assumed spatial homogeneity, this would merely be the elementary relation Ω = p thm V that connects thermodynamics with statistical physics (Ω = T W is the grand potential). Thus p thm is the grand potential density which, in the absence of the magnetic field, would coincide with the pressure that appears in the stress. We define local energy, entropy, particle and magnetization densities by dp thm = sdT + ndµ + MdB and ǫ + p thm = T s + µn, which are merely the fundamental thermodynamic relations (31). It's now a simple matter to calculate the equilibrium j 0 , ε 0 and T ij . To clarify the Φ dependence, parameterize the time circle by some interval x 0 ∈ (0, 1 T 0 ). We then have Varying A 0 , Φ and g ij we find j 0 = e Φ n, ε 0 = e Φ ǫ, T ij = (p thm − MB)g ij (120) The magnetization contribution to the internal pressure arises due to the magnetic flux density's metric dependence B = 1 √ g (∂ 1 A 2 − ∂ 2 A 1 ).

B. WEYL INVARIANCE
In special cases the theory may exhibit Weyl invariance. This happens, for example, when the interaction is a purely contact interaction [17]. In this case the functional form of the transport coefficients considered above will be constrained. We derive these constraints in this appendix.
To satisfy the scaling rule (122) the bulk viscosity must vanish: ζ = 0 [38]. Furthermore, the equation of state and viscosities must be homogeneous functions of the thermodynamic variables