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Relativistic Gravity and Parity-Violating Non-Relativistic Effective Field Theories

We show that the relativistic gravity theory can offer a framework to formulate the non-relativistic effective field theory in a general coordinate invariant way. We focus on the parity violating case in 2+1 dimensions which is particularly appropriate for the study on quantum Hall effects and chiral superfluids. We discuss how the non-relativistic spacetime structure emerges from relativistic gravity. We present covariant maps and constraints that relate the field contents in the two theories, which also serve as the holographic dictionary in context of gauge/gravity duality. A low energy effective action for fractional quantum Hall states is constructed, which captures universal geometric properties and generates non-universal corrections systematically. We give another holographic example with dyonic black brane background to calculate thermodynamic and transport properties of strongly coupled non-relativistic fluids in magnetic field. In particular, by identifying the shift function in the gravity as minus of guiding center velocity, we obtain the Hall viscosity with its relation to Landau orbital angular momentum density proportional to Wen-Zee shift. Our formalism has a good projection to lowest Landau level.

Introduction Symmetry plays an important role in almost every aspect of modern physics. Combined with effective field theory (EFT) techniques, it is a powerful tool for understanding strongly correlated quantum phenomena in systems from high energy to condensed matter and atomic physics. One preeminent example is the fractional quantum Hall (FQH) effect, where interactions are crucial and defy perturbative approaches. Nonrelativistic (NR) general coordinate invariance (GCI) was introduced in [1]. In the aid of Newton-Cartan geometry, [2] constructed an EFT for FQH states that respects this symmetry. Of particular interest is that this theory has a good projection to the lowest Landau level (LLL) and encodes universal geometric properties of FQH states such as Hall viscosity [3][4][5][6][7]. A key ingredient in [2] to generate the geometric features is to couple the composite particles to curved space through spin connection, which was also considered recently in [8] from flux attachment. Further developments along this line can be found in [9][10][11][12][13][14][15][16]. It is worthy to note that speaking of EFTs, we also include holography [17], which can be viewed as EFTs constructed in the aid of higher dimensional geometries and holographic dictionaries.
For NR field theories, a fundamental feature is the existence of a global time, a requirement of the NR causality. Thus to build up NR GCI EFTs, it is natural to employ NR gravity theories such as Newton-Cartan geometry [2,11,13] and Hořava-Lifshitz gravity [10,[18][19][20]. Relativistic gravity theories do not have a built-in notion of global time a priori. However, this does not exclude the possibility that they can be used as a framework to construct NR GCI EFTs, provided that the background isometry or certain imposed condition selects a preferred time foliation. The pioneering works of [21,22] and [23] show that this can be achieved in relativistic gravity with Schrödinger and Lifshitz backgrounds. Furthermore, [24] shows that for Lifshitz holography, the boundary geometry is of various types of Newton-Cartan geometry, depending on the boundary condition the time-like vielbein satisfies. Actually such boundary condition may exist independently of the holographic bulk structure. Our first step in this letter is to argue that relativistic gravity can be used for general NR EFTs even without the aid of holography, provided that the time-like vielbein satisfies the hypersurface orthogonality condition.
A second common feature in the recent geometric formalism of NR EFTs is a velocity field. In NR field theories, the U (1) gauge field transforms under diffeomorphism not just as a Lie derivative but with additional anomalous terms only dependent on metric. This is a consequence of Galilean boost invariance and gives rise to the well-known relation between momentum density and conserved current [25] and its variations [13]. In EFTs the velocity field is necessary to cancel the anomalous terms and covariantize the U (1) gauge field. In holography, this covariant map serves as part of the holographic dictionary [26]. The precise form of this relation may vary depending on how the microscopic theory is coupled to curved space. In this letter, in addition to local U (1) gauge symmetry and spatial diffeomorphism wildly studied in NR EFTs, we also consider homogeneous time reparametrization and local anisotropic Weyl rescaling. These almost completely determine the covariant map.
As an example, we show a single Chern-Simons term together with the covariant map can reproduce all correlation functions obtained in [2] using Newton-Cartan formalism.
A NR EFT formalism built on relativistic gravity is particularly convenient for applications of holography, which is developed mostly within the frame of relativistic gravity theories. As a prerequisite for NR holography, a notion of global time must exist at the boundary. In [18,20] the global time is extended to the whole bulk by employing Hořava gravity [28]. However, Hořava gravity is notorious for complications involving black hole event horizon [29] and difficulty of finding hyperbolic black hole solutions [30], thus its holographic applications are limited. In this letter we offer an alternative: the bulk is still relativistic without a preferred time foliation. The global time is only realized at the boundary by imposing the hypersurface orthogonality condition for vielbein there. The holographic dictionary ensures the dual field theories are NR, while the relativistic bulk allows black hole solutions previously well studied in relativistic holography. This facilitates holographic study of thermal effects and phase transitions of NR systems. As an example, we employ dyonic black brane model of [31] to study NR Hall effects with finite temperature and magnetic field. The Hall viscosity in this case still satisfies the well-known relation with the charge density and shift found in [5,6].
NR Symmetries We consider a (2+1)-dimensional NR field theory described by the following microscopic action in curves spacetime where ". . ." denotes the interactions, g ij is spatial metric, g = det(g ij ) and

The magnetic field is
√ g ǫ ij with ǫ ij the Levi-Civita symbol and ǫ xy = 1. g is the gyromagnetic factor. The microscopic field ψ has intrinsic spin s 0 and couples to curved space through spin connection: where e a i is the vielbein for metric g ij with tangent space index a. The action is invariant under NR diffeomorphism and anisotropic Weyl transformations (parametrized by ξ µ and σ, with ∂ i ξ t = 0) as shown in [20], with a slightly different transformation rule for A t : because we do not include Ricci scalar in D t . The Ward identities resulting from the NR spacetime symmetries derived in [13] are still applicable here.
Global Time For the purpose toward NR EFTs, it is convenient to discuss geometry in term of vielbein rather than metric. For (2+1)-dimensional relativistic gravity, ds 2 = −τ 2 + δ ab e a e b . The time-like vielbein is τ = e −Φ (dt − C i dx i ) and space-like ones e a = e a i (dx i + N i dt). Notice g ti = e −2Φ C i + N i where i is lowered by g ij . C i is the source to NR energy flux [13,18,21], which can be seen by matching its diffeomorphism with that of the source. However, C i does not appear in the above NR field theory because it is written in global time coordinates (GTC) where C i = 0. The condition for existence of a global time as required by NR causality is τ ∧ dτ = 0 [13], which corresponds to the twistless torsion condition for Newton-Cartan geometry in [24]. For the application of relativistic gravity on NR EFTs, this hypersurface orthogonality condition for vielbein must be imposed. Then we can always work in GTC with C i = 0. Under diffeomorphism, This implies the allowed diffeomorphism in GTC must satisfies ∂ i ξ t = 0. This is called foliation preserving diffeomorphism (FPD) and is exactly the assumption made to ensure the diffeomorphism invariance of the above NR field theory. Now Φ and e a i can be identified with their counterparts in NR field theory because they have the same symmetry transformations. To compute energy flux, C i dependence has to be restored. This can be done by performing a ∂ i ξ t = 0 diffeomorphism and going away from GTC. For the rest of this letter, we will work in GTC for simplicity.
Holography and Covariant Map For holography in 3+1 dimensions, the bulk theory includes relativistic graviton described by vielbein (τ ,ê a ,n) and a U (1) gauge fieldV =V M dx M , among others. We assume the background near boundary r ⇒ 0 is asymptotic Lifshitz with AdS radius L: and choose gauge condition for radial vielbeinn = (L/r)dr andτ r =ê a r = V r = 0. These conditions do not completely fix the bulk gauge freedom. The residual diffeomorphism near boundary isξ µ ⇒ξ µ ,ξ r ⇒ −rσ. Then near boundaryτ ⇒ (L/r) zτ ,ê a ⇒ (L/r)ē a and V µ ⇒V µ . Under (ξ µ ,σ),τ ,ē a andV µ transform in the same way as their counterparts in (2+1)-dimensional relativistic gravity, hence are identified with above τ , e a and a V µ . Now the global time condition becomes a boundary condition in holography:τ ∧ dτ = 0. We will work in the stronger conditionτ i = 0, which forces ∂ iξ t = 0 at boundary but allows ∂ iξ t = 0 in the bulk. The map for the U (1) fieldV µ = V µ is non-trivial: after matching the symmetry transformations with those of A µ in the NR field theory, we obtain where s ′ is an arbitrary constant. Being part of the holo-graphic dictionary, this is an extension of that in [18] to parity violating case. It is also the covariant map for (2+1)-dimensional EFTs (independent of holography), an extension of [2,13,16]. A nice feature is for FQH effect, it has a good LLL projection when g = 2 and m → 0. In holography the mass m is dual to a bulk scalar whose near-boundary behavior matches its Weyl transformation in NR field theory. Shift Vector The only remaining problem in our formalism is the shift vector N i , which has not been interpreted nor determined from NR field theory point of view. According to its symmetry transformations, it corresponds to the velocity field in Newton-Cartan formalism [2,13]. In relativistic theories, it sources the momentum density, but this is not the case here, because in NR theories, momentum density p i is completely determined in term of charge current J µ by Galilean symmetry [25]. In the current case, it is in flat spacetime. Thus N i does not source p i . To determine it in terms of other fields, a constraint has to be imposed. There is no universal prescription in the literature. [20] has a detailed discussion on how to impose a diffeomorphism invariant constraint with smooth LLL limit for FQH effect. Similarly here, there are two viable choices: The former is essentially a path-integral in the spirit of [2] to integrate out the velocity field, while the latter also appears recently in [16]. For FQH effect, they yield the same universal features in Chern-Simons EFT [20]. Effective Action for FQH states As an example, we show how to build low energy effective action for FQH states using our formalism. We work in the limit when the magnetic field B is large compared to the electric field E i and derivatives. At leading order in derivative expansion, the gauge Chern-Simons term encodes the universal properties. To build a NR GCI action, we start with a relativistic Chern-Simons term S CS = ν 4π d 3 x V ∧ dV and apply the map (1), then plug into the constraint (3) and solve for N i . We get N i = −ǫ ij E j /B + O(∂B, ∂Φ), thus −N i is the drift velocity. Then we obtain the NR GCI Chern-Simons action. For constant B and E i , it has a simple form: where we have set Φ = 0 and O (∂ µ ) denotes derivative corrections to the local Lagrangian. These corrections can be systematically calculated to any higher order using (1) and (3), and the fist few are given in [20]. All correlators obtained from this action (for their explicit expressions see also [20]), including the derivative corrections omitted in the above expression, agree with those computed in [2]. From this action, we can easily recognize s ′ is half of the Wen-Zee shift [32] and Hall viscosity η H = (ν/4π)Bs ′ . Total momentum density can be calculated using (2), while the guiding center one is the part conjugate to drift velocity −N i . Subtracting the latter from the former, we obtain the Landau orbit momentum density, from which the orbital angular momentum density can be calculated: ℓ orb = −(ν/2π)Bs ′ . This justifies the relation η H = −ℓ orb /2 in [5,6,33]. There are two more topological terms in (2+1)dimensional relativistic theories beside V ∧ dV . One is the gravitational Chern-Simons term tr ω ∧ dω + 2 3ω ∧ω ∧ω , whereω denotes the non-Abelian spin connection constructed from the full spacetime vielbein. Its contribution to the NR effective action after applying the covariant map has been calculated in [20]. Its primary role is to shift the coefficient of ω ∧ dω by a constant, which is related to the central charge of chiral conformal field theory on the boundary and the thermal Hall conductivity. A third relativistic topological term which mixes the U (1) gauge field with spacetime curvature had recently been constructed in [34]. By applying our covariant map, its contributes to NR effective action is equivalent to a shift of the constant s ′ . Local terms in the relativistic parent theory will generate non-universal features related to the interactions of the microscopic theory. We will not discuss these terms here.
A Holographic Model We now give another example in term of relativistic holography [31], which is dual to strongly coupled quantum fluids in external magnetic fields. The bulk action includes Einstein-Maxwell termŝ The background metric is a dyonic black brane .
withV t = −qα(1 − r),V y = hα 2 x, where q and h are electric and magnetic charges. h is related to a constant magnetic field B 0 = hα 2 on the boundary. The blackening function is f (r) = 1 − (1 + h 2 + q 2 )r 3 + h 2 + q 2 r 4 , with the horizon located at r = 1. The mass parameter α is related to the Hawking temperature by 4πT = α 3 − h 2 − q 2 . The renormalized action is given by subtracting the Gibbons-Hawking term and a counterterm of the boundary volume. To calculate correlation functions, we solve all the nine metric and gauge fluctuations in the bulk up to linear order in momentum (ω, k). By rotational symmetry we can set k x = k, k y = 0. We will not list the full expressions of the solutions nor the action here, but only give results of correlators computed from them. The procedure is similar to that in [31]. After obtaining the on-shell boundary action, still in relativistic form, we solve the constraint equation (3) (we use the first one) and apply the holographic dictionary (1) to calculate the NR GCI effective action. The non-vanishing 1-point functions are the charge density ρ, energy density ǫ and internal pressure P : Thermodynamic pressure (i.e. the grand potential density) can be obtained from the background action: The chemical potential µ = −qα − (g − 2) B 0 /(4m) is identified with the background of A t in the NR field theory. The magnetization density is defined as M = ∂P thm /∂B 0 | T,µ . Bekenstein-Hawking law gives entropy density s = 2πL 2 α 2 /κ 2 4 . All these thermodynamic quantities satisfy the following fundamental relation: The system has local thermodynamic stability: Some of the 2-point functions are where definitions of retarded correlators G A,B ra follow [13]. From G i,j ra , the longitudinal conductivity vanishes and Hall conductivity equals ρ/B 0 as expected. G i,00 ra shows current response to inhomogeneous gravitational field ∂ i Φ, with a transport coefficient σ G H = (ǫ + P ) /B 0 that agrees with [35] from hydrodynamic analysis on LLL. The above correlators satisfy Ward identities given in eqs. (47) and (48) in [13].
The shear, Hall and bulk viscosities are The covariant map (1) is crucial to the above nonvanishing Hall viscosity, whose form in terms of charge density and the shift agrees with [5,6]. [13,36] pointed out that there is a zero-frequency divergent term proportional to the inverse internal compressibility κ −1 int in the bulk viscosity. For our case, it is Other than this contact term, the bulk viscosity is vanishing as required by Weyl invariance. At the end we have two comments on the mass m. (1) The dyonic black brane background is asymptotic AdS with dynamical exponent z = 1. From our holographic dictionary, for z = 2, m is dual to a bulk scalar with non-trivial profile and r z−2 asymptote. Here we do not consider this profile explicitly because we work in the probe limit where this scalar sector can be engineered such that it decouples, similar as in [18]. (2) Instead of working in the probe limit, we can also project to LLL, where m → 0 and g = 2. In this limit the cyclotron frequency ω c = B/m diverges which forbids higher Landau level mixing. In this case all (g − 2)/m terms in the above expressions drop off, and this holographic model becomes one for LLL.
Conclusions We have shown that relativistic gravity theories can be used as a framework to build effective theories for NR systems that respect all NR spacetime symmetries, holographically or not, for any dynamical exponent z. In order to adapt to the global time, the time-like vielbein must satisfy hypersurface orthogonality condition. Under this condition, we present a covariant map that relates the relativistic gauge field to the NR one, which can also serve as part of the holographic dictionary. Additional constraints are given to fix the shift vector. Our formalism is particular suitable for spin-polarized NR particles, including the FQH fluids and chiral superfluids [37]. Low energy effective actions for these systems are then constructed from purely (2+1)dimensional Chern-Simons field theory and from (3+1)dimensional holographic theory with a dyonic black brane background. They have a good LLL projection and capture the linear response properties of these systems.