Large Charge Sector of 3d Parity-Violating CFTs
EEFI-21-1
Large Charge Sector of 3d Parity-Violating CFTs
Gabriel Cuomo a,b , Luca V. Delacrétaz c and Umang Mehta c a Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794, USA b C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794, USA c Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, IL 60637, USA
Abstract
Certain CFTs with a global U (1) symmetry become superfluids when coupled to achemical potential. When this happens, a Goldstone effective field theory controls thespectrum and correlators of the lightest large charge operators. We show that in 3d,this EFT contains a single parity-violating 1-derivative term with quantized coefficient.This term forces the superfluid ground state to have vortices on the sphere, leading to aspectrum of large charge operators that is remarkably richer than in parity-invariantCFTs. We test our predictions in a weakly coupled Chern-Simons matter theory. a r X i v : . [ h e p - t h ] F e b ontents κ = 1 . . . . . . . . . . . . . . . . . . . . . . . 183.5 Quantum OPE coefficients and transport . . . . . . . . . . . . . . . . . . . 203.6 Large κ and vortex lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 A General parity-violating superfluids 32
A.1 General EFT without conformal symmetry . . . . . . . . . . . . . . . . . . 32A.2 Gravitational stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33A.3 Comparison with parity-violating hydrodynamics . . . . . . . . . . . . . . . 37A.4 Linearized stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
Introduction and Results
Universality, or the stringent constraint of scale invariance and unitarity, implies that anumber of quantum field theories and other many body systems are described by the sameConformal Field Theory (CFT) when tuned to a fixed point. Strikingly, different CFTsthemselves share ‘super-universal’ features, such as the spectrum and correlators of largespin [1, 2, 3, 4], large charge [5, 6, 7, 8] and heavy [9, 10, 11, 12] operators. Some of thesefeatures can be established and studied in the absence of a small parameter in the underlyingCFT.In CFTs with an internal global symmetry, operators of large charge Q (cid:29) µ . Doing so can drive the CFTinto a number of phases, including a superfluid, a Landau Fermi liquid, an extremal blackhole, or possibly a non-Fermi liquid. In any of these examples the chemical potential sourcesboth a finite energy and charge density in the thermodynamic limit , implying a scaling ofthe lightest operators at fixed charge ∆ min ( Q ) ∼ Q d/ ( d − in d spacetime dimensions.In this paper, following Refs. [5, 6] we study the large charge sector of CFTs in d = 3 witha global U (1) symmetry, assuming they enter a superfluid phase. Superfluids are describedby a local effective field theory (EFT), which provides a simple tool to make controlledpredictions about the large charge spectrum of the underlying CFT. We focus on CFTs thatdo not have parity (or time-reversal) symmetry. In this case, we find that the EFT containsa single parity-violating 1-derivative correction. It is best formulated in dual language interms of a gauge field a µ , and takes the form S EFT [ a ] = − α Z d x | f | / + κ π Z d x √− g a µ (cid:15) µνλ (cid:15) αβγ u α (cid:18) ∇ ν u β ∇ λ u γ − R νλβγ (cid:19) + O ( ∂ ) , (1.1)where f µν = ∂ µ a ν − ∂ ν a µ , and u µ is a unit vector proportional to (cid:15) µνλ f νλ . The leading termwith coefficient α is the superfluid stiffness and preserves parity; it controls the trajectoryof the lightest large charge operators ∆ min ( Q ) = √ παQ / + O ( Q / ). The second termhas a quantized coefficient κ ∈ Z , and was first introduced in Ref. [13]. It is O ( ∂ ) and hencesuppressed compared to the stiffness, but more relevant than the O ( ∂ ) parity-preservinghigher-derivative corrections [6]. Despite its derivative suppression, it has crucial consequences There are also situations where this does not happen, such as a free complex scalar, or in theories with amoduli space. a) ∆ ∆ min ( Q ) ∆ min ( Q, J )∆ min ( Q ) + q J ( J +1)2 J (b) Figure 1: Classically, the superfluid ground state has κ vortices arranged to minimize theirpotential energy, shown in (a) for κ = 2. Quantum mechanically, the true ground state hasvanishing spin but vortex excitations are soft. For κ = 2, the spectrum of vortex excitationsis given in Eq. (1.3) and shown in (b) in red with the spectrum of single phonon states inblue.for the spectrum of lightest operators at fixed large charge, which are described as finitedensity states on the sphere R × S . We show that the EFT (1.1) can only be consistentlyplaced on the sphere with vortices, with total vorticity κ (in the dual picture (1.1), thenew term introduces a background charge density on the sphere which must be neutralizedby charged particles). Classically, the ground state is then described by | κ | vortices in aconfiguration which minimizes their Coulomb energy – an extremization problem known asWhyte’s problem in the mathematical literature. Although the solution is not known forgeneral κ , it can be shown to have vanishing angular momentum when | κ | 6 = 1, so that thespin of the corresponding lightest operator vanishes. When | κ | = 1, the ground state containsa single vortex with angular momentum J = Q/ Q in CFTs with | κ | = 1 therefore also have large spin J = Q/
2. In all of these cases, the vortexfugacities give a contribution to the dimension of the lightest large charge operators∆ min ( Q ) = √ παQ / + κ √ πα p Q log Q + O ( p Q ) . (1.2)Quantizing the system leads to a rich low-lying spectrum of vortex excitations, which aresofter than superfluid phonons and hence describe the lightest operators of large charge andfinite spin 0 < J (cid:46) √ Q . For example, when κ = 2 one finds∆ min ( Q, J ) ’ ∆ min ( Q ) + 16 √ πα J ( J + 1) Q / . (1.3)4he spectrum of large charge states is illustrated in Fig. 1. Quantization also shows thatalthough the vortex configuration is inhomogeneous classically, in the true quantum groundstate the vortices are delocalized.These results should hold in any 3d parity-violating CFT that becomes a superfluid atfinite density. We show that these predictions are borne out in a weakly coupled Chern-Simons matter theory, consisting of a single Dirac fermion coupled to a dynamical gaugefield S anyon CFT = Z d x ¯ ψi (cid:26)(cid:26) Dψ − k π (cid:15) µνλ a µ ∂ ν a λ , (1.4)with level k (cid:29)
1. A non-relativistic cousin of this theory goes under the name of anyonsuperfluid [14], and we show that the CFT (1.4) also becomes a superfluid at finite density.Weak coupling ∼ /k of this CFT allows one to derive the corresponding superfluid EFT(1.1) directly from (1.4), thereby obtaining expressions for the EFT parameters α and κ .Large charge operators in this theory are monopoles, dressed with fermions occupying thefirst k Landau levels. The degeneracy of Landau levels on the sphere (or monopole harmonics)implies that κ fermions are missing from the k th Landau level – these are the κ vorticesexpected from the EFT.The rest of this paper is organized as follows: the EFT (1.1) for parity-violating conformalsuperfluids is constructed and studied classically in Sec. 2. It is quantized in Sec. 3, where thespectrum of lightest operators at large charge is discussed, and Eqs. (1.2), (1.3) are obtained.As the leading parity-violating effect in the EFT, κ also gives the leading contributionto parity-odd heavy-heavy-light OPE coefficients in the large charge limit, which are alsodiscussed in Sec. 3. Finally, we test these predictions in the weakly coupled CFT (1.4) anddiscuss further applications in Sec. 4. We briefly review the construction of the EFT for conformal superfluids, referring thereader to Ref. [6] for details. In a superfluid phase, a Goldstone φ nonlinearly realizes theinternal U (1) symmetry (as well as certain spacetime symmetries); the most general invariantLagrangian may be easily obtained requiring U (1) and Weyl invariance. Retaining terms up5o second order in derivatives, one obtains: S = − c d ( d − Z d d x √ g | ∂φ | d + c Z d d x √ g | ∂φ | d ( R| ∂φ | + ( d − d −
2) [ ∇ µ | ∂φ | ] | ∂φ | ) (2.1)+ c Z d d x √ g | ∂φ | d ( R µν ∂ µ φ∂ ν φ | ∂φ | + ( d − d −
2) [ ∂ µ φ ∇ µ | ∂φ | ] | ∂φ | +( d − ∇ µ (cid:20) ∂ µ φ∂ ν φ | ∂φ | (cid:21) ∇ ν | ∂φ || ∂φ | (cid:27) + · · · where R µν and R ≡ R µµ are the Ricci tensor and scalar for the background metric g . Wewill only need the leading term with coefficient c in this paper. The terms beyond the firstline are two-derivative suppressed compared to the leading term, and · · · denotes higherderivative terms. In the next section, we will see that the leading parity-violating terms areonly one-derivative suppressed compared to c .The microscopic U (1) symmetry is nonlinearly realized on the scalar field as φ → φ + c .In addition, the superfluid phase has an emergent ( d − U (1) ( d − thatcounts the winding of φ . The currents for both of these symmetries are j µ = c d − | ∂φ | d − ∂ µ φ + · · · , J µ ··· µ d − = 12 π (cid:15) µ ··· µ d ∂ µ d φ , (2.2)where · · · denotes higher derivative terms. The currents are normalized such that theirintegrals over closed manifolds produce integer valued charges R M d ?j and R M ?J . Parity-violation is better studied after dualizing, namely replacing the scalar degree offreedom φ with a gauge field (see e.g. [15] where this is done in a similar context). This canbe achieved by replacing ∂ µ φ → v µ and introducing a Lagrangian parameter i π R a ∧ dv thatforces v µ to be longitudinal. Integrating v µ out then leads to the effective action S = − α Z d d x √− g | f | dd − + · · · (2.3)with f µ ≡ (cid:15) µλ ··· λ d − ∂ λ a λ ··· λ d − and | f | ≡ p − f µ f µ , and · · · denotes higher derivative termscoming e.g. from c and c in (2.1). The coefficient α is given in terms of c by (2.6) below.The currents (2.2) are now given by j µ = 12 π f µ , J µ ··· µ d − = α dd − (cid:15) µ ··· µ d f µ d | f | d − d − . (2.4)6nd the stress tensor is T µν = dd − α | f | dd − (cid:18) u µ u ν + 1 d g µν (cid:19) + · · · , (2.5)where we have introduced a unit vector u µ ≡ f µ / | f | satisfying u µ u µ = −
1. Around finitedensity backgrounds h j µ i = π h f µ i = ρδ µ , the energy density is ε = α (2 πρ ) dd − , whichleads to the relation ∆ min ( Q ) ∼ αQ dd − for large charge operators studied in Sec. 3. Thedimensionless charge susceptibility of the superfluid is given by χ ≡ µ d − dρdµ = c = d − π ) d (cid:18) d − d α (cid:19) d − . (2.6)We will interchangeably use χ or α in the rest of the paper.We now specialize to d = 3 spacetime dimensions. In this dual picture it was found inRef. [13] that one can write two parity-violating terms consistent with symmetries that areonly one-derivative suppressed compared to the leading term (2.3) S = − α Z d x √− g | f | / + ζ Z d x | f | (cid:15) µνλ u µ ∂ ν u λ + κ π Z d x √− g a µ (cid:15) µνλ (cid:15) αβγ u α (cid:18) ∇ ν u β ∇ λ u γ − R νλβγ (cid:19) + O ( ∂ ) , (2.7)We are focusing here on parity-violating superfluids with conformal symmetry, the moregeneral case is discussed in Appendix A. The two new terms ζ, κ are less relevant thanthe leading term α , but more relevant than the subleading corrections c , c in (2.1). Theadvantage of working in the dual picture is now clear: the ζ term would vanish if written interms of u µ = ∂ µ χ/ | ∂χ | , and the κ term would be non-local . Note that this term is gaugeinvariant because a µ is contracted with an identically conserved current J µ Euler ≡ π (cid:15) µνλ (cid:15) αβγ u α (cid:18) ∇ ν u β ∇ λ u γ − R νλβγ (cid:19) , (2.8)called ‘Euler current’ in Ref. [13]. The name stems from the fact its integral on a spatialmanifold is proportional to its Euler charateristic – see e.g. Eq. (2.16) below. Invariance The action (2.7) can presumably also be obtained directly using an appropriate coset construction, alongthe lines of Ref. [16]. One advantage of this somewhat more tedious approach is that all terms can be obtained‘algorithmically’, including the κ term which arises as a Wess-Zumino term following Refs. [17, 18]. Said differently, the κ term explicitly breaks the emergent winding (1-form) symmetry, which acts onthe gauge field as a µ → a µ + c µ . Non-conservation of the 1-form current will have crucial consequences inSec. 2.3. κ ∈ Z . (2.9)In the conformal context, the EFT (2.7) can in fact be simplified. Since the ζ term vanisheson the leading equations of motion (because of conservation of J µν in (2.4)), one mightexpect that it can be removed with a field redefinition. Although this is not quite correctin the general case (cf. Appendix A), it is for the conformal superfluid where the fieldredefinition is a µ → a µ + 2 ζ α f µ | f | / . (2.10)Now this field redefinition does not preserve the normalization H a π ∈ Z , unless ζ = (cid:16) α (cid:17) n with n ∈ Z . In this case (2.10) reads a µ → a µ + n ( ?J ) µ , (2.11)which preserves the normalization because of the quantization of the higher form charge (2.4).The part of ζ that cannot be removed is therefore circle-valued ζ = (cid:16) α (cid:17) θ π with θ ∈ [0 , π )and has similar properties to a θ -term in gauge theory. The final EFT for parity-violatingconformal superfluids hence takes the form S = − α Z d x √− g | f | / + θ Z ( ?J ) ∧ d ( ?J ) + κ Z d x √− g a µ J µ Euler + O ( ∂ ) , (2.12)The θ term is a total derivative and it does not contribute in perturbation theory around thehomogeneous background. Nonetheless, it has simple nonperturbative effects; for example,it will split the energy of positive and negative winding vortices on the plane R . Since wewill be working on a compact spatial manifold, namely the sphere S , we can drop all totalderivatives and simply work with the action (1.1).The stress tensor of the theory (2.12) is given by T µν = 32 εu µ u ν + ˜ η h u α (cid:15) αβµ ∇ β u ν + ( µ ↔ ν ) i − trace + · · · . (2.13)with ε = α | f | / and ˜ η = κ π | f | . This expression fits into the general form a stress tensortakes in a parity-violating fluid [19]; a more thorough comparison with the hydrodynamics isgiven in appendix A, where details for obtaining the stress tensor from (2.12) are also given.In hydrodynamics, ˜ η is referred to as the Hall viscosity, because it describes parity-odd8omentum transport in finite density states h f µ i = 2 πρδ µ through the Kubo formula (seee.g. [20]) ˜ η = lim ω → iω G RT xx T xy ( ω, k = 0) . (2.14)Eq. (2.9) implies that ˜ η is quantized in units of ρ/ In order to study the large charge spectrum of local operators in CFTs, we place the EFT(2.7) on a spatial sphere S . Strikingly, it turns out that, despite its gradient suppression,the leading parity-odd term κ has an important qualitative effect when studying the theoryon the sphere: it forbids a homogeneous finite density solution h f µ i = 2 πρδ µ [13]. This canbe seen from the equation of motion for a µ , which reads as a non-conservation of the 1-formcurrent (or simply as charged electric matter in dual language) ∇ µ J µν = κJ ν Euler . (2.15)Looking for a solution close to the homogeneous finite density profile h f µ i = 2 πρδ µ , whichis a solution of the leading order equation of motion, one finds that the right-hand side isproportional to the Euler density on the sphere J ν Euler ’ δ ν R π = δ ν πR . (2.16)Eq. (2.15) then requires a velocity field with a constant vorticity on the sphere, which is notpossible without producing vortices . In dual language, (2.15) introduces a constant electriccharge density, which on the sphere must be neutralized by charged matter for Gauss’slaw to hold. Although we have here expanded around the finite density background, theseconclusions hold more generally, see Ref. [13].We therefore generalize our EFT to include vortices. The spacetime trajectories of thevortices can be parametrized with worldlines X µp ( τ ). The most general action compatiblewith the symmetries of the system reads: S = S superfluid + S vortices , (2.17) This vorticity can also be neutralized with a background flux of magnetic field for the U (1) symmetry. Inthis context κ is related to the ‘shift’ of superfluids and quantum Hall systems [21, 22, 23, 13]. Since CFToperators map to states on the sphere without any fluxes, we do not turn on any background fields in thefollowing. S superfluid given by (2.7) and [24, 15] S vortices = − X p w p Z X p a + γ p Z X p | f | / dX p + · · · ! = − X p Z dτ (cid:18) w p a µ ˙ X µp + γ p | f | / q − g µν ˙ X µ ˙ X ν + · · · (cid:19) , (2.18)where w p ∈ Z are the windings and γ p the dimensionless tensions of the vortices p =1 , , . . . , N vortices . Equation (2.15) now instead reads ∇ µ J µν = κJ ν Euler − j ν vortex . (2.19)A natural derivative counting scheme in the superfluid and vortex system is f ∼ ∂a ∼ ∂X ∼
1. In this scheme the α term is O ( ∂ ), the κ term and Wilson line term are O ( ∂ ) (note that δ ( x − X p ) ∼ ∂ ), and the contribution from the vortex velocities in thetension term is O ( ∂ ). In this paper, we only study the leading effects of parity violation –we therefore only keep terms up to O ( ∂ ) and work with the action S = − α Z d x √− g | f | / + Z d x √− g a µ (cid:16) κJ µ Euler − X p w p δ ( x − X p ) ˙ X µp (cid:17) + O ( ∂ ) . (2.20)Quantum mechanically, the spectrum of a point particle on the sphere in a monopole magneticfield consists of Landau levels [25]. Dropping the mass term for the vortex is equivalent torestricting the description to the lowest Landau level [26, 27]. Indeed, the gap separation ofhigher Landau levels is given by the cyclotron frequency formula ρ/m vortex ∼ √ ρ/γ , whichcoincides parametrically with the EFT cutoff.We now can compute the energy and angular momentum deriving from the action (2.20)for an arbitrary classical state; our analysis will be almost identical to the one in Ref. [15].The results obtained here will then be used in the next section to study the CFT spectrum.We begin expanding the fields around a solution with finite density ρ : h j µ i = 12 π h f µ i = ρδ µ , (2.21)and write the electric and magnetic fields as f = 2 πρ − b , f i = − (cid:15) ij e j . (2.22)To first order in perturbation theory, the equations of motion reduce to the ones of electrostaticfor point charges on the sphere in the presence of a homogeneous charge density: α / √ πρ ∇ · e = κ πR − X p w p δ ( x − X p ) , e i = f ij ( ˙ X p ) j . (2.23)10he first equation is just Gauss’s law on the sphere; to this order, the magnetic fieldperturbation b is unsourced. The second equation in (2.23) implies that vortices movewith small drift velocities | ~ ˙ X p | ∼ / √ ρ in trajectories of vanishing Lorentz force. This isconsistent with the absence of fast cyclotron degrees of freedom in the EFT. Integrating thefirst equation in (2.23) we obtain X p w p = κ . (2.24)As expected, consistency of Gauss’s law on the sphere implies the net charge of the vorticesmust neutralize the homogeneous contribution from the Euler term. Finally, it is easy tosolve equation (2.23) – for example, for a single vortex ( κ = 1) placed on the north pole,one finds α / √ πρ e sol ( θ ) = − θ πR sin θ ˆ θ . (2.25)Solutions for κ > /R = α Z d x √− g (cid:20) (2 πρ ) / + 3 / √ πρ e (cid:21) , (2.26)with the electric e field given by the solution to Gauss’s law (2.23). Solving in terms of thevortex worldlines, we obtain:∆ = 23 √ πχ Q / + √ πχ p Q " κ (log 4 − − X p,r w p w r log( ~R p − ~R r ) , (2.27)where ~R p = (sin θ p cos φ p , sin θ p sin φ p , cos θ p ) is the position of the p th vortex in the R embedding of the unit sphere, and where the dimensionless susceptibility is χ = π α (2.6).The first term in parenthesis is the electrostatic energy due to the homogeneous chargeinduced by the Euler current, while the second one is the electrostatic potential for theinteracting vortices. The terms with p = p diverge, but they will be cutoff at distancesof order the vortex size δX ∼ α − / √ ρ . These correspond to the usual fugacities of thevortices, which give a constant contribution ∼ √ ρ log ρ to the energy. Notice that becauseof the self-energy term, vortices with charge w p = ± This estimate is obtained considering the distance from the vortex core at which the electric field (2.25)becomes comparable with the leading monopole magnetic field. ~J = 3 α p πρ Z d x √ g ~n i (cid:15) ij √ ge j = Q X p w p ~R p , (2.28)where we denoted collectively ~n i = ( n ix , n iy , n iz ) the Killing vectors associated with rotationsand we used Gauss’s law to obtain the right hand side. When all vortices have the samecharge, Eq. (2.28) implies that the angular momentum is proportional to the center-of-massof the system. This property will be important in the following.Finally we mention that in the EFT (2.17) we neglected the possible existence of additionaldegrees of freedom living in the vortex cores. In particular, vortices may carry internalspin degrees of freedom. These may be included in the EFT straightforwardly introducingGrassmanian fields on the wordlines and would provide additional O (1) contributions to theangular momentum, as well as a O ( √ Q ) contribution to the energy due to Pauli interactionbetween the spin and the monopole magnetic field (see [28] for a more detailed analysisin a similar context). Both of these contributions are subleading with respect to the oneswe consider in this paper, and therefore, in what follows, we shall consistently neglectany additional dynamics of the vortices other than the motion of the worldlines. We willnonetheless return to this point when considering microscopic realizations of the superfluidvortices for specific theories in Sec. 4. In this section we apply the EFT discussed in the previous section to the study of thespectrum of conformal field theories, following the strategy presented in Ref. [5]. Moreprecisely, we consider generic non-parity preserving CFTs with a U (1) global symmetrywhich enter a superfluid phase when coupled to a chemical potential µ (cid:29) /R on R × S ,where R is the sphere radius. Such CFTs have a large charge spectrum of operators controlledby a superfluid EFT description. The EFT, combined with the state-operator correspondence,then allows to extract the CFT data of the corresponding operators in a systematic expansionin inverse powers of the charge. To leading order in 1 /Q , the dimension of the lightest operator of charge Q is independentof the presence of vortices or higher derivative terms. It can be found simply from Eq. (2.5)12nd reads: ∆ min ( Q ) ’ √ πχ Q / + · · · , (3.1)where χ is the dimensionless charge susceptibility of the CFT. The notion of a chargesusceptibility is meaningful regardless of the phase of matter that the CFT enters at finitedensity. It is a property of the CFT that cannot be revealed by light operators only (it issimilar in that respect to the coefficient b T in d > However, subleading corrections to (3.1), the angular momentum of the ground state,and the spectrum of nearby operators are drastically affected by the parity-violating termsin the second line of (2.20), as we now discuss.Let us now consider the classical ground state for the theory including vortices. Thisamounts to finding the classical configuration minimizing the expression (2.27). We recallfirst that, because of their logarithmic self-energy, vortices with | w p | = 1 are energeticallyfavored. Therefore we consider κ identical charges interacting through the electrostatic field.The problem of finding the configuration of charges minimizing the logarithmic potential(2.27) on the sphere, or equivalently the product of their distances, is known as Whyte’sproblem [32] in the mathematical literature. The solution is known for a few values of κ , butnot in general . However, we shall need only the following general property of the solution:the minimal energy configuration for κ > κ >
1. For κ = 1 instead,Eq. (2.28) implies that the ground state has spin J = Q/
2. In summary the classical spin ofthe lightest operator of charge Q reads: J ( Q ) = Q/ | κ | = 1 , | κ | 6 = 1 . (3.2) For example the charge susceptibility of the O (2) Wilson-Fisher CFT was obtained from Monte-Carlosimulations in Ref. [29] to be χ = ( c / ) ≈ . More precisely, Eq. (3.1) follows from the assumption that the scaling dimension ∆ min ( Q ) admits anon-trivial macroscopic limit [7]; this assumption might be violated when additional symmetries require theexistence of flat directions, as in free theories or theories with extended supersymmetry [31]. Here are some examples: for κ = 2 the charges are at opposite poles; for κ = 3 they are on the corners ofan equilateral triangle; for κ = 4 they are on the corners of a tetrahedron. min ( Q ) (3.1) comes from the fugacities of thevortices, which can be obtained from (2.27) without solving the interactions:∆ min ( Q ) = 23 √ πχ Q / + κ √ πχ p Q log Qχ + O ( p Q ) . (3.3)In words, each vortex gives a √ Q log Q contribution to the dimension of the large chargeoperator. This is more singular than the non-universal O ( √ Q ) corrections, which arise fromthe vortex tensions and interactions, as well as from the subleading terms in the EFT (2.1).We stress that unlike in Ref. [15], where it was found that the lightest state with charge Q and spin √ Q (cid:46) J < Q also contains vortices (with again a √ Q log Q contribution to ∆), weare here discussing the lightest operator at any spin in a parity violating CFT. When | κ | 6 = 1,this state has no angular momentum but still contains vortices. Higher spin operators withthe same charge Q will also exist but have larger dimension – they correspond to phonon orvortex excitations on top of the ground state and will be discussed in Sec. 3.3. We have kepttrack of the charge susceptibility inside the logarithm in Eq. (3.3); although one expects χ = O (1) for strongly coupled CFTs, in weakly coupled theories χ can take parametricallylarge values as we will see in Sec. 4 (see also [33]).The present classical analysis leaves some open questions. For instance, though the groundstate for | κ | 6 = 1 has angular momentum J = 0, it is classically inhomogeneous – what is thenits quantum mechanical spin? What is its degeneracy? And are there quantum corrections toEqs. (3.2) and (3.3)? Addressing these questions requires a quantum-mechanical descriptionof the vortex worldlines. We shall illustrate how to proceed in Sec. 3.3, where we show thatquantum corrections provide only subleading corrections to our results (3.2) and (3.3), anddiscuss the low-lying spectrum of vortex excitations. We study in particular the dimension∆ min ( Q, J ) of the lightest operator at large charge Q and finite spin J – lightest states atfixed quantum numbers are more easily accessible via numerical methods .Before doing so, we briefly review the spectrum of superfluid phonons, which arises evenwhen κ = 0 and whose treatment is almost identical to the parity-preserving case [5]. For example, it should be straightforward to extend the Monte-Carlo simulations of Ref. [29] to obtain∆ min ( Q, J ) for the first few spins. Although their current results are consistent with any phase of matter atfinite charge and energy density, observing the (close to) linear dispersion ∆ min ( Q, J ) = ∆ min ( Q ) + q J ( J +1)2 would essentially confirm superfluidity. On a cubic lattice, spins J = 0 , , , , SO (3) has 5 irreps. In parity-violating CFTs with κ = 0, we find that ∆ min ( Q, J ) isinstead very different and not controlled by phonons, see e.g. Eq. (3.14). .2 Superfluid phonons The dynamics of the superfluid phonons can be described expanding Eq. (1.1) to quadraticorder in the gauge field fluctuations: S ’ α / √ πρ Z d x (cid:20) e − b (cid:21) + O ( b /ρ )+ κ π πρ Z d x (cid:15) ij e i ˙ e j + O ( ∂b /ρ ) + · · · . (3.4)If coefficients α, κ are order unity, both derivatives and interactions are suppressed by thesame scale Λ ∼ √ ρ . The theory could be quantized as such, with κ giving a contribution tothe Goldstone propagator. However we prefer to remove the κ term with a field redefinition;the quadratic action will then match that of parity-preserving superfluids so that thequantization of Refs. [5, 6] applies, at the cost of introducing new terms in operators suchas the currents j µ and T µν . The appropriate field redefinition is a µ → a µ + δa µ with δa = κ πα / √ πρ b , δa i = κ πα / √ πρ (cid:15) ij e j . (3.5)Note that for the purposes of doing perturbation theory, the normalization H a π ∈ Z neednot be preserved. The action is now given by only the α term in (3.4), i.e. it is identical tothat of parity-preserving conformal superfluids, which were quantized on the sphere in [5, 6].The canonically normalized Goldstone degree of freedom is mode-expanded on the sphere ofradius R as π c ( t, ˆ n ) = π + π t + X J> X | m |≤ J √ J (cid:16) a Jm Y Jm (ˆ n ) e − i Ω J t/R + h.c. (cid:17) , (3.6)with creation and annihilation operators satisfying [ a Jm , a † J m ] = δ JJ δ mm , and Ω J = q J ( J +1)2 . The Hilbert space of the EFT in the sector of charge Q = 4 πR ρ consists ofphonon Fock states | Q, { n Jm }i = Y J> Y | m |≤ J √ n Jm ! ( a † Jm ) n Jm | Q i , (3.7)with energy R P J ≥| m | n Jm Ω J above the ground state | Q i . States with n m > P µ ∝ a † m + h.c.. The spectrum receives corrections due tointeractions; however at large Q these are suppressed by 1 /Q . The fields b, e i appearing in(3.4) are related to the canonically normalized scalar as e i = √ πρ α ! / (cid:15) ij ∂ j π c + · · · , b = − √ πρ α ! / ˙ π c + · · · . (3.8)15 .3 Quantization and the spectrum of vortices We now turn to the vortices. At a quantum level, the spectrum of vortices consists of Landaulevels on the sphere, or monopole harmonics [25]. The lowest Landau level has spin J = 12 Z S f π = Q , (3.9)in agreement with the classical angular momentum obtained in Eq. (2.28). Higher Landaulevels are separated by a gap given by the cyclotron frequency ω c ∼ ρ/m vortex ∼ √ ρ/γ whichis at the cutoff. This implies that the EFT describes κ particles in the lowest Landau level.Quantization then imposes the following algebra for the vortex coordinates [ J pa , J p b ] = i(cid:15) abc J pc δ pp , ~J p = Q ~R p . (3.10)Because of the non-commutativity of the vortex coordinates, this system is sometimesreferred to as a “fuzzy sphere” [34]. In the following, we will use Eq. (3.10) to study thespectrum of vortex excitations at a quantum level.We start with the case | κ | = 1, which is special in several aspects. Since there is onlya single vortex, Eq. (2.27) only produces the fugacity contribution accounted for in (3.3).The spectrum simply consists of a single spin J = Q/ Q (cid:29) J = Q/
2. It is also interesting that this spin is half-integerfor half the values of Q – a CFT that enters a superfluid phase with | κ | = 1 is thereforenecessarily fermionic. Notice that whether the ground state is fermionic for even or oddvalues of Q depends on the internal spin of the vortex, and thus might depend on the specifictheory under consideration. More generally, CFTs with κ odd are necessarily fermionic.Let us now turn to the case | κ | >
1. The κ vortices each have spin J = Q/ κ spins with thefollowing all-to-all interaction: H = − √ πρ α X p
J (cid:28) Q shows that they have energy∆ min ( Q, J ) = ∆ min ( Q ) + √ πχ J ( J + 1) Q / + · · · . (3.14)These excitations are much softer than the superfluid phonons discussed in the previoussection, which have ∆ Q, J − ∆ min ( Q ) = p J ( J + 1) /
2. The lightest operator of charge Q andspin 0 < J (cid:46) Q is therefore an excitation of the vortices. The full spectrum consists of atensor product of the vortex spectrum and the phonon Fock states – part of this low-lyingspectrum is shown in Fig. 1. Notice also that we expect the two vortices to be identical,hence their wave-function should be properly symmetrized or antisymmetrized dependingon their internal spin. For instance, assuming bosonic vortices, this implies that only evenvalues of J are allowed in Eq. (3.14).We were not able to solve the Hamiltonian (3.11) exactly for larger values | κ | >
2. Exactdiagonalization of the Hamiltonian (3.11) for low values of κ and Q suggests that the groundstate minimizes its spin – i.e. it has J = 1 / κ · Q is odd and vanishes otherwise. Ingeneral, the commutation relations (3.10) imply that the non-commutativity of the vortexcoordinates is proportional to 1 /Q , which hence controls quantum corrections. This impliesthat the classical angular momentum (2.28) may receive at most O (1) corrections due toquantum effects. This may be seen more formally as follows: since the Hamiltonian (3.11)describes κ interacting spins with spin Q/ Q (cid:29) Z [ β ] = Tr e − βH = Tr (cid:20) e β P p
1) exp β X p
2, we expect low-lying vortex excitations to still besofter than phonons as we found for | κ | = 2; this expectation can in fact be confirmed in the κ → ∞ limit studied in Sec. 3.6. κ = 1 The EFT can also be used to study matrix elements of light local operators in betweenthe superfluid ground state and/or its excitations. In the following we shall analyze thepredictions for some of the OPE coefficients involving the U (1) current or the stress tensor,mostly focusing on the leading parity violating effects. When doing so, it will be convenientto distinguish between semiclassical correlators, whose leading value can be obtained bysimply considering the classical expectation values of specific operators, and quantum ones,which are instead controlled by the (small) quantum fluctuations of the fields. We study theformer here and the latter in Sec. 3.5.In this section we first demonstrate how to compute classical OPEs, focusing on matrixelements of the U (1) current in between the | κ | = 1 ground state with J = J z = Q/ θ = 0). For instance, from the expressionfor the U (1) current (2.4), we obtain the following matrix elements: h j i = − Q πR , h j φ i = − α √ πρ π R (1 + cos θ ) . (3.16)Here we used f µ ’ πρδ µ and Eq. (2.25).The predictive power of the EFT is easily illustrated comparing Eq. (3.16) with the most18eneral possible structure for the CFT matrix elements: h vor J | j ( x ) | vor J i = P Jm =0 a m sin m θ + cos θ P J − m =0 b m sin m θ , J = integer , P J − / m =0 a m sin m θ + cos θ P J − / m =0 b m sin m θ , J = half-integer , h vor J | j φ ( x ) | vor J i = P Jm =1 c m sin m θ + cos θ P Jm =1 d m sin m θ , J = integer , P J +1 / m =1 c m sin m θ + cos θ P J − / m =1 d m sin m θ , J = half-integer , (3.17)where | vor J i is the J = J z ground state. The coefficients a m and c m multiply parity-evenstructures, while the coefficients b m and d m parity odd ones. To compare this Eq. with theEFT result (3.16), we notice that not all the structures in Eq. (3.17) can be predicted, assome of them will be peaked at the vortex core or they might involve Fourier componentswith frequency larger than the EFT cutoff ∼ √ ρ . It is in turn convenient to use half-angleformulas to rewrite the matrix elements in Eq. (3.17) as: h vor J | j ( x ) | vor J i = ∼√ Q X m =0 α m cos m θ , h vor J | j φ ( x ) | vor J i = ∼√ Q X m =1 β m cos m θ , (3.18)The α m and β m in Eq. (3.18) are linear combinations of the coefficients in Eq. (3.17).Terms with m (cid:38) √ Q contain Fourier components with frequency larger than the cutoff andare exponentially suppressed away from the vortex core, therefore we neglected them inEq. (3.18). We may now compare with Eq. (3.16) to obtain: α m = − Q πR δ m , β m = − √ χ Q √ πR δ m , for m (cid:28) p Q . (3.19)We may similarly compute the matrix elements for the stress-energy tensor. UsingEq. (2.5) the EFT provides h T i ’ α (2 πρ ) / , h T φ i ’ πρ (1 + cos θ )4 πR , h T φφ i ’ α πρ ) / , (3.20)where we neglected subleading orders. Proceeding as before, we compare Eq. (3.20) with the Eq. (3.17) can be obtained imposing rotational invariance on the sphere; the conformal group additionallyrelates these matrix elements to different ones involving descendant states, but it does not constraint furtherthe expressions (3.17). h vor J | T ( x ) | vor J i = ∼√ Q X m =0 ¯ α m cos m θ , h vor J | T φ ( x ) | vor J i = ∼√ Q X m =1 ¯ β m cos m θ , h vor J | T φφ ( x ) | vor J i = ∼√ Q X m =0 ¯ γ m cos m θ . (3.21)As a result we obtain:¯ α m = 2¯ γ m = 2 Q / √ πχ R δ m , ¯ β m = Q πR δ m , for m (cid:28) p Q . (3.22)Finally we comment that a purely classical approach cannot be used to compute OPEcoefficients in the ground state for κ = 2. This is because it does not have macroscopic spin; infact it has J = 0, and there is no semiclassical approximation for the homogeneous quantumwave-function of the vortices. Instead, one must properly integrate over the zero-modes ofthe saddle-point configuration in the corresponding path-integral. A semiclassical approachdoes however allow to compute the OPE coefficients of the excited states (3.14) with J (cid:29) κ = 2. The procedure is analogous to the one described above and we do not report theresults here. Phonon excitations lead to transport properties – namely two-point functions of currents j µ , T µν in the finite density state – that are characteristic of superfluids. One such featurefor parity-violating superfluids is the Hall viscosity (2.14), proportional to κ . In a CFT,transport can be studied with 4-point functions involving two heavy operators, e.g. h QT T Q i .At a more basic level, transport signatures should be visible in off-diagonal heavy-heavy-lightOPE coefficients h QT Q i [11]. When the large charge sector is controlled by the superfluidEFT, we can make this manifest: the intermediate states in h QT T Q i that give the dominantcontribution to transport correspond to the ground state dressed with a single superfluidphonon – choosing Q = Q J ≡ a † Jm | Q i to be such a state (see Eq. (3.7)), one expects OPEcoefficients h QjQ J i and h QT Q J i to capture the salient features of transport in the superfluidstate. This is illustrated in Fig. 2. Similar OPE coefficients were studied in Refs. [6, 7]; wefollow the strategy presented there to compute heavy-heavy-light OPE coefficients from A similar procedure involving the zero mode of the Goldstone field ensures charge conservation incorrelation functions [6]. ∆ j µ , T µν Q ∼ Q d +1 d Figure 2: Spectrum in the large charge sector of a CFT. The triangle shows an example ofa heavy-heavy-light OPE coefficient that can be computed from the superfluid EFT.the EFT. We are focusing on privileged light operators – the U (1) current j µ and stresstensor T µν – which are universally present in the CFTs of interest. Moreover, their fixednormalization implies that matching these operators to the EFT only involves, to leadingorder, the charge susceptibility χ (or equivalently α (2.6)) which can already be read offfrom the spectrum (3.1).We therefore proceed to computing h QjQ J i and h QT Q J i , focusing in this section onphonon physics. At small wavevectors k ∼ /R on the sphere, phonons will be sensitiveto the presence of the κ vortices. To cleanly separate the phonon dynamics from that ofvortices, which were studied in the previous sections, we focus on wavevectors k (cid:29) /R ,i.e on operators Q J with spin 1 (cid:28) J (cid:46) p Q . (3.23)In this regime, we can ignore curvature corrections, e.g. to the stress tensor (2.13). Asdiscussed in Sec. 3.2, in order to utilize the phonon algebra of Refs. [5, 6, 7], we needed toperform a field redefinition (3.5). After this field redefinition the U (1) current is no longergiven by j µ = π (cid:15) µνλ ∂ ν a λ but rather, to linear order in b, e i , by j = ρ − π b , j i = − π (cid:15) ij (cid:18) e j + 2 / √ πρ κ πα ∂ j b (cid:19) . (3.24)The stress tensor also receives an additional contribution shown in Appendix A.4. It is thenstraightforward to compute matrix elements of the current between single phonon states21nd the vacuum, by expressing e i and b in terms of the canonical phonon field (3.8) andusing the algebra (3.6). For the time component of the current, one obtains: h Q | j ( t, ˆ n ) | Q, J m i = − π h Q | b a † Jm | Q i’ − i p Ω J ( χ ρ/ / R Y Jm (ˆ n ) e − i Ω J t/R . (3.25)where in the second step we used (3.8) and (3.6). For the spatial component, h Q | j i ( t, ˆ n ) | Q, J m i| odd ’ − π / √ πρ κ πα h Q | ∂ j b a † Jm | Q i’ − iκ p Ω J (2 χ /ρ ) / R ∂ j Y Jm (ˆ n ) e − i Ω J t/R , (3.26)where we focused on the parity-odd part, proportional to κ ; the parity-even part can becomputed similarly and is related to (3.25) by the Ward identity. Matrix elements for thestress tensor are obtained in Appendix A.4.These expressions can be recast into OPE coefficients after choosing a basis for CFTthree-point functions. On general grounds we expect the three-point functions to involveone parity-even and one parity-odd OPE coefficient, see Ref. [36]. In their notation, thethree-point function of O = Q † , O = j µ and O = Q J takes the form G ( P , P , P ; Z , Z , Z ) = h Q † j Q J i| even J X α =0 c α ( J , J ) V J − α H α V J − α P τ , / P τ , / P τ , / + h Q † j Q J i| odd J − X α =0 ˜ c α ( J , J ) (cid:15) V J − α − H α V J − α − P τ , / P τ , / P τ , / , (3.27)with J = 1 and J = J (recall that J > O = T µν is similar, with J = 2. The coefficients c α and ˜ c α areentirely fixed up to two overall constants c and ˜ c from the Ward identities. In order not tolose the reader to technical details, we will not specify the overall normalization conventions(which depend on the spin J ) – the OPEs listed below therefore have an overall J -dependentnormalization which we are not keeping track of (their relative J dependence is meaningfulhowever). We are now left with the dynamics, in the OPE coefficients h Q † j Q J i| even , h Q † j Q J i| odd , h Q † T Q J i| even , h Q † T Q J i| odd . (3.28)We only report the result to leading order in Q (cid:29) √ Q (cid:38) J (cid:29)
1, droppingnumerical factors and emphasizing instead the parametric dependence on the quantumnumbers
Q, J and CFT properties χ , κ . The OPE coefficients involving the current j µ can22e obtained from Eqs. (3.25) and (3.26): h Q † j Q J i| even ∼ ( χ Q ) / J / , h Q † j Q J i| odd ∼ κχ / Q − / J / . (3.29)For the stress tensor, one finds h Q † T Q J i| even ∼ Q / χ / J / , h Q † T Q J i| odd ∼ κχ / Q / J / . (3.30)To leading order in 1 /Q , both parity-even OPE coefficients only depend on the quantumnumbers of the operators ( Q and J ) and on the susceptibility χ – they are entirely fixedonce χ is measured through (3.1). The leading parity-odd OPE coefficients involve a newparameter: the quantized coefficient κ ∈ Z appearing in the EFT (2.12). Eqs. (3.29) and(3.30) exemplify how transport can be captured by off-diagonal heavy-heavy-light OPEcoefficients – for example the product of the two lines in (3.30) computes the Hall viscosity(2.14) of the superfluid. κ and vortex lattices So far, we implicitly considered κ as an O (1) parameter. It is natural to ask what happensin the limit κ (cid:29)
1. Though we expect such situation to be mostly relevant in weakly coupledtheories, such as the ones we discuss in sec. 4, here we provide some general comments fromthe EFT perspective.Let us focus first on the properties of the ground state. The limit κ → ∞ of the Whyte’sproblem received a considerable attention in the mathematical literature [37]. It is simple tounderstand the main feature of the solution in our setup. For κ (cid:29) ρ v ( x ) ≡ X p δ ( x i − X ip ) κ (cid:29) ≈ κ πR (3.31)where the last equality holds in the sense of distributions clearly. In this way their con-tributions cancels the charge sourced by the Euler term in the Gauss’s law (2.23), whichthus gives e i ≈
0, hence minimizing the electrostatic energy. It may be shown that thecontribution to the electrostatic energy from pairwise interactions in this limit reads [37]: − X p = r log( ~R p − ~R r ) = − κ (log 4 − − κ log κ + O ( κ ) . (3.32)23he leading term in this formula cancels the electrostatic energy due to the homogeneouscharge sourced by the Euler current in eq. (2.27), as expected. The logarithmic − κ log κ correction then recasts our result (3.3) for the ground state energy in the form:∆ min ( Q ) = 23 √ πχ Q / + κ √ πχ p Q log Qκ χ + O ( p Q ) . (3.33)For the EFT to hold, we need to require that the one derivative terms in the second lineof the action (2.20) be subleading with respect to the leading order action; this leads tothe condition κ (cid:28) α √ Q ∼ p Q/χ . In practice Eq. (3.33) might be of interest in weaklycoupled theories, where the Wilson coefficients also depend explicitly on κ . In that case,it might happen that the subleading contribution from the vortex masses and the higherderivative terms are numerically more important than the logarithmic fugacity of the vorticesin eq. (3.33) for non-exponentially large charge..The large κ limit provides us with a handle on the otherwise complicated spectrum ofexcited states. It was indeed observed numerically that the configuration minimizing thepotential energy takes a regular structure in this limit [37], analogous to the usual triangularlattice which is observed in superfluids [38, 39]. Therefore, at distances much larger thanthe typical vortex separation 1 / √ ρ v ∼ R/ √ κ , we may resort to a simplified description interms of the collective lattice coordinates of the vortices to study the CFT spectrum.Rather than constructing the generic vortex lattice action, we prefer to adapt the vortexlattice EFT of [40] to our problem; we do not expect significant differences in the mostgeneral case. We therefore consider the following quadratic Lagrangian: L / √ g ’ α / √ πρ (cid:18) e i − b (cid:19) + κ πR e i u i − κ (2 πρ )8 π u i ˙ u j (cid:15) ij − C p πρ ( ∇ i u i ) − C p πρ h ( ∇ i u k ) − R ij u i u j i , (3.34)where u i is a vector representing the displacement from equilibrium of the vortex latticecoordinates. The first term is just the phonon action, whose spectrum is unmodified toleading order. The terms proportional to κ arise from the Euler term and from the minimalcoupling of the vortices to the gauge field. The coefficients C i ’s parametrize the elasticityof the solid. We can estimate their value from the expression (2.27) for the electrostaticpotential; this gives C i ∼ κ/α . Finally notice the appearance of the curvature R ij in thelast term, whose role will become clear in a moment. This condition is generically stronger than the one which follows from the requirement that vortex coresare not overlapping: κ (cid:28) χ Q (recall the comments below Eq. (2.27) on the vortex size). Here we neglected an higher derivative term − κ π (2 πρ ) e i (cid:15) ij ˙ e j from the expansion of the Euler current aswell as a ∼ √ πρ ˙ u i ˙ u i contribution to the kinetic term of the lattice coordinate fluctuations. This arises when expanding solid invariants – see [41] for details about the solid EFT on the sphere. ω J = 2 C α (2 πρR ) (cid:20) J ( J + 1) R − R (cid:21) (cid:20) O (cid:18) κ (cid:19)(cid:21) , J ≥ . (3.35)The requirement J ≥ R = 2 /R implies that ω J vanishes for J = 1. This was to be expected,since the lattice coordinates can be thought as the Goldstone bosons of the rotations brokenby the lattice. This implies that, analogously to the descendants created by the J = 1 modeof the phonon, the rotation generators are proportional to the J = 1 mode of the latticefluctuations, which should hence be gapless. Finally, we notice that the sound speed of thismode is much smaller than the one of the phonons. Therefore, as we found previously for κ = 2, the lightest mode with non-zero angular momentum is provided by fluctuations ofthe vortices and its energy reads:∆ min ( Q, J ) = ∆ min ( Q ) + C r χ κQ q J ( J + 1) − , ≤ J (cid:28) κ , (3.36)where C is a O (1) Wilson coefficient. The prediction (3.36) holds as long as the angularmomentum does not exceed the scale set by the vortex density.A remark is in order. The reader familiar with spinning superfluids might find thespectrum we just discussed rather bizarre. In particular, it is known that the low energyspectrum of a vortex lattice usually consists of a single gapless mode, with quadraticdispersion relation: the Tkachenko mode [42]. However, that result applies in a regime offrequency which is inaccessible on the sphere for us, namely ω (cid:28) m Kohn where m Kohn setsthe gap of the so called Kohn mode [43]. In our notation m Kohn ∼ ρ v α √ ρ , hence m Kohn (cid:28) /R in our setup (see the discussion below Eq. (3.33)). The two modes we find may instead beconsidered as the extrapolation of the Tkachenko and the Kohn mode to higher frequencies m Kohn (cid:28) ω (cid:28) ρ v .Finally we notice that a similar EFT may also describe excitations of the large spin statesstudied in [15], with appropriate modifications since the vortex density is not homogeneousin that situation . We expect our results to apply to a number of 3d CFTs, both strongly and weakly interacting.Candidates without parity symmetry include QED with an odd number N f of fermions, We thank Angelo Esposito for discussions on this topic.
25r Chern-Simons–matter theories. Below we study a theory that has another perturbativehandle, where our predictions can be tested and extended to regimes beyond the EFTdiscussed in the previous section. A number of perturbative checks have been made forparity preserving theories, see e.g. [44, 45, 33].
We consider a weakly coupled parity-violating CFT: a single Dirac fermion coupled to U (1) − k + Chern-Simons term S = Z d x ¯ ψi (cid:26)(cid:26) Dψ − k π (cid:15) µνλ a µ ∂ ν a λ , (4.1)with (cid:26)(cid:26) D = γ µ ( ∂ µ − ia µ ). When the level k is large, 1 /k acts as a loop counting parameterand the theory is weakly coupled. For k = 0, this theory has time-reversal symmetry (whichacts non-trivially on the fermion field, mapping it to a monopole operator) and has beenconjectured to flow to the U (1) Wilson-Fisher CFT [46, 47, 48]. For generic values of k it ishowever not time-reversal invariant. Moreover, non-relativistic versions of this theory areknown to enter a superfluid at finite density, at least at large k [14]. We will in fact showthat in this limit the EFT (1.1) can be derived from the microscopic weakly coupled CFT(4.1) at finite density, confirming the validity of the framework in this context and providingthe EFT coefficients χ and κ . Anyons on the plane
The theory (4.1) has a global U (1) symmetry carried by the current j µ = 12 π (cid:15) µνλ ∂ ν a λ . (4.2)The equation of motion for a is a constraint equation that attaches flux to fermion density¯ ψγ ψ = kj . (4.3)We start by studying the theory on the plane R , at finite density h j i = ρ . Eq. (4.3)then implies that the theory consists of a finite density n ψ ≡ h ¯ ψγ ψ i = kρ of fermionsin a background magnetic field b = 2 πρ . When k → ∞ , the fermions decouple from thefluctuating gauge field a µ so that the finite density ground state consists of k filled Landau The half-integer level refers to the fact that the theory is regularized such that gapping out the fermionwith + m ¯ ψψ ( − m ¯ ψψ ) drives the system to a topological phase S = − k π R ada with level k = k − k = k ).Since we will work to leading order in k (cid:29)
1, we will neglect this subtlety in what follows. a µ . Specifically, the action will admit a natural derivative expansion for momentamuch smaller than p ρ/k , which sets the gap of the lightest particle-hole excitations of thefermionic field. The Hall response of k filled Landau levels is σ xy = k π – this implies thatthe leading term coming from integrating out the fermions precisely cancels the CS term in(4.1) , so that the effective action for a µ starts at two-derivatives and the photon is massless.This is the superfluid Goldstone boson. Since this is a CFT which enters a superfluid phaseat finite density, we expect it to be described by the EFT in Sec. 2, with κ = 0 since the CFTis parity-violating. To derive the EFT, we need to integrate out the Landau level fermions.This was done in Ref. [49], which obtained to leading order in fields and derivatives S eff = α k / √ πρ Z d x (cid:18) e − b (cid:19) + · · · (4.4)with α k = √ π k / + O ( k / ). This matches the leading term in the expansion of the EFT(3.4) (in particular the speed of sound c s = 1 / √
2, as expected for conformal superfluids),with susceptibility (2.6) given by χ = 19 π α k = 12 πk + O (1 /k ) . (4.5)The fact that χ is small at large k implies that the strong coupling scale of the EFT isparametrically larger than the scale set by the densityΛ sc ∼ √ ρχ / ∼ p ρk (cid:29) √ ρ . (4.6)However, as already commented, we do expect to see ‘new physics’ – e.g. in the form ofroton states [50] – at energies Λ ∼ p ρ/k . The large charge spectrum of the CFT willtherefore contain operators with dimension ∼ ∆ min ( Q ) + p Q/k corresponding to thesegapped excitations on top of the superfluid ground state.Extending the calculation of [49] to higher point-functions and higher derivatives, onecould determine all the coefficients of the EFT (1.1) in flat space. By Weyl invariance thiswould determine also the bulk action for the theory in R × S as an expansion in 1 /R Λ,where Λ ∼ p ρ/k is the EFT cutoff. In particular, we could find the parity-odd coefficient κ in this way. However this is tedious to do in practice; we will instead discuss a simpler andmore direct method to match its value below, where we discuss the theory on the sphere. The sign can be checked at the classical level from the Lorentz force: ∂ µ T µi = f iν j ψν = 0 ⇒ j ψi = k π (cid:15) ij f j leading to a term in the effective action + k π (cid:15) µνλ a µ ∂ ν a λ . nyons on the sphere Let us now consider the theory (4.1) on R × S . The previous discussion immediately allowsus to obtain the leading order dimension of monopole operators of large charge Q in thetheory (4.1). Indeed using the value (4.5) for the susceptibility and (3.1), we obtain:∆ min ( Q ) ’
23 ( kQ ) / + · · · (4.7)On general grounds, we expect that this equation will receive 1 /k relative corrections fromhigher loops in perturbation theory, as well as k/Q relative corrections from higher derivativeterms in the Lagrangian. We shall see that to leading order in the coupling Eq. (4.7) isactually exact. To show this, let us directly compute the dimension of monopole operators inthe theory (4.1) to leading order in coupling k , but all orders in Q/k , i.e. in the double-scalinglimit k → ∞ , Q → ∞ with Q/k fixed, in analogy with [33, 51]. As in those works, this isachieved by means of the state-operator correspondence. To leading order, the gauge fieldis set to a background value ¯ a µ , which provides the homogeneous magnetic field on thesphere. The Dirac field in a monopole background is then quantized using spinor monopoleharmonics (see e.g. [53] for a definition). The constraint (4.3) implies that the lowest energystate of charge Q consists of kQ particles organized in Landau levels. Summing their energiesgives: ∆ min ( Q ) = 23 k (cid:18) Qk (cid:19) / (cid:20) O (cid:18) k (cid:19)(cid:21) . (4.8)This equation coincides with (4.7); this result is exact in Q/k to leading order in thecoupling. Fluctuations of the gauge field couple to particle-hole excitations of this state,whose gap depends only on
Q/k up to O (1 /k ) corrections. Since this gap does not scalewith k , the weak coupling of the theory ensures that fluctuations are indeed suppressed.Of course, in principle one can compute all corrections systematically integrating out theDirac field on a finite charge state as in [49]. In practice, as we showed before, the gaugefield propagator receives large corrections in this limit. These make it non-local at shortdistances, and the calculation is technically challenging.We now turn to the effect of higher derivative corrections in the EFT. In particular, wewould like to determine the value of the parity-odd coefficient κ in perturbation theory. Wefound in Sec. 2.3 that κ controls the number of vortices (or charges, in the dual picture) in A similar double-scaling limit can be studied by a straightforward generalization of the ideas discussedin [33] in Chern-simons theories coupled to bosonic matter; similar ideas were explored in [52] in a SU (2)Chern-Simons theory coupled to a scalar field. This result agrees with [54, 53], where the authors worked in the limit of Q fixed. κ .To leading order in 1 /k , we can treat a µ as a background field. The constraint (4.3)implies that the state consists of N ψ = kQ Dirac fermions on the sphere in a monopolebackground with flux Q . The fermions will populate monopole harmonics with degeneracy Q + 2 | p | , with p ∈ Z . Filling up the levels p = 1 , , . . . , k therefore requires kQ + k ( k + 1)fermions. The missing fermions (or holes) in the k th Landau level are precisely the vorticesthat were expected in the EFT. Their number fixes the parity-odd coefficient: κ = k ( k + 1) . (4.9)This result can also be found in Ref. [23]; note that κ is even, as expected in CFTs with nogauge invariant fermionic operators. With a microscopic model in hand, we can investigatethe internal structure of the EFT vortices (cf. comments at the end of Sec. 2): since particlesin higher Landau levels have angular momentum J > Q/
2, we find that the vortices herewill carry non-zero internal spin s = J − Q/ ∼ k .The large number of vortices κ (cid:29) − ∆ min ( Q ) ∼ p J/kQ for J (cid:28) k .The fermionic nature of the vortices has a striking consequence for the theory at hand.Indeed we find that when κ > Q + 1 they will spill out to the next Landau level which isnot captured by the EFT, as discussed in Sec. 2.3. The EFT (2.20) therefore only correctlycaptures operators of large charge Q satisfyingEFT regime: Q (cid:38) κ ’ k . (4.10)This window is much smaller than the one in which the superfluid EFT applies in flat space,where the gap of the heavy states scales like p ρ/k . It is then natural to wonder if there isan alternative low energy description for k (cid:38) Q (cid:29) k . To answer this question we will makeuse of the weak coupling of this CFT.We work to leading order in k in the following for simplicity. Now from the perspective ofthe CFT, it is clear what happens when Q is taken below this threshold: the κ ’ k missing Landau levels for Dirac fermions have one more available state than regular monopole harmonics [25],which have spin Q/ p and degeneracy Q + 2 p + 1, see e.g. [23, 53]. k th Landau level, and start depleting the ( k − ’ Q , the number of filled Landau levels isnow no longer k , but rather k − $ k Q % . (4.11)Integrating out the gapped Landau levels now no longer cancels the Chern-Simons term: S EFT = − π $ k Q % Z ada + S EFT , (4.12)where S EFT is still given by (2.20) with coefficients α, κ unchanged to this level of precision(we are assuming that number of depleted Landau levels is small j k Q k (cid:28) k ). However, thenumber of vortices in S EFT is now no longer given by the total number of holes κ ’ k , butrather by the number of holes in the partially filled Landau level N vortices = k − $ k Q % Q . (4.13)How is this consistent with Eq. (2.19) for the nonconservation of the higher-form current,which ties the number of vortices to κ ? The resolution is that the Chern-Simons term alsobreaks the higher-form symmetry, and therefore gives an additional contribution to thedivergence of the current ∇ µ J µν = − $ k Q % j ν + κJ ν Euler − j ν vortices (4.14)Integrating the ν = 0 component over the spatial sphere gives (4.13). We therefore find thatthe upgraded EFT (4.12) with Chern-Simons term extends the regime of validity of the EFTto EFT regime: k (cid:38) Q (cid:29) k . (4.15)The physics in this regime is very similar. For instance, the spectrum of phonons is unaffectedto leading order by the Chern-Simons term. In flat space, this would give them a gap m phonon ∼ q kρ ; however, this is much smaller than 1 /R and it is hence invisible on thesphere. The main difference is the reduced number of vortices in the ground state (4.13).This affects the spectrum of their excitations as well as the coefficient in front of the fugacityterm in the scaling dimension of the monopole operator (3.3).Finally, for Q/k (cid:46) /R , hence the EFT breaks down. As explainedaround Eq. (4.8), it may still be possible to exploit the weak coupling of the theory toexplore this regime. We leave this task for future work.30 .2 Further applications and prospects Large N c Chern-Simons matter theories
Chern-Simons theories with fundamental fermion matter at infinite N c and finite chemicalpotential have the thermodynamic properties of a (renormalized) Fermi liquid [55, 56] ,even though there are no gauge-invariant fermionic operators for N c even. However, these areunlikely to be the true ground states at any finite but large N c . Indeed, 1 /N c corrections inthe presence of a Fermi surface may be more relevant than the terms leading in N c , changingthe physics at low energies (similar effects even spoil large N f expansions in the presenceof a Fermi surface [59]). Consider for example the gluon propagator, which in axial gaugesreceives no loop correction at leading order in 1 /N c and takes the form G ( p ) ∼ p [60]. Atorder 1 /N c , it receives a self-energy correction from the Fermi surface (Landau damping) G ( p ) ∼ p + Π( p ) , with Π( p ) ∼ µN c p p . (4.16)We expect gauge invariant observables to receive similar corrections, leading to a breakdownof the large N solution at energies p (cid:46) µ/N c . It may be possible to reorganize the perturbativeexpansion (see for example [61]) and establish whether the true ground state is a superfluidor a non-Fermi liquid; we leave this for future work. IR instabilities due to quantum effectsmay also plague near extremal black holes, and lead to a qualitatively different ground state. Non-relativistic CFTs
It was suggested in a non-relativistic context in Ref. [62] that superfluid EFTs may describethe large charge sector in scale invariant theories of anyons. We believe the superfluid EFTcan indeed be derived in these situations as well following arguments similar to those outlinedin this section, see in particuliar Ref. [50]. A term similar to κ also appears in non-relativisticsuperfluids, see e.g. Refs. [63, 64]. Acknowledgements
We thank Clay Córdova, Angelo Esposito, Ilya Esterlis, Andrey Gromov, Zohar Komargodski,Emil Martinec, Márk Mezei, Shiraz Minwalla, Dung X. Nguyen and Dam T. Son for inspiringdiscussions (alas mostly virtual). We also thank Domenico Orlando for comments on anearlier version of the manuscript. UM is partially supported by NSF grants No. PHY1720480 In [57, 58] it was also suggested that Fermi liquid states in N = 4 SYM at infinite N c might be dual tolarge charge extremal black holes in AdS . A General parity-violating superfluids
We review in this section the effective field theory for superfluids in 3 spacetime dimensionswithout conformal symmetry, following Ref. [13]. Similar EFTs have been studied in thenon-relativistic context of chiral superfluids, see e.g. [63, 64]. As in the main text, we workin the dual formalism where the action is a function of an abelian gauge field a µ . The stresstensor is then computed, and compared to the one obtained in Ref. [19] from a hydrodynamicapproach. A.1 General EFT without conformal symmetry
The most general action for parity violating relativistic superfluids in 3 spacetime dimensionswas constructed in [13], which we reproduce here S = S α + S ζ + S κ + · · · = σ Z d x √ σg α ( | f | )+ Z d x √ σg ζ ( | f | ) (cid:15) µνλ u µ ∂ ν u λ + κ π Z d x √ σg a µ (cid:15) µνρ (cid:15) αβγ u α (cid:18) ∇ ν u β ∇ ρ u γ + σ R νρβγ (cid:19) + · · · , (A.1)where σ is the signature of the metric, and we have defined f µ = (cid:15) µνλ ∂ ν a λ , | f | = p σf µ f µ and u µ = f µ / | f | is a unit vector. As in the main text, R νρβγ is the Riemann tensor of thespacetime manifold that the superfluid lives on. Without conformal symmetry α ( | f | ) and ζ ( | f | ) are generic functions of | f | ; in the conformal case in equation (2.7) they were given by α ( | f | ) = α · | f | / , ζ ( | f | ) = ζ · | f | , (A.2)where α and ζ are now constants. The last term term in (A.1) is automatically scaleinvariant, and has no generalization to the non-conformal case since it is constructed fromthe identically conserved ‘Euler’ current J µE = 18 π (cid:15) µνρ (cid:15) αβγ u α (cid:18) ∇ ν u β ∇ ρ u γ + σ R νρβγ (cid:19) . (A.3)32n Euclidean manifolds with a spatial factor of S , the total charge R d x J measures thewinding number of the map u µ : S → S , and is hence an integer. Invariance under largegauge transformations imposes κ ∈ Z as in the conformal case.The ζ term can be eliminated in the conformal case by the field redefinition (2.10), so itis natural to ask if the same can be done in the general case. The equation of motion fromthe leading term in the action is (cid:15) µνλ ∂ µ (cid:18) α ( | f | ) f ν | f | (cid:19) = 0 . (A.4)So a field redefinition of the type a µ → a µ + δa µ , δa µ = c α ( | f | ) f µ | f | , (A.5)would leave local gauge invariant observables unchanged on-shell, since δf µ = (cid:15) µνλ ∂ ν δa λ = c(cid:15) µνλ ∂ µ (cid:18) α ( | f | ) f ν | f | (cid:19) = 0 . (A.6)The change in the leading term in the action under this field redefinition is δS α = − c Z d x √− g ( α ) (cid:15) µνλ u µ ∂ ν u λ . (A.7)Hence, if ( α ) ( | f | ) ∝ ζ ( | f | ) as a function, this field redefinition can be used to eliminate the ζ term altogether by setting c ≡ ζ ( α ) (A.8)In the conformal case α ( | f | ) = α | f | / and ζ ( | f | ) = ζ | f | , so this is indeed true and we find c = 2 α/ ζ as in (2.10). However, ζ/ ( α ) isn’t a constant in the general case and the ζ termcannot be eliminated, even perturbatively. A.2 Gravitational stress tensor
We compute the gravitational stress tensor for the superfluid EFT in the general, non-conformal case using the action (A.1). For simplicity, we restrict ourselves to Euclideansignature. The gravitational variation of the leading order term is straightforward and resultsin the expression δS α = 12 Z √ g (cid:2)(cid:0) α ( | f | ) − | f | α ( | f | ) (cid:1) g µν + | f | α ( | f | ) u µ u ν (cid:3) δg µν = − Z √ g T µν δg µν (A.9)from which we can read off its contribution to the gravitational stress-tensor to find T µνα = | f | α ( | f | ) u µ u ν + (cid:0) α ( | f | ) − | f | α ( | f | ) (cid:1) g µν (A.10)33n the conformal case α ( | f | ) = α | f | / this expression simplifies to T µνα | conformal = 3 α | f | / (cid:18) u µ u ν − g µν (cid:19) . (A.11)Moving on to the general ζ term we can use the variational identities δ | f | = 12 | f | ( u µ u ν − g µν ) δg µν δu α = u µ ( δ να − u ν u α ) δg µν (A.12)to simplify the variation down to δS ζ = − Z √ g " | f | ζ ( | f | ) g µν udu + (2 ζ ( | f | ) − | f | ζ ( | f | )) u µ u ν udu − ζ ( | f | ) u µ (cid:15) ναβ ∇ α u β + 2 ζ ( | f | ) u µ (cid:15) ναβ u α ∇ β | f | δg µν (A.13)where udu is short-hand for (cid:15) µνλ u µ ∂ ν u λ . The contribution of this term to the stress tensoris then given by T µνζ = | f | ζ ( | f | ) g µν udu + (cid:2) ζ ( | f | ) − | f | ζ ( | f | ) (cid:3) u µ u ν udu − ζ ( | f | ) u ( µ (cid:15) ν ) αβ ∇ α u β + 2 ζ ( | f | ) u ( µ (cid:15) ν ) αβ u α ∇ β | f | (A.14)In the conformal case, this simplifies to T µνζ | conformal = ζ | f | g µν udu + ζ | f | u µ u ν udu − ζ | f | u ( µ (cid:15) ν ) αβ ∇ α u β + 2 ζu ( µ (cid:15) ν ) αβ u α ∇ β | f | (A.15)Moving on to the Euler term, using the fact that δu α = − ( u α / u µ u ν δg µν and δ R βσνρ = ∇ ν δ Γ βρσ − ∇ ρ δ Γ βνσ we arrive at the following expression for the gravitational variation of theaction 8 πκ δS κ = 12 Z √ ga µ (cid:15) µνρ (cid:15) αβγ u α (cid:18) ∇ ν u β ∇ ρ u γ + 12 R βγνρ (cid:19) g αβ δg αβ − Z √ ga µ (cid:15) µνρ (cid:15) αβγ u α (cid:18) ∇ ν u β ∇ ρ u γ + 12 R βγνρ (cid:19) u σ u λ δg σλ − Z √ ga µ (cid:15) µνρ (cid:15) αβγ u α R βλνρ g γσ δg σλ + 2 Z √ ga µ (cid:15) µνρ (cid:15) αβγ u α u σ ∇ ρ u γ δ Γ βνσ + Z √ g | f | (cid:15) µνρ (cid:15) αβγ u α g γσ u ν δ Γ βνσ + Z √ ga µ (cid:15) µνρ (cid:15) αβγ g γσ ∇ ρ u α δ Γ βνσ (A.16)Next, we can simplify the last three lines further using the variational identity δ Γ βνσ = − g βλ ( ∇ ν δg βσ + ∇ σ δg βν − ∇ β δg νσ ) (A.17)34implifying further, we find that the variation of the action organizes itself into a sum oftwo terms δS κ = δS κ,f + δS κ,a (A.18)where δS f contains terms depending directly on f µ that are local in the dual scalar language,while δS a contains terms that explicitly depend on the gauge field and lead to non-local,gauge dependent contributions to the stress tensor. We find, for the explicitly gauge invariantpart δS κ,f = κ π Z √ g | f | u µ (cid:15) ναβ u α ( u · ∇ ) u β − u µ (cid:15) ναβ u α ∇ β | f | + | f | u µ (cid:15) ναβ ∇ α u β − | f | u α (cid:15) αβµ ∇ β u ν ! δg µν (A.19)The gauge dependent part, on the other hand, is given by the following sum8 πκ δS κ,a = 12 Z √ g a µ (cid:15) µνρ (cid:15) αβγ u α ∇ ν u β ∇ ρ u γ (cid:16) g σλ − u σ u λ (cid:17) δg σλ + 12 Z √ ga µ (cid:15) µνρ (cid:15) αβγ u α R νρβγ (cid:16) g σλ − u σ u λ (cid:17) δg σλ − Z √ ga µ (cid:15) µνρ (cid:15) αβσ u α R λνρβ δg σλ − Z √ g(cid:15) µνρ (cid:15) αβγ a µ ∇ ν [( P k ) σγ ∇ ρ u α ] δg βσ + Z √ g(cid:15) µνρ (cid:15) αβγ a µ ∇ β [( P k ) σγ ∇ ρ u α ] δg νσ + Z √ g(cid:15) µνρ (cid:15) αβγ a µ ∇ σ [( P ⊥ ) σγ ∇ ρ u α ] δg βν + Z √ g(cid:15) µνρ (cid:15) αβγ ∇ σ a µ ( P ⊥ ) σγ ∇ ρ u α δg βν + Z √ g(cid:15) µνρ (cid:15) αβγ ∇ β a µ ( P k ) σγ ∇ ρ u α δg σν (A.20)where P µν k = u µ u ν and P µν ⊥ = g µν − u µ u ν . To simplify these further, we use the followingstrategy: each factor with a lower α, β or γ index is multiplied by a Kronecker delta expandedin projectors parallel and transverse to u . Since the product of these factors is multiplied by (cid:15) αβγ , the only terms that survive upon expanding are those that have exactly one factorof P k and two factors of P ⊥ , by virtue of the properties of scalar triple products. We alsouse the fact that ( P k ) λα ∇ δ u λ = 0. This trick can be used, for instance, to show that the lasttwo lines in the expression above cancel with each other. Finally, we arrive at the following35xpression8 πκ δS κ,a = 12 Z √ ga µ (cid:15) µνρ (cid:15) αβγ u α R νρβγ (cid:16) g σλ − u σ u λ (cid:17) δg σλ − Z √ ga µ (cid:15) µνρ (cid:15) αβσ u α R νρβδ (cid:16) g δλ − u δ u λ (cid:17) δg σλ + 12 Z √ ga µ (cid:15) µνρ (cid:15) αβγ u α ∇ ν u β ∇ ρ u γ (cid:16) g σλ − u σ u λ (cid:17) δg σλ + Z √ ga µ (cid:15) µνρ (cid:15) αβγ u α ∇ β u σ ∇ ρ u σ δg γν − Z √ ga µ (cid:15) µνρ (cid:15) αβγ ∇ ρ u α h δ λβ ∇ ν ( P k ) σγ + δ σβ δ λν ∇ δ ( P k ) δγ − δ λν ∇ β ( P k ) σγ i δg σλ (A.21)The terms in the integrand that multiply δg in the first two lines are linear in Goldstonefluctuations while those in the last three lines are at least quadratic. Only terms linear inthe Goldstone are needed to study the transport properties of the superfluid at leadingorder in small frequencies and momenta in Secs. 3.5 and A.4; we will therefore drop the lastthree lines. The first two lines give contributions to transport on curved manifolds, such asthe sphere at small momenta k ∼ /R , and consequently will affect the OPE coefficients inSec. 3.5 at small spin J ∼
1. However, phonons with wavelengths of the order of the sphereradius will be sensitive to the | κ | vortices distributed on the sphere – correlation functionswith k ∼ /R therefore require a careful treatment of the vortex-superfluid system (2.17).We expect that such a treatment will also resolve the apparent gauge non-invariance of thestress tensor on the sphere. In the remainder of this section, we focus on the leading stresstensor on the plane, where all non-gauge invariant terms (A.21) can be ignored.From (A.19) we find the gauge invariant part of the stress tensor T µνκ,f = κ π " − | f | u h µ (cid:15) ν i αβ u α ( u · ∇ ) u β + u h µ (cid:15) ν i αβ u α ∇ β | f |− | f | u h µ (cid:15) ν i αβ ∇ α u β + | f | u α (cid:15) αβ h µ ∇ β u ν i (A.22)where the angular brackets stand for symmetrization and trace subtraction. Putting thistogether with (A.11) and (A.15), the gauge invariant part of the stress tensor is given by T µνf = 3 α | f | / u h µ u ν i + ζ ( t µν − t µν + 2 t µν ) κ π ( − t µν + t µν − t µν + t µν ) (A.23)in a basis of gauge invariant symmetric, traceless tensors constructed from a single factor of36 , one derivative and u µ ’s t µν = | f | u h µ u ν i (cid:15) αβγ u α ∇ β u γ t µν = | f | u h µ (cid:15) ν i αβ u α ( u · ∇ ) u β t µν = | f | u α (cid:15) αβ h µ ∇ ν i u β t µν = | f | u h µ (cid:15) ν i αβ ∇ α u β t µν = u h µ (cid:15) ν i αβ u α ∇ β | f | t µν = | f | u α (cid:15) αβ h µ ∇ β u ν i (A.24)These tensors are not linearly independent and we find the following relations between them t µν = t µν + t µν , t µν = t µν + t µν + t µν (A.25)so that the gauge invariant part of the stress tensor becomes T µνf = 3 α | f | / u h µ u ν i + ζ ( − t µν − t µν + 2 t µν ) κ π ( − t µν − t µν + t µν + t µν ) (A.26)Finally, to leading order in the derivative expansion in the EFT these terms can be simplifiedfurther on-shell using the leading equations of motion, whose projections parallel andperpendicular to u µ are given, respectively, by udu = 0 | f | ( u · ∇ ) u α = 12 ( P ⊥ ) βα ∇ β | f | (A.27)These result in the on-shell relations t µν = 0 , t µν = 12 t µν (A.28)We then find that the ζ term vanishes and the κ term simplifies to give T µνf = 3 α | f | / u h µ u ν i + κ π | f | u α (cid:15) αβ h µ ∇ β u ν i . (A.29) A.3 Comparison with parity-violating hydrodynamics
Ref. [19] obtained the most general constitutive relations for a U (1) current and stress tensorof a parity-violating fluid in 2 + 1 dimensions, up to first order in gradients. These arelocal expressions for the currents, in a derivative expansion, in terms of the fluid degreesof freedom: fluctuations in temperature T , chemical potential µ and a velocity vector u µ u = −
1. These expressions should apply to our superfluid as a special case,with the following restrictions: T is not a degree of freedom in a zero-temperature QFT,dissipative terms such as the bulk and shear viscosities are set to zero, and finally in thesuperfluid the vorticity vanishes (cid:15) µνλ u µ ∂ ν u λ = 0 (we ignore vortices in this section). Theconstitutive relations of Ref. [19] in Landau frame are then given by T µν = ( (cid:15) + P ) u µ u ν + P g µν − ˜ η ˜ σ µν + · · · , (A.30a) j µ = ρu µ − ˜ σ(cid:15) µνλ u ν ∂ λ µ + · · · , (A.30b)with ˜ σ µν = u µ (cid:15) ναβ u α ( u · ∇ ) u β + u α (cid:15) αβµ ∇ β u ν + ( µ ↔ ν ) . (A.31)We used (cid:15) µνλ u µ ∂ ν u λ = 0 to simplify terms. The theory contains two parity-odd 1-derivativecoefficients, the Hall conductivity ˜ σ and Hall viscosity ˜ η . In studying the EFT (1.1), wedefined the unit vector to point in the direction of the current, so that j µ ≡ ρu µ . From theperspective of hydrodynamic constitutive relations, this amounts to working in Eckart frame,which can be reached with the redefinition u µ → u µ + δu µ with δu µ = ˜ σρ (cid:15) µνλ u ν ∂ λ µ . (A.32)After using the thermodynamic identities (cid:15) + P = µn , d(cid:15) = µdn and the continuity relationto leading order ∂ µ T µν , the 1-derivative stress tensor in Eckart frame can be expressed T µν = ( (cid:15) + P ) εu µ u ν + P g µν + 2˜ ηu α (cid:15) αβ ( µ ∇ β u ν ) + (cid:16) ˜ η − µ ˜ σ (cid:17) u ( µ (cid:15) ν ) αβ u α ( u · ∇ ) u β , (A.33)where we denoted symmetrization by A ( µν ) ≡ ( A µν + A νµ ). This agrees with the stresstensor obtained from the EFT (2.13), with˜ η = κn , ˜ σ = κn µ . (A.34) A.4 Linearized stress tensor