Spontaneously Broken Boosts in CFTs
SSpontaneously Broken Boosts in CFTs
Zohar Komargodski, a M´ark Mezei, a Sridip Pal, b Avia Raviv-Moshe a a Simons Center for Geometry and Physics, SUNY, Stony Brook, NY 11794, USA b School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
Conformal Field Theories (CFTs) have rich dynamics in heavy states. We de-scribe the constraints due to spontaneously broken boost and dilatation symmetries in suchstates. The spontaneously broken boost symmetries require the existence of new low-lying primaries whose scaling dimension gap, we argue, scales as O (1). We demonstrate these ideasin various states, including fluid, superfluid, mean field theory, and Fermi surface states. Weend with some remarks about the large charge limit in 2d and discuss a theory of a singlecompact boson with an arbitrary conformal anomaly. a r X i v : . [ h e p - t h ] F e b ontents I Bounding the Gap in the Operator Spectrum 6
II Examples 19
2d CFTs at Large Charge 40
A The Zoo of Correlation Functions 51B ∆ Q in the Free Fermionic Phase 53C Contact Terms in Energy-Momentum Correlators 55 Nambu-Goldstone theorems are some of our strongest non-perturbative constraints on thedynamics of Quantum Field Theories (QFTs). Let us review the setup for continuous internal(non space-time) symmetries in a QFT in d space-time dimensions. By Noether’s theorem,there exists a charge Q which is an extended operator depending topologically on a co-dimension 1 surface Σ. If in some state | Ω (cid:105) we have for some local operator O (0) (cid:104) Ω | [ Q, O (0)] | Ω (cid:105) (cid:54) = 0 , then it can be shown by a deformation of Σ that, roughly speaking, (cid:104) Ω | j ( x ) O (0) | Ω (cid:105) cannotdecay faster than 1 / | x | d − as we take | x | → ∞ . This algebraic decay of a correlation functionimplies gapless excitations of | Ω (cid:105) in infinite volume. Furthermore, under various additionalassumptions about the nature of the state | Ω (cid:105) , the existence of an ordinary massless boson(or a superfluid mode) can be established. Incidentally, if (cid:104) Ω | j ( x ) O (0) | Ω (cid:105) (cid:54) = 0 holds true,this means that also (cid:104) Ω | O † ( x ) O (0) | Ω (cid:105) must decay algebraically at most. And under someassumptions it cannot decay faster than 1 / | x | d − . This is unacceptable in d = 2 since itleads to a violation of clustering, due to the connected correlator not decaying. Hence nosuch states | Ω (cid:105) can exist in d = 2, which is the familiar statement of the Coleman-Mermin-Wagner theorem [1, 2].In finite volume (in the absence of boundaries) no state | Ω (cid:105) can have the property that (cid:104) Ω | [ Q, O (0)] | Ω (cid:105) (cid:54) = 0. This is simply because we can diagonalize Q in finite volume. Thus, thephenomenon of symmetry breaking is really due to the infinite volume limit. When symmetrybreaking occurs, the Hilbert space of the finite volume theory becomes closer and closer to a The argument for this invokes inserting a complete set of states and recalling that the matrix elementsof j µ are suppressed by a factor of momentum at small momentum. It would be nice to understand to whatextent this argument about the matrix elements is general. – 1 –irect sum of sub-Hilbert spaces which do not communicate via the action of local operators.When we take the infinite volume limit we only keep one of these sub-Hilbert spaces. Butthese sub-Hilbert spaces can still communicate by the action of extended operators such as Q . This is why symmetry breaking may occur in infinite volume and this is why the notionof superselection sectors exists. These comments will be important below.The situation for space-time symmetries is, in principle, similar. There are however someinteresting differences. Given the energy-momentum (EM) tensor of the theory, T µν , we canconstruct the space-time symmetries of the infinite volume theory living in R d − , from theconserved currents J ( ξ ) µ = ξ ν T µν , where ξ is a Killing vector satisfying as usual ∂ ( ν ξ µ ) = 0. The currents J ( ξ ) µ of course leadto the usual translations, rotations, and boosts. We denote these charges by Q ( ξ ) . There canexist a state | Ω (cid:105) and a local operators O (0) (which may or may not have spin indices, whichwe suppress for now) such that (cid:104) Ω | [ Q ( ξ ) , O (0)]Ω (cid:105) (cid:54) = 0 . (1.1)For constant ξ , namely the translation symmetry, this can be easily achieved in anystate which is not translationally invariant. Similarly, for rotations the commutator wouldbe generally nonzero in a non-isotropic state for any operator O with spin indices. A generaldiscussion of various allowed symmetry breaking patterns can be found in [3]. A rathercommon situation is the spontaneous breaking of boost symmetry in states | Ω (cid:105) which arehomogeneous and isotropic. This is one of the focal points of our paper. From dimensional analysis, a nonzero commutator for the boost Killing vector (1.1)(which we can take to be ξ = x , ξ = x , with the rest of the components vanishing) meansthat correlators of some components of the EM tensor and O decay not faster than 1 / | x | d .This does not lead to any particular problems in d = 2 and hence there is no obstruction forthe spontaneous breaking of boost symmetry in two space-time dimensions. We will indeedsee some examples later.The boost symmetry differs conceptually from ordinary (non space-time) symmetries inthat it cannot be preserved at finite volume. Indeed, if we compactify space while keeping timeintact, the symmetry between space and time is manifestly destroyed. Therefore, the spon-taneous breaking of boost symmetry does not necessarily mean that super-selection sectorsmust arise! Thus, the boost symmetry Nambu-Goldstone theorem does lead to an algebraicdecay and hence a gapless spectrum of excitations of | Ω (cid:105) , but it does not imply super-selectionsectors since there is no sense in diagonalizing the boost symmetry in compact space.As we have argued above, the boost symmetry Nambu-Goldstone theorem leads to asomewhat faster decay of correlation functions compared to the case of spontaneous breaking There are important consequences of the spontaneous breaking of boost symmetry also in states whichbreak the spatial translation symmetry. See for example [4, 5]. Here our focus is on translationally invariant,isotropic states. – 2 –f ordinary global symmetries. Therefore, one-particle excitations are not necessary, butrather, composite massless excitations could play a role. This was realized in the beautifulrecent paper [6].While there is no boost symmetry in compact space, it is still true that if we focusour attention on a small enough patch of our compact space, there is an approximate boostsymmetry. Depending on the state of the system, this approximate boost symmetry mayappear to be spontaneously broken. The algebraic decay and gapless excitations in the flatspace limit therefore entail some constraints on the spectrum of the finite volume theory. Thenature of these constraints can be understood on dimensional grounds. Take space to be ahypercube V = L d − where L is the length and V is the volume. Then to see a state with finiteenergy density (cid:15) in a small patch of the hypercube we have to start from a state with totalenergy E = (cid:15)V . We assume that on distances (cid:15) − /d (cid:28) ∆ x (cid:28) L we see an algebraic decay.Indeed, (cid:15) − /d is a length scale in the infinite volume theory and since we see an algebraicdecay in infinite volume it must be true that there is an algebraic decay at finite volume inthe range (cid:15) − /d (cid:28) ∆ x (cid:28) L . An algebraic decay in the range (cid:15) − /d (cid:28) ∆ x (cid:28) L is not to betaken for granted. It means that the gap above our state, E gap , must be much smaller than (cid:15) /d . So the energy of the excitations of | Ω (cid:105) should scale as (cid:15)V + E gap with E gap (cid:28) (cid:15) /d .Clearly, for the consistency of an algebraic decay, the gap has to be arbitrarily smaller than (cid:15) /d . This means that the gap has to go to zero as L → ∞ , in agreement with the gaplessnature of the excitations in infinite volume. We can roughly speaking say that therefore E gap ∼ L − γ , with γ >
0. We can think of γ as a certain critical exponent that measures how fast the gapcloses in finite space as we increase the volume.Of course, the most natural choice is γ = 1 but this does not follow from our generalconsiderations. That γ = 1 is natural can be motivated based on Wilsonian considerations.Let us take space to be infinite. The deep infrared theory should be a fixed point of therenormalization group (perhaps in the Lifshitz sense) and hence there is a scale-free effec-tive field theory description of the gapless degrees of freedom that give rise to the algebraiccorrelators at infinite volume. In that case, γ = 1 follows from dimensional analysis. Theabove argument assumes that the long correlators are captured by some scale-free infraredtheory where the energy density is a cutoff scale. In practice, it could be that the gap iseven smaller than what is predicted by dimensional analysis if the low-energy modes haveadditional degeneracy (which in finite volume is only approximate). Indeed, consider statesthat are generic in the Eigenstate Thermalization Hypothesis (ETH) sense (see [7] for a recentreview and references). Those states have a macroscopic entropy and hence an exponentiallysmall gap. Our bound on the gap holds in all theories and appropriate states, regardless ofwhether the states are generic. In particular it holds for states such as ground states at fixedcharge, Fermi surfaces etc.Furthermore, one can already at this stage say a few words about the density of these low-– 3 –ying states. For instance, if there was just one such state with energy above | Ω (cid:105) scaling like L − γ then the correlator in the range (cid:15) − /d (cid:28) ∆ x (cid:28) L would have behaved non-algebraically.The same argument holds for any finite collection of states. We therefore need infinitely manystates that become gapless as L → ∞ .The breaking of the boost symmetry is not a rare phenomenon. As we will see, in aunitary theory, any state that in the infinite volume limit has a non-vanishing energy densitywill break the boost symmetry. Therefore, the above constraints on the spectrum hold forquite generic states in finite volume.Our purpose here is to apply these ideas to Conformal Field Theories (CFTs). CFTs canbe studied on the cylinder S d − × R , where the radius of the sphere is R . The energy spectrum is related to the spectrum of scalingdimensions E = ∆ R . (1.2)We consider states with a nontrivial macroscopic limit, i.e. states with nonzero energy density.This means that we must take ∆ Ω = s d − (cid:15) R d with (cid:15) the energy density and s d − the volumeof the unit S d − . So as we take the macroscopic limit, we are discussing states that correspondto operators with a large scaling dimension. From our general considerations above we foundthat there are infinitely many states with energy going to zero as R − γ . This means that thereare infinitely many states with energy below E = s d − (cid:15) R d − + c (cid:15) (1 − γ ) /d R − γ , (1.3)where c is some dimensionless constant and the power of (cid:15) in the second term is adjusted sothat the result makes sense dimensionally. This translates to having operators with scalingdimensions ∆ = ∆ Ω + c ∆ − γd Ω . (1.4)This means that the gap around heavy operators with scaling dimension ∆ Ω is at most ∆ − γd Ω with γ >
0. As we have explained above, with some additional physical input it follows that γ = 1 (or larger) and hence the gap around heavy operators scales like O (1). This bound onthe scaling dimension gap around heavy operators should hold very generally and it does notrequire the genericity or typicality that is assumed in ETH. (Several of the applications wewill study here are in fact concerned with large charge ground states, which are very atypicalstates.)In CFTs on the cylinder (1.2) the gap constraint (1.4) may appear trivial given that thereare descendant states. (For a review of CFTs on the cylinder see for instance [8].) Indeed,for any primary state | Ω (cid:105) there is a family of states | ∂ N Ω (cid:105) (where the index contractionsin the derivatives are not explicitly displayed) with scaling dimension ∆ Ω + N and henceenergy E = ∆ Ω /R + N/R for any non-negative integer N . There is therefore an interesting– 4 –wist in the story: we can ask if the descendant states are those responsible for the low-energy theorems. The question of whether the descendant states are those responsible for thelow-energy theorems is the essential new question addressed in this paper.We will see that, perhaps surprisingly, the answer is negative. One must have new primary states with dimension (1.4). Furthermore, in all examples we study, γ = 1 forthese new primary states, in agreement with the general arguments. The boost symmetryNambu-Goldstone theorem is not sufficient to completely determine the low lying spectrum ofprimary excitations of | Ω (cid:105) and indeed the structure of excitations differs in different exampleswe study.A situation where the existence of new low-lying primaries as above is a nontrivial con-straint arises in the study of ground states at fixed, large charge Q for some U (1) symmetry.From general considerations (which however do not apply in mean field theory) one expects [9](for a review and more references see [10])∆ ∼ Q dd − . These ground states are very non-generic heavy states and the constraints arising from thespontaneous breaking of boost symmetry are satisfied quite differently in different exampleswe study. Sometimes the excitations needed for the boost Nambu-Goldstone theorem shouldbe considered as a Regge trajectory of (primary) excitations of | Ω (cid:105) – the Regge trajectoriesappearing in these cases are reminiscent of [11]. This occurs in the superfluid case and 2d,where these are one-particle states, and in mean field theory, where the excitations are tobe thought of as a particle and a (zero momentum) hole in the Bose-Einstein condensate.Sometimes we find that the primary excitations are two-particle states, as in the free Fermisurface.The case of 2d CFTs provides an interesting testing ground for our considerations. Asbriefly alluded to above, spontaneous breaking of boost symmetry is possible in 2d. The statesresponsible for the algebraic decay of correlators are the Virasoro (but not global conformal)descendants of the heavy state. Hence, we do not find any constraint on the gap above aheavy operator. Nevertheless, the relevant Virasoro descendants form a Regge trajectory.We also examine the fate of the large charge effective field theory of [9] in 2d. We find thatfor compact CFTs, it just becomes the free compact scalar representation of the u (1) × u (1)Kaˇc-Moody algebra. And the theory only makes sense for c = 1. For situations where the U (1) symmetry does not get enhanced to a current algebra, the effective theory is nontrivialand it describes a single compact boson with arbitrary conformal anomaly. It resembles theeffective string theory of Polchinski-Strominger [12].To put the present paper in context, let us point out that there has been a monumentaleffort in recent years to understand the spectrum of scaling dimensions in CFTs. The nu-merical bootstrap constraints on the low scaling dimension operators are beautifully reviewedin [8] and [13], among others. There is then a complementary effort to understand the scalingdimensions of various special heavy operators, e.g. the ground states at large fixed charge, as– 5 –eviewed in [10], large fixed spin [14, 15], and various combinations of large charge and largespin [16, 17]. The present paper is aimed at understanding the general constraints from thespontaneous breaking of boost symmetry. This is relevant for the study of heavy operatorsquite generally. But except for some brief discussion of the hydrodynamic regime, our focushere will be exclusively on the large charge ground states.The outline of the paper is as follows. The paper consists of two parts. In Part I wepresent an abstract argument for the bound on the gap in the operator spectrum of CFTsaround heavy operators. In section 2 we write the precise consequences of the spontaneousbreaking of boost symmetry (reviewing and very slightly extending [6]) and dilatation sym-metry. This results in some constraints on the low momentum, low frequency behavior ofEnergy-Momentum correlation functions. We show how these constraints are satisfied instates that obey the assumptions of hydrodynamics: the ballistic sound mode in hydrody-namics exists essentially because of the spontaneous breaking of boost symmetry. In section 3,we discuss in more detail CFTs on the cylinder and review how the operator-state correspon-dence can be used to relate matrix elements on the cylinder and the CFT data. We explainsome important properties of conformal blocks in heavy states and conclude that descendantscannot play a role in the Nambu-Goldstone theorem for boosts. We argue that the Nambu-Goldstone theorem implies a bound on the gap in the CFT operator spectrum. In Part IIwe examine a variety of examples, show that our bound is obeyed and identify special sets ofoperators that saturate the Nambu-Goldstone theorem. In section 4 we discuss the case ofthe superfluid. We identify in detail the primary states that are responsible for the Nambu-Goldstone theorem. In section 5 we repeat the exercise for mean field theory. In section 6 werepeat it for a free Fermi surface. In section 7 we show that in d = 2 the Nambu-Goldstonesum rule is satisfied by a Regge trajectory of one-particle-like states in the Verma module.We also discuss the role of the large charge effective field theory in 2d. Three appendicescontain technical details about Green’s functions in nontrivial states, an extension of the freefermion discussion to d = 4, and, finally, a derivation of some contact terms in correlationfunctions of the EM tensor. Part I
Bounding the Gap in the OperatorSpectrum
While Nambu-Goldstone theorems for internal symmetries are thoroughly understood, thecase of spacetime symmetries provides us with new lessons to this day. One difference ex-plained in the introduction, is that breaking of the boost symmetry does not lead to superse-lection sectors. Another difference is that while it is impossible to break continuous internal– 6 –ymmetries in d = 2, boosts can be spontaneously broken in d = 2. Finally, as mentionedabove, in [6] it was shown that the boost Nambu-Goldstone theorem can be saturated bymultiparticle states; this is a novel phenomenon. In this section we review the formulation of the boost Nambu-Goldstone theorem intranslationally invariant energy eigenstates as a sum rule as discussed in [6], and extend itto dilatations in CFTs. We demonstrate how these sum rules are obeyed in generic ETHstates, i.e. by hydrodynamics. Throughout the paper we use mostly minus signature, g µν =diag(+1 , − , − , . . . ).The logic of the Nambu-Goldstone theorem requires us finding an order parameter forwhich (cid:104) Ω | δ K i O ( t, x ) | Ω (cid:105) (cid:54) = 0, where K i is a boost in the i th direction. Let us choose O = T j ,for which we have (cid:104) Ω | δ K i T j ( t, x ) | Ω (cid:105) = i (cid:104) Ω | [ K i , T j ( t, x )] | Ω (cid:105) = (cid:104) Ω | T ( t, x ) δ ij + T ij ( t, x ) | Ω (cid:105) = ( (cid:15) + P ) δ ij . (2.1) (cid:15) is the energy density and P is the pressure.We can rewrite it as a sum rule obeyed by the following correlator( (cid:15) + P ) δ ij = i (cid:104) Ω | [ K i , T j (0)] | Ω (cid:105) = − i (cid:90) d d − x x i (cid:10) [ T ( t, x ) , T j (0)] (cid:11) , (2.2)which we conveniently rewrite in momentum space as (see appendix A for our notation anddefinition of the Green’s functions) πδ ( ω ) ( (cid:15) + P ) δ ji = lim k → ∂∂k i G ( comm ) T ,T j ( ω, k ) , (2.3)where G ( comm ) T ,T j ( ω, k ) is the commutator Green’s function, or by using (A.4), δ ( ω ) ( (cid:15) + P ) δ ji = lim k → ∂∂k i ρ T ,T j ( ω, k ) . (2.4)( ρ T ,T j stands for the spectral density.) Our task is then to find states whose contribution tothe spectral density saturates the above sum rule. As usual, the Nambu-Goldstone theoremis a constraint on the low-frequency behavior of Green’s functions. It would be nice to understand whether this can happen for internal symmetries away from the vacuumstate – see footnote 1. The Fourier transform is defined through f ( ω, k ) = (cid:90) d d x e i ( ωt − (cid:126)k · (cid:126)x ) f ( t, x ) . – 7 – .1 The Example of Hydrodynamics In the standard treatment of hydrodynamics, we obtain the retarded correlators G ( R ) AB ( ω ),see e.g. [18]. We want to extract the spectral density from this data. Using time reflectionsymmetry to deduce G ( R ) T ,T j ( ω, k ) = G ( R ) T j ,T ( ω, k ), using (A.4) we end up with the formula ρ T ,T j ( ω, k ) = 1 π Im (cid:104) G ( R ) T ,T j ( ω, k ) (cid:105) . (2.5)We then calculate G ( R ) T ,T j ( ω, k ) = ( (cid:15) + P ) ωk j ω − ( c s k ) + iγ s ωk ρ T ,T j ( ω, k ) = ( (cid:15) + P ) π γ s ω k k j ( ω − ( c s k ) ) + ( γ s ωk ) . (2.6)If we plot this function for fixed k , we see two peaks at ω = ± c s k . As we decrease k theyget narrower, but also get closer. To understand this better, let us examine them in a newvariable ˆ ω ≡ ω/ ( c s k ), in terms of which, we have ρ T ,T j ( ω, k ) = ( (cid:15) + P ) πc s ˜ k j ˆ ω (ˆ ω − + ˜ k ˆ ω , ˜ k ≡ γ s kc s . (2.7)Now it is a standard result that for small ˜ k π ˜ k j ˆ ω (ˆ ω − + ˜ k ˆ ω ≈ ˜ k j ˆ ω k [ δ (ˆ ω −
1) + δ (ˆ ω + 1)]= ˜ k j k [ δ (ˆ ω −
1) + δ (ˆ ω + 1)]= c s k j δ ( ω − c s k ) + δ ( ω + c s k )] ≈ c s k j δ ( ω ) . (2.8)Putting the factors together, we learn that for small kρ T ,T j ( ω, k ) ≈ ( (cid:15) + P ) k j δ ( ω ) . (2.9)Plugging this into the RHS of (2.4), we verify that the equation indeed holds.One wonders what states gave us the saturation of the sum rule. Hydrodynamics differsfrom the other EFTs we study in Part II in that the sound mode is a collective excitation,and there is no preferred set of states | Ω (cid:48) (cid:105) that gives rise to the above spectral density. Inharmony with ETH, all states | Ω (cid:48) (cid:105) close in energy to | Ω (cid:105) contribute (see [19] for a detailed– 8 –nalysis). Yet, one can say that it is the ballistic pole in hydrodynamics which is responsiblefor the spontaneously broken boost symmetry. One can argue from conservation that ρ T ,T ( ω, k ) should go to − ( (cid:15) + P ) k δ (cid:48) ( ω ) at verysmall momentum. One argument goes as follows: we have determined in (2.4) that for k (cid:28) ω : ρ T ,T j ( ω, k ) ≈ ( (cid:15) + P ) k j δ ( ω ) , (2.10)hence the Green’s function according to (A.2) G T ,T j ( ω, k ) = (cid:90) ∞−∞ dω (cid:48) ρ T ,T j ( ω (cid:48) , k ) ω − ω (cid:48) ≈ ( (cid:15) + P ) k j ω (2.11)Conservation of the stress tensor gives G T ,T ( ω, k ) = − k j ω G T ,T j ( ω, k ) ≈ ( (cid:15) + P ) k ω , (2.12)which is reproduced by ρ T ,T ( ω, k ) ≈ − ( (cid:15) + P ) k δ (cid:48) ( ω ) . (2.13)Let us see how this comes about in hydrodynamics. G ( R ) T ,T ( ω, k ) = ( (cid:15) + P ) k ω − ( c s k ) + iγ s ωk ρ T ,T ( ω, k ) = ( (cid:15) + P ) π γ s ωk ( ω − ( c s k ) ) + ( γ s ωk ) ≈ ( (cid:15) + P ) c s c s k δ ( ω − c s k ) − δ ( ω + c s k )] ≈ − ( (cid:15) + P ) k δ (cid:48) ( ω ) . (2.14)Note that − δ (cid:48) ( ω ) is positive for ω >
0, as required for the spectral density.
Let us ask what sum rules follow from the breaking of boosts K i , dilatation D , and specialconformal transformations S µ in a homogeneous and isotropic finite energy density CFT state | Ω (cid:105) , which we imagine as the macroscopic limit of some CFT state on S d − that correspondsto a heavy scalar operator. The idea is to find an operator for which (cid:104) Ω | δ O ( t, x ) | Ω (cid:105) (cid:54) = 0,where δ O is the transformation of the operator under the generator K i , D, S i .– 9 –et us first consider K i . We have already worked out the case O = T j , and obtainedthe sum rule (2.4). Similarly for O = J j , by using δ K i J j = J δ ij , we get δ ( ω ) ρ δ ji = lim k → ∂∂k i ρ T ,J j ( ω, k ) , (2.15)where ρ is the charge density.Next we ask about D , for which we have δ D O = ∆ O O . We then write∆ O (cid:104) Ω | O | Ω (cid:105) = i (cid:104) [ D, O (0)] (cid:105) = i (cid:90) d d − x x µ (cid:104) [ T µ ( t, x ) , O (0)] (cid:105) = − i (cid:90) d d − x x i (cid:10) [ T i ( t, x ) , O (0)] (cid:11) . (2.16)This is then almost identical to (2.2), and we getfor O = T : δ ( ω ) d(cid:15) = lim k → ∂∂k i ρ T i ,T ( ω, k )for O = J : δ ( ω )( d − ρ = lim k → ∂∂k i ρ T i ,J ( ω, k ) . (2.17)The right hand side of the first equation here is known from (2.4) to be equal to( d − (cid:15) + P ) δ ( ω ), so we only learn that (cid:15) = ( d − P , which is true in CFT. The secondequation is a new sum rule. We will check it for the superfluid in section 4.Finally we discuss S µ . Since for a primary O , [ S µ , O ] = 0 we have to work with descen-dants. Let us consider the variation δ S µ ( i [ P ν , O ]) = − [ S µ , [ P ν , O ]]= [[ P ν , S µ ] , O ]= − i [ η µν D − M µν , O ]= − η µν δ D O + 2 δ M µν O , (2.18)where in the second line we used the Jacobi identity and [ S µ , O ] = 0, while in the thirdthe conformal algebra. Since the last line is a linear combination of terms we have alreadyconsidered, we conclude that we do not learn any new constraint from the sum rule associatedto the breaking of S µ .In appendix C we derive identities for the contact terms in certain products of the energy-momentum tensor. The constraints above follow from these contact terms.– 10 – Conformal Field Theory
The purpose of this section is to explore the implication of sum rules for the spectrum ofCFTs. The sum rules are obeyed by a state with finite energy and/or charge density. Thestrategy is to start from a state of a CFT on S d − × R . The state on the cylinder S d − × R corresponds to an operator of the CFT in the plane with scaling dimension ∆ Ω . The energy E of the state is given by E = ∆ Ω R , (cid:15) = ∆ Ω s d − R d , where (cid:15) is the energy density and s d − is the volume of unit sphere S d − . Now we takethe macroscopic limit [11, 20] by considering a family of operators with ∆ Ω → ∞ and take R → ∞ while keeping (cid:15) fixed. Likewise the theory may have a conserved U (1) symmetryand we can construct states with fixed charge density ρ = Q/ ( s d − R d − ). We emphasize thatthe states | Ω (cid:105) correspond to heavy operators in the CFT. In particular, we consider correlators of light operators of the form (cid:104) Ω |OO| Ω (cid:105) on thecylinder and take the required limit. Since we are considering a family of operators in themacroscopic limit, we are actually looking at a family of correlators, as we take R → ∞ . Oneof the underlying assumptions is that the correlators lead to a nice function of ∆ Ω and/or Q so that it makes sense to take the limit. In this limit the positions of the light operators O remain fixed on the cylinder.The question that we are interested in is what states are responsible for saturating thesum rules discussed in section 2. The sum rules are obtained as limits of two point functionson the cylinder, which we can rewrite by inserting a complete set of states (cid:104) Ω | OO | Ω (cid:105) = (cid:88) (cid:104) Ω |O| (cid:105)(cid:104) |O| Ω (cid:105) . (3.1)In the limits prescribed by the sum rule, only certain states | (cid:105) give contributions. Forexample as we discuss in Part II, in the infinite volume limit, for the superfluid the | (cid:105) ’scontributing are one particle states, while for free fermions they are particle-hole states [6]. Ifwe trace them back to the states of CFT on the cylinder at finite R and then via radial quan-tization to the operators on the plane, they correspond to certain operators appearing in the s -channel conformal block decomposition of the four point correlator (cid:104) Ω(0) O ( z ) O (1)Ω( ∞ ) (cid:105) (by s -channel we always mean the channel where we consider the OPE of Ω(0) O ( z )). Every s -channel conformal block sums up the contribution of a primary and its descendants in (3.1). The scale invariance of CFTs imply that R is just a convenient auxiliary parameter. We could set R = 1and discuss the macroscopic limit equally well: it would involve zooming in onto a small patch of the cylinderin correlation functions. In cases where it is possible to construct EFT to describe the correlator in these heavy states, the heavystate effectively acts like vacuum for the EFT modes. – 11 –t the end of the day, we will be interested in spinning conformal blocks where O carriesspin index. Nonetheless in the macroscopic limit, the difference between scalar blocks andspinning blocks is inconsequential. So, let us illustrate the basic concepts of conformal blocksusing scalar operators O(cid:104)
Ω(0) O ( z ) O (1)Ω( ∞ ) (cid:105) = ( z ¯ z ) − (∆ Ω +∆ O ) (cid:88) | C Ω O ∆ | G ∆ Ω +∆ ,(cid:96) ( z, ¯ z ) , (3.2)where G ∆ Ω +∆ ,(cid:96) ( z, ¯ z ) is the conformal block. Here we have parametrized the scaling dimensionof the operators appearing in the intermediate channel in a way such that ∆ denotes theexcitation over the state Ω i.e C Ω O ∆ is the OPE coefficient involving the operator Ω, O andthe operator appearing in the s -channel with scaling dimension ∆ Ω + ∆. This is a convenientchoice as one of the main results of our paper involves putting a bound on this gap comparedto ∆ Ω , i.e putting a bound on ∆. We will come back to the discussion of gap in due time.Starting from (3.2), one can transform to the cylinder and write down the correlator of O in the heavy state | Ω (cid:105) . (cid:104) Ω |O ( τ, (cid:126)n ) O (0 , (cid:126)n ) | Ω (cid:105) = (cid:88) | C Ω O ∆ | g ∆ ,(cid:96) ( z, ¯ z ) , (3.3)where we defined the O operator on the cylinder by conformally transforming it from theplane. Furthermore, we have defined g ∆ ,(cid:96) as g ∆ ,(cid:96) ( z, ¯ z ) ≡ ( z ¯ z ) − ∆ Ω / G ∆ Ω +∆ ,(cid:96) ( z, ¯ z ) . (3.4)Note, the LHS of (3.3) is defined on the cylinder and thus the cross ratio z, ¯ z on the RHSshould be understood as a function of cylinder coordinates. In particular, the conformallytransformed operators O are inserted at ( τ, (cid:126)n ) and (0 , (cid:126)n ) with (cid:126)n · (cid:126)n = cos θ . The crossratio z, ¯ z is related to τ and θ in following way √ z ¯ z = e τ/R , z + ¯ z √ z ¯ z = cos θ . (3.5)The limit R → ∞ is taken in a way, so that θR ≡ x and τ are kept fixed and identified withthe coordinate in the macroscopic limit. The macroscopic limit reads in terms of u = τ + ix and ¯ u = τ − ix z = 1 + uR , ¯ z = 1 + ¯ uR , R → ∞ , u, ¯ u fixed . (3.6)In what follows, we will be establishing that in the macroscopic limit, the descendantsare suppressed i.e the conformal blocks take a very simple form. This will be followed by thediscussion on the implication of this suppression for the spectrum of primaries appearing inthe s -channel. Along the way, we will explore how and which of these primaries survive themacroscopic limit and eventually saturate the sum rule.– 12 – .2 Supression of Descendants In this subsection we show that the primaries dominate the s -channel expansion in the macro-scopic limit. We will first consider the heavy operator limit which amounts to ∆ Ω → ∞ . (Thislimit is not the macroscopic limit since we do not yet scale the coordinates as required inthe macroscopic limit. The suppression of descendants in this limit was studied in [11].)Afterwards, we will keep the energy/charge density fixed and take R → ∞ limit, i.e. themacroscopic limit.Let us start with the blocks showing up in (cid:104) Ω OO Ω (cid:105) , where Ω is the heavy primarywith ∆ Ω → ∞ and O has O (1) scaling dimension. We write the block as an expansion inGegenbauer polynomials C (cid:96) ≡ C ( d/ − (cid:96) (cid:16) z +¯ z √ z ¯ z (cid:17) : g ∆ ,(cid:96) ( z, ¯ z ) = (cid:88) m,n r m,n ( z ¯ z ) (∆+ m + n ) / C (cid:96) + m − n ,r = 1 ,r = 12(2 (cid:96) + d −
2) ( (cid:96) + 1)(∆ + ∆ O + (cid:96) ) ∆ Ω + ∆ + (cid:96) ,r = 12(2 (cid:96) + d −
2) ( (cid:96) + d − O − (cid:96) − d + 2) ∆ Ω + ∆ − (cid:96) − d + 2... (3.7)(The sum is restricted to (cid:96) + m − n ≥ r m,n = (cid:101) r m,n ∆ m + n Ω + . . . , (cid:101) r m,n ≈ m ! n ! , for 1 (cid:28) m, n (cid:28) ∆ Ω , (3.8)where there are some powers of m, n that we suppressed. We conclude that the contributionof the k = m + n level descendants is suppressed by 1 / ∆ k Ω . This suggests that we should beable to approximate the conformal block by the first term ( m = n = 0) in (3.7). Since we willbe eventually interested in setting z, ¯ z close to 1 as in (3.6) we need to make sure that thedescendants corresponding to m, n (cid:38) ∆ Ω continue to be suppressed. It can be shown againfrom the recursion relations that the contribution from m, n (cid:29) ∆ Ω is suppressed comparedto that of the primary. Instead of going though a meticulous analysis of this kind, we use theknowledge of the 4 d blocks to demonstrate the absence of anomalously large resummationeffects. – 13 –he 4 d conformal blocks are known in closed form [22]: G δ,(cid:96) ( z, ¯ z ) = ( − (cid:96) z ¯ zz − ¯ z [ k δ + (cid:96) ( z ) k δ − (cid:96) − (¯ z ) − k δ − (cid:96) − ( z ) k δ + (cid:96) (¯ z )] k β ( z ) = z β/ F (cid:18) β − ∆ − , β − ∆ − , β ; z (cid:19) (3.9)As a consistency check, we recover (3.7). Let us now investigate the macroscopic limit. Letus first set ∆ = O (1) , (cid:96) = O (1), and use z = 1 + uR . We obtain using the definition (3.4) g ∆ ,(cid:96) ( z, ¯ z ) = ( z ¯ z ) − ∆ Ω / G ∆ Ω +∆ ,(cid:96) ( z, ¯ z ) = ( (cid:96) + 1) + ( (cid:96) + 1)(2∆ − ∆ O )( u + ¯ u )2 R + . . . . (3.10)In order for our analysis to apply for the macroscopic limit we need to simultaneouslytake z = 1 + uR along with keeping the energy density fixed, in other words, we must studythe double scaling limit of the conformal blocks: z ≡ uR , ∆ = E R , (cid:96) ≡ p R , ∆ Ω ≡ (cid:15) R d . (3.11)Then w becomes a coordinate in the macroscopic limit, E is a fixed O (1) coefficient (whoseinterpretation is the energy of the intermediate state above the energy of | Ω (cid:105) ), p is the modulusof the momentum of the intermediate state and (cid:15) is the energy density in | Ω (cid:105) . The reasonthat we scale ∆ and (cid:96) as above is not immediately obvious but it will soon become clear thatthis leads to an interesting macroscopic limit. We get g ∆ ,(cid:96) ( z, ¯ z ) = R exp (cid:18) E u + ¯ u ) (cid:19) (cid:0) p ( u − ¯ u ) (cid:1) u − ¯ u + . . . . (3.12)Noting that in 4d, C (1) (cid:96) (cos( x )) = sin (( (cid:96) + 1) x )sin( x ) (3.13)and that z + ¯ z √ z ¯ z = cos (cid:18) u − ¯ u iR (cid:19) + . . . , (3.14)we realize that in the two limits discussed above C (1) (cid:96) (cid:18) z + ¯ z √ z ¯ z (cid:19) = ( (cid:96) + 1) + . . . for (cid:96) = O (1) , R ( p ( u − ¯ u ) ) u − ¯ u for (cid:96) = O ( R ) . (3.15)These are exactly the leading pieces of (3.10) and (3.12). Hence we find that the conclusionthat the primaries dominate the blocks is indeed correct, i.e. it is not spoiled by resummation– 14 –ffect. These conclusions remain true for spinning blocks, which are obtained by differentialoperators acting on scalar blocks [23, 24].Before we proceed it is worth noting that from (3.15) we readily see that the interestingintermediate blocks have (cid:96) = O ( R ) and ∆ = O ( R ), which makes sense, since these correspondto intermediate states with finite momentum and frequency in infinite volume.Thus far we have found that the contributions in the macroscopic limit are solely dueto the primary operators in the intermediate channel and hence the macroscopic conformalblocks are exceedingly simple (3.12). The factor exp (cid:0) E ( u + ¯ u ) (cid:1) is simply the time dependencethat follows from translating the operators in energy eigenstates. The spatial dependence ∼ sinh ( p ( u − ¯ u ) ) u − ¯ u contains the absolute value of the momentum p = | (cid:126)p | . It reflects the sum overall momentum eigenstate with fixed | (cid:126)p | . Using the fact that O is a scalar operator, the matrixelements have a simple dependence on (cid:126)p and the Gegenbauer polynomial arises through anintegral of the form C ( d/ − (cid:96) = pR (cos( x/R )) ∼ (cid:82) d d − k δ ( | (cid:126)k | − p ) e i(cid:126)k · (cid:126)x (for details see (3.29)). For completeness let us present an intuitive argument for the suppression ∆ − k Ω found in (3.8)that also applies to spinning blocks. One can argue for this suppression by computing matrixelements on the cylinder directly in the ∆ Ω → ∞ limit. This computation can be done ina straightforward way even for spinning operators. For example, let us derive that the firstdescendant is indeed suppressed. To proceed, we recall that (cid:104) Ω (cid:48) ( x ) T αβ ( z )Ω( y ) (cid:105) = ( y − z ) ∆ − d ( y − x ) Ω +∆ − d f ( x, z ) , (3.16)where f ( x, z ) is the OPE coefficient C Ω T Ω (cid:48) times a kinematical function that carries thespin indices and is independent of the operator dimensions. Ω (cid:48) is the primary with scalingdimension ∆ Ω + ∆ appearing in the internal channel. We assume ∆ scales slower than ∆ Ω ,hence Ω (cid:48) and Ω has the same scaling dimension to leading order in ∆ Ω . From (3.16) it followsthat (cid:104) ∂ µ Ω (cid:48) ( x ) T αβ ( z )Ω( y ) (cid:105) = 2∆ Ω ( x − y ) µ ( y − z ) ∆ − d ( y − x ) Ω +∆ − d +2 f ( x, z ) + ( y − z ) ∆ − d ( y − x ) Ω +∆ − d ∂ µ f ( x, z ) . (3.17)Now we map (3.16) and (3.17) from the plane onto the cylinder ( T ( z ( τ, θ )) should beunderstood as the operator on the cylinder and T rr ( z ) should be understood as an operatoron the plane) via (cid:104) Ω (cid:48) | T ( z ( τ, θ )) | Ω (cid:105) = (cid:18) Rr (cid:19) − d lim y →∞ y Ω (cid:104) Ω (cid:48) (0) T rr ( z )Ω( y ) (cid:105) ∼ (cid:18) Rr (cid:19) − d f (0 , z ) , (cid:104) ∂ µ Ω (cid:48) | T ( z ( τ, θ )) | Ω (cid:105) = (cid:18) Rr (cid:19) − d lim y →∞ y Ω (cid:104) ∂ µ Ω (cid:48) (0) T rr ( z )Ω( y ) (cid:105) ∼ (cid:18) Rr (cid:19) − d ∂ µ f (0 , z ) , (3.18)– 15 –nd it follows from the above (cid:104) ∂ µ Ω (cid:48) | T | Ω (cid:105)(cid:104) Ω (cid:48) | T | Ω (cid:105) (cid:39) O (1) , (3.19)as the ratio of function f ( x, z ) and its derivative with respect to x at x = 0 is order one.Note that we are not making any assumption on the OPE coefficient C Ω T Ω (cid:48) , since it cancelsout in the ratio. Now the contribution to the correlator coming from an s-channel conformalblock corresponding to a primary Ω (cid:48) is given by (cid:104) Ω | T T | Ω (cid:105) (cid:51) (cid:88) α,β =Ω (cid:48) ,P Ω (cid:48) ,P P Ω (cid:48) ··· (cid:104) Ω | T | α (cid:105)N − αβ (cid:104) β | T | Ω (cid:105) , (3.20)where N αβ = (cid:104) α | β (cid:105) . We follow the notation of [8] and by P Ω (cid:48) we mean the operator ∂ Ω (cid:48) . Inparticular, for the first descendant P Ω (cid:48) , we have N P µ Ω (cid:48) ,P ν Ω (cid:48) = (cid:104) Ω (cid:48) | K µ P ν | Ω (cid:48) (cid:105) (cid:39) (∆ Ω + ∆) δ µν , (3.21)where we used the commutation relation of K and P . Altogether, we find that the contributioncoming from the first descendant is suppressed, i.e. (cid:104) Ω | T | ∂ Ω (cid:48) (cid:105) N − P Ω (cid:48) ,P Ω (cid:48) (cid:104) ∂ Ω (cid:48) | T | Ω (cid:105) (cid:39) (cid:104) ∂ µ Ω (cid:48) | T | Ω (cid:105)| ∆ − (cid:39) (cid:104) Ω (cid:48) | T | Ω (cid:105) ∆ − . (3.22)A similar argument implies that the level k descendant is suppresed by ∆ − k Ω . The factors m ! n ! in the conformal block in (3.8) can be accounted for from the number of distinct statesat level k = m + n with spin (cid:96) + m − n . In this subsection, we combine the Nambu-Goldstone boost sum rules, the macroscopic limitof CFTs and the form of conformal blocks in this limit to derive constraints on the gap inthe CFT operator spectrum. We will be studying the correlator (cid:104) Ω | T T | Ω (cid:105) , (3.23)use the s -channel decomposition, and aim to put a bound on the gap ∆ = ∆ Ω (cid:48) − ∆ Ω , where | Ω (cid:48) (cid:105) is an excited state (above the “vacuum” state | Ω (cid:105) ) exchanged in the correlator. The sumrule corresponding to the correlator in (3.23) was given in (2.13) and in CFTs takes the form ρ T T ( ω, p ) p = − d(cid:15)d − δ (cid:48) ( ω ) + O ( p ) . (3.24)We chose to focus on a spectral density of two identical operators, since every contributionto such a spectral density is positive definite for ω >
0. The obvious next step is to expressthe LHS of (3.24) with CFT data from the cylinder.To this end, we write the Euclidean correlator (for τ >
0) in the large charge limit at– 16 –nite R using the above results for conformal blocks: (cid:104) Ω | T ( τ, (cid:126)n ) T (0 , (cid:126)n ) | Ω (cid:105) = (cid:15) (cid:34) (cid:88) (cid:96) (cid:96) !∆ (cid:96) Ω e − (cid:96) | τ | /R C ( d/ − (cid:96) (cos θ ) (cid:35) + (cid:88) ∆ C T ∆ R d (cid:104) e − ∆ | τ | /R C ( d/ − (cid:96) (∆) (cos θ ) + O (1 / ∆ Ω ) (cid:105) , (3.25)where in the first line we wrote the contribution from the conformal block of Ω itself (schemat-ically including the descendants and using that its OPE coefficient C Ω T /R d = (cid:104) Ω | T | Ω (cid:105) = (cid:15) ), while in the second we included the blocks corresponding to other exchanged operators(and only wrote explicitly the contributions of the primaries of dimension ∆ Ω + ∆ and spin (cid:96) (∆)).While the expression (3.25) is correct in the macroscopic limit, for the macroscopic limitto actually exist, one must make some assumptions about the OPE coefficients C T ∆ . (Asimilar logic holds in the discussion of ETH in CFT [20].) Indeed, the most importantexchanged operators which will contribute in the macroscopic limit are clearly such that∆ ∼ R and (cid:96) ∼ R . In addition, there is a factor of R d − from the Gegenbauer polynomialsas in (3.15) (where d = 4). Let us introduce the density of states by per unit energy and perunit momentum ρ ( ω, p ; R ) = (cid:88) O δ (cid:18) ω − ∆ O R (cid:19) δ (cid:18) p − (cid:96) O R (cid:19) , (3.26)This is a finite volume, un-smeared object. In terms of ρ ( ω, p ; R ) we obtain an expression forthe macroscopic limit of the the correlator (3.25): (cid:104) Ω | T ( τ, x ) T (0) | Ω (cid:105) = (cid:15) + (cid:90) dω dp ρ ( ω, p ) C T ∆ R d +3 e − ω | τ | F p ( x ) , (3.27)where F p ( x ) is the Gegenbauer polynomial with the R -dependence stripped out (we will soonwrite a concrete expression for it in any dimension; in d = 4 this can be read out from (3.15)and we find F p ( x ) = sin( px ) /x ).For the macroscopic limit to exist the combination K ( ω, p ) ≡ ρ ( ω, p ) C T ∆ /R d +3 mustbecome R -independent in the appropriate sense. We note that by mapping from the | z | < τ < T (0 , (cid:126)n ) T ( τ, (cid:126)n ). Here we instead wrote the answer for τ > To make the required averaging procedure precise we introduce the smearing: K ( ω, p ) = 14 δ (cid:88) O∈I ( ω,p ; R,δ ) C T ∆ R d +3 , I ( ω, p ; R, δ ) ≡ (cid:26) O (cid:12)(cid:12)(cid:12) ∆ O R ∈ ( ω − δ, ω + δ ) , (cid:96) O R ∈ ( p − δ, p + δ ) (cid:27) , (3.28)where the set I contains operators whose energy and momentum in the macroscopic limit agrees with ω, p – 17 –inally, to give an expression for F p ( x ) that is valid in any dimension we use the integralrepresentation of the Gegenbauer polynomials:lim R →∞ C ( d/ − pR (cos( x/R )) = lim R →∞ Γ ( pR + d − d − Γ (cid:0) d − (cid:1) Γ( pR + 1) × (cid:90) π dϑ sin d − ( ϑ ) (cos( x/R ) + i sin( x/R ) cos( ϑ )) pR = ( pR ) d − d − Γ (cid:0) d − (cid:1) (cid:90) π dϑ sin d − ( ϑ ) e ipx cos( ϑ ) = 12 d − π ( d − / Γ (cid:0) d − (cid:1) R d − p (cid:90) d d − k δ (cid:16)(cid:12)(cid:12)(cid:12) (cid:126)k (cid:12)(cid:12)(cid:12) − p (cid:17) e i(cid:126)k · (cid:126)x . (3.29)The last relation is very intuitive: if we zoom in onto a small patch of the sphere, we get planewaves with fixed | (cid:126)p | , averaged over all directions. In d = 4 this precisely agrees with (3.12).Therefore, F p ( (cid:126)x ) = 12 d − π ( d − / Γ (cid:0) d − (cid:1) p (cid:90) d d − k δ (cid:16)(cid:12)(cid:12)(cid:12) (cid:126)k (cid:12)(cid:12)(cid:12) − p (cid:17) e i(cid:126)k · (cid:126)x (3.30)The expression (3.30) allows us to rewrite (3.27) in a simpler form, after doing the p integral (cid:104) Ω | T ( τ, x ) T (0) | Ω (cid:105) = (cid:15) + 12 d − π ( d − / Γ (cid:0) d − (cid:1) (cid:90) dω d d − k K ( ω, k ) e − ω | τ | e i(cid:126)k · (cid:126)x k . (3.31)It is now straightforward to obtain an expression for the spectral density in the macroscopiclimit ρ T T ( ω, p ) ∼ p ( K ( ω, p ) − K ( − ω, p )) . (3.32)We must now make contact with (3.24). We learn that, at the very least, K ( ω, p ) musthave support at ω = 0 and p = 0. This means that there must be operators with ∆ gap /R → R → ∞ . In more conventional CFT terms, if we denote the dimension of the ground stateby ∆ Ω , we conclude that the gap ∆ gap must be smaller than ∆ /d Ω in the sense that theremust be operators with ∆ gap / ∆ /d Ω → Ω → ∞ . If we in addition recall the physical reasoning advocated in the introduction,namely, that there is a scale invariant low energy theory describing the massless excitations inthe small ω regime, we would conclude that ∆ gap = O (1). In addition, we can constrain theangular momentum of these operators in a similar fashion since the sum rule (3.24) implies respectively. Here we have introduced an infinitesimal window width δ ; K ( ω, p ) should be independent of δ toleading order in the macroscopic limit, and hence we do not include δ as its argument. – 18 –hat we need to take infinitesimal p . This means that the angular momentum must satisfy (cid:96)/ ∆ /d Ω → O (1).This implies a surprisingly small primary gap of O (1) (for dimension and angular mo-mentum) around any heavy state. One can say more about the density of operators or OPEcoefficients using (3.24) if one makes additional assumptions about the density or OPE coef-ficients separately. Incidentally, since the density ρ ( ω, p ; R ) cannot decay as R → ∞ (becausethe number of participating operators cannot go to zero) we get a rather general bound onthe OPE coefficient C T ∆ = O ( R d +3 ). (This bound can be strengthened by requiring amore realistic density of states.)Recalling that the sum rule (3.24) is linear in (cid:15) , we can refine this bound slightly. Sincethe macroscopic limit is a double scaling limit, we can choose (cid:15) ∼ ∆ Ω /R d at will, while R, ∆ Ω → ∞ . We can then resolve R d +3 as ∆ Ω R (cid:0) ∆ Ω /R d (cid:1) α , and show that α = 0 bycontradiction: were α >
0, we take (cid:15) large, while for α <
0, we take (cid:15) small to derive aviolation of the sum rule. Hence we conclude C T ∆ = O (∆ Ω R ) . (3.34) Part II
Examples
In the superfluid phase the ground state is homogenous, isotropic, has a finite charge density,and breaks U (1) spontaneously. As in the previous section, we imagine this state to bethe macroscopic limit of a family of large charge states on the cylinder S d − × R . Thesestates correspond to scalar operators of the underlying CFT. In the Q → ∞ limit, thereis a separation of energy scales; ρ − / ( d − (with ρ ∼ Q/R d − the charge density) is a UVscale while the IR scale is given by R . For distances much larger than the UV scale butmuch less than R , the system is described by an effective field theory with 1 /Q being theexpansion parameter [9, 25]. As we will see the UV scale is precisely related to (cid:15) − /d (where (cid:15) is the energy density) mentioned in the introduction while R plays the role of L . In theinfinite volume limit, the state with finite energy density breaks SO ( d + 1 , × U (1) down to a SO ( d ) and a linear combination of U (1) and time translation. The action of the effective fieldtheory can be constructed in the CCWZ way [26, 27] in terms of a field χ and its fluctuation π around the symmetry breaking saddle. The field π is identified with the Goldstone mode,corresponding to the aforementioned spontaneous breaking.In this section, we will elaborate on the general ideas described in the previous section– 19 –sing the explicit example of superfluid EFT, which captures the large charge sector of anunderlying CFT. The aim is to identify the primary states that saturate the sum rule forbroken boost symmetries and discuss the connection to what we have found based on generalarguments in the previous section. The effective field theory is described by the Euclidean action [9, 25] S = − c (cid:90) d d x √ g | ∂χ | d + · · · + i (cid:90) d d x √ g ρ ˙ χ (4.1)(Here we assume that the underlying CFT preserves parity, hence the parity violating termsof [28] are forbidden.) The saddle point that describes a ground state with finite homogenouscharge density ρ is given by χ = − iµτ + πc dµ d − = ρ (4.2)Expanding around this saddle, we obtain the effective action for the Goldstone field πS π = d ( d − c µ d − (cid:90) d d x √ g (cid:18) ˙ π + 1 d − ∂ i π∂ i π (cid:19) + · · · . (4.3)For the purpose of examining the sum rules, we will eventually be interested in thecorrelator involving the stress-energy tensor (in particular the components T , T i ) andcurrent J i . From (4.1), we obtain the stress-energy tensor (below we consider the Minkowskisignature metric and we use τ = it ) T = (cid:15) + i d(cid:15)µ dπdτ + · · · , T i = d(cid:15) ( d − µ ∂ i π + · · · , T ij = − η ij (cid:15)d − · · · , (4.4)where we used that the (leading order in µ ) energy density is (cid:15) = c ( d − µ d . (4.5)The two-point correlator of π on the cylinder in the large Q limit is given by D ( τ, x ) = µ s d − R d − d(cid:15) (cid:34) −| τ | + (cid:88) (cid:96) =1 (cid:96) + d − d − e − ω (cid:96) | τ | /R ω (cid:96) /R C d/ − (cid:96) (cos θ ) (cid:35) , (4.6)where θ = arccos( (cid:126)n · (cid:126)n ) and τ are the angle and time separation between insertion of two π fields respectively. Here ω (cid:96) = (cid:112) (cid:96) ( (cid:96) + d − / ( d −
1) . Note, in Euclidean signature, we have Euclidean stress energy tensor defined as T µν = √ g δSδg µν . TheMinkowski T is related via T ττ = − T and T τi = − iT i . In the text, we use the index 0 and work with theMinkowski T operator. The same applies for J . – 20 –nother relevant quantity for our purpose is the current J µ corresponding to U (1) thatacts as a shift symmetry on the field χ : J = ρ + i ( d − ρµ dπdτ , J i = ρµ ∂ i π . (4.7)In rest of this section, we will be heavily using (4.4), (4.6), (4.7). The three major sum rules that we have been discussing in this paper involves looking at (cid:104) T T (cid:105) , (cid:104) T T i (cid:105) and (cid:104) T J k (cid:105) in the large charge state. For now, we will focus on the twopoint correlator of T in the large charge state and will perform the analysis in detail byworking on the cylinder and then taking the macroscopic limit while tracking the set of states,that are eventually going to saturate the corresponding sum rule. We will come back to theother sum rules involving (cid:104) T T i (cid:105) and (cid:104) T J k (cid:105) later and verify them by working directly inthe macroscopic limit without performing the computation on the cylinder. (cid:104) T T (cid:105) The two point correlator in T in the large charge state is given by for τ > (cid:104) T ( τ ) T (0) (cid:105) = (cid:15) (cid:20) d d −
2) 1∆ Ω e −| τ | /R C d/ − (cos θ ) + · · · (cid:21) + d(cid:15)s d − R d − (cid:34)(cid:88) (cid:96) =2 (cid:18) (cid:96) + d − d − (cid:19) (cid:16) ω (cid:96) R (cid:17) e − ω (cid:96) | τ | /R C d/ − (cid:96) (cos θ ) + · · · (cid:35) (4.8)For τ <
0, the right hand side provides us with (cid:104) T (0) T ( τ ) (cid:105) . We will keep this mind andfor brevity use (cid:104) T T (cid:105) . We can compare the above with (3.25). In the first line on theRHS, we have two terms corresponding to the contribution coming from exchange of Ω andits descendant. We have omitted the contributions coming from higher descendants. Theyare suppressed in ∆ Ω → ∞ limit. In the previous section, we have argued that there is nocumulative effect coming from considering all the descendants together. In the second line,we have a contribution from a single Regge trajectory, one primary for each given integer (cid:96) ≥ Ω + ∆, where we have ∆ = (cid:112) (cid:96) ( (cid:96) + d − / ( d −
1) . Herewe denote ∆ as ω (cid:96) since we have single Regge trajectory and sum over spin (cid:96) suffices. TheOPE coefficients can be read off as C T ∆ = d ∆ Ω s d − (cid:18) (cid:96) + d − d − (cid:19) (cid:112) (cid:96) ( (cid:96) + d − / ( d − . (4.9)In what follows we will show that the states with (cid:96) ∼ pR contribute to the macroscopiclimit and then in the p → ρ T T ( ω, p ) = d(cid:15) √ d − p (cid:20) δ (cid:18) ω − p √ d − (cid:19) − δ (cid:18) ω + p √ d − (cid:19)(cid:21) . (4.10)In the p → ρ T T ( ω, p ) (cid:39) p → − d(cid:15)d − p δ (cid:48) ( ω ) = − ( (cid:15) + P ) δ (cid:48) ( ω ) , (4.11)and the sum rule (2.13) is satisfied. We note that p → (cid:96) = o ( R ), thus itis clear that the states with ∆ = (cid:96) = o ( R ) from the Regge trajectory saturate the sum rule.The OPE coefficients for these states C T ∆ = O (cid:0) ∆ Ω (cid:96) (cid:1) = o (cid:0) ∆ Ω R (cid:1) . (4.12)Comparing with the bound from (3.34) in sec. 3.4, we see that this is down by a factor of 1 /R .The reason for this is that that there are o ( R ) states in the superfluid in this kinematicalregime, whereas the bound (3.34) allows only one state (as a worst case scenario). The rest of the sum rules
The two-point correlators of T µ and J µ in the macroscopic limit can be found from themacroscopic limit of the two-point correlator of the π field: (cid:104) π ( τ, (cid:126)x ) π (0) (cid:105) macro = µ d ( d − s d − (cid:15) √ d − (cid:16) d − τ + (cid:126)x · (cid:126)x (cid:17) d/ − . (4.13)From (4.4) and (4.7) we immediately realize that all we need to know for these computationsis that for τ > ∂ τ ∂ i (cid:104) π ( τ, (cid:126)x ) π (0) (cid:105) macro = µ s d − (cid:15) τ √ d − x i (cid:16) d − τ + (cid:126)x · (cid:126)x (cid:17) d/ ≡ h i ( x ) . (4.14)In terms of this correlator: (cid:104) T i ( x ) T (0) (cid:105) macro = − i d (cid:15) ( d − µ h i ( x ) , (cid:104) T i ( x ) J (0) (cid:105) macro = (cid:104) T ( x ) J i (0) (cid:105) macro = − i d(cid:15)ρµ h i ( x ) . (4.15)– 22 –he Fourier transformed function (cid:101) h ( ω, (cid:126)p ) is defined through: (cid:101) h i ( ω, (cid:126)p ) = (cid:90) d d x e iωτ − i(cid:126)p · (cid:126)x h i ( x ) . (4.16)By rescaling τ = √ d − τ (cid:48) , ω = ω (cid:48) / √ d −
1, we get a standard Lorentz invariant integral (cid:101) h i (cid:18) ω (cid:48) √ d − , (cid:126)p (cid:19) = √ d − µ s d − (cid:15) (cid:90) dτ (cid:48) d(cid:126)x e iω (cid:48) τ (cid:48) − i(cid:126)p · (cid:126)x τ (cid:48) x i (cid:0) τ (cid:48) + (cid:126)x · (cid:126)x (cid:1) d/ = √ d − πµ d(cid:15) ∂∂ω (cid:48) ∂∂p i (cid:104) Θ( ω (cid:48) )Θ( ω (cid:48) − p ) (cid:16) ω (cid:48) − p (cid:17)(cid:105) . (4.17)We use (A.13) and (A.4) to write ρ A,B ( ω, (cid:126)p ) = 12 π (cid:16) ˜ G A,B ( ω, (cid:126)p ) − ˜ G ∗ A,B ( − ω, − (cid:126)p ) (cid:17) , (4.18)and get the following expressions for the spectral densities from (4.17): ρ T i T ( ω, p ) = 12 ( (cid:15) + P ) p i (cid:20) Θ( ω ) δ (cid:18) ω − p √ d − (cid:19) + Θ( − ω ) δ (cid:18) ω + p √ d − (cid:19)(cid:21) (cid:39) p → ( (cid:15) + P ) p i δ ( ω ) ,ρ T J i ( ω, p ) = ρ T i J ( ω, p ) (cid:39) p → ρp i δ ( ω ) . (4.19)Hence the sum rules (2.4), (2.15), and (2.17) are all satisfied.We end this section with a remark about three-dimensional parity violating superflu-ids [28]. The ground state contains vortices, but it is homogenous and isotropic, hence oursum rules apply. The low lying excitations are phonons (the π excitations that we have beenstudying) and other softer vortex excitations with spin (cid:96) and a gap of ∆ ∼ (cid:96) ( (cid:96) + 1) Q − / above the ground state. (For parity preserving superfluids, the excitations with spin (cid:96) abovethe ground state have a gap ∆ ∼ (cid:96) ( (cid:96) + 1). Vortex excitations in parity preserving fluids onlyappear for (cid:96) > √ Q [16, 17]. A natural question is what saturates the sum rules in parityviolating superfluids. The answer is that it is still the phonons, because [28] found that theOPE coefficients involving the phonon modes, the ground state Ω and T (or other relevantcomponents of T ) stay unchanged compared to the parity preserving one. Thus the softervortex modes should not contribute in the macroscopic limit and should not play any role inthe sum rule. This is corroborated by the fact that there are only finitely many vortices onthe cylinder, hence they disappear in the macroscopic limit. It would be nice to check theseclaims explicitly. – 23 – Free Scalar
In this section, we study the saturation of the sum rules associated with broken boosts andscale invariance described in section 2 for the case of a free relativistic complex scalar field ina finite charge density state in dimension d > ∼ Q rather than ∆ ∼ Q d/ ( d − . This means thatthe effective field theory description of (4.1) is inappropriate, and should be replaced by theapproach [29–34]. In fact sometimes these two types of effective theories are connected [35].In our analysis of the boost symmetry realization on the large charge states in free field theory,we will not use an effective theory approach, rather, we will pursue a more straightforwardanalysis of the correlation functions. The Euclidean two-point correlation function in a theory of a free complex scalar field is givenby (we use the normalization of [36]): G ( x − y ) ≡ (cid:104) ¯ φ ( x ) φ ( y ) (cid:105) = 1( d − s d − | x − y | φ , (5.1)where s d − = 2 π d / Γ( d ) and ∆ φ = d − . We define the lightest operator of charge Q by thefollowing: O Q ≡ (( d − s d − ) Q/ √ Q ! ¯ φ Q . (5.2)Its scaling dimension scales like ∆ Q ∼ Q . Using radial quantization, the ground state ofcharge Q on the cylinder is: | Ω (cid:105) ≡ O Q (0) | (cid:105) . For the purpose of examining the sum rules, we will eventually be interested in the correlatorsinvolving the U (1) current J µ and the stress-tensor T µν . On the plane, they are given by thefollowing expressions: T µν = ∂ µ ¯ φ∂ ν φ + ∂ ν ¯ φ∂ µ φ − g µν g αβ ∂ α ¯ φ∂ β φ + T µν imp , T µν imp = χ (cid:2) g µν ∂ − ∂ µ ∂ ν (cid:3) ¯ φφ, (5.3) J µ = i (cid:0) ∂ µ ¯ φφ − ¯ φ∂ µ φ (cid:1) , (5.4)where χ = d − d − is the coefficient of the curvature coupling term R ¯ φφ (with R the Ricciscalar). In the next subsection, we will argue that the sum rules associated with the brokenboosts and dilatations are not affected by the improvement term, and therefore we can ig-nore it when calculating correlation functions for the purpose of verifying the sum rules (seesubsection 5.2.1 for further details). For details about Wick rotating tensors to Minkowskisignature see footnote 9. – 24 – .2 Sum Rules We would like to check how the sum rules described in section 2 are satisfied for the caseof the free bosonic field theory. For this purpose, one can take the following strategy: first,calculate the correlators which are of interest for the saturation of the sum rules on the plane.Then, map it onto the cylinder S d − × R and take the radius of the S d − sphere to infinity.This last step amounts to taking the infinite volume limit. In practice, taking the macroscopiclimit (as described in subsection 5.2.2) directly on the flat space correlators is equivalent toperforming the procedure described above.In what follows, we will be using x µ , y µ as the coordinates on the plane. On the cylinder,we want to evaluate (cid:104) Ω | J i (0) T ττ ( τ, θ j ) | Ω (cid:105) , where ( τ, θ j ) refer to the coordinate on the cylinder. We can choose a coordinate systemwhere we have only one angle θ . This correlator is related to a four-point correlator on theplane (cid:104) Ω(0) T rr ( z, ¯ z ) J i (1)Ω( ∞ ) (cid:105) by conformal transformation where we have r = e τ/R , z ¯ z = r , z + ¯ z √ z ¯ z = 2 cos θ . (5.5)We follow the usual convention (where x d denotes the Euclidean time on the plane) z = x d + ix , ¯ z = x d − ix . In the macroscopic limit, as we let z → T rr on the plane essentially becomes T dd . Therefore,we will leverage that and on the plane we will calculate (cid:104) Ω(0) T dd ( z, ¯ z ) J i (1)Ω( ∞ ) (cid:105) . We deal with the right hand side of the sum rule in a similar way, going from the plane tothe cylinder.The organization of this subsection is as follows: in subsection 5.2.1, we calculate thecorrelation functions on the plane. In subsection 5.2.2, we define the macroscopic limit asso-ciated with the free bosonic theory. In subsection 5.2.3, we calculate the spectral density andshow that the sum rules are satisfied.
The free field correlators are obtained using Wick contractions. We concentrate on (cid:104)O Q (0) T µν ( x ) J ρ ( y ) O − Q ( ∞ ) (cid:105) . – 25 –ventually, when taking the macroscopic limit, we will choose to work a particular configu-ration where T, J lie in the ( x d , x ) plane as mentioned previously and get the macroscopiccorrelator and then covariantize to obtain the correlator for arbitrary insertion points of T and J .We first evaluate the contribution to (cid:104)O Q (0) T µν ( x ) J ρ ( y ) O − Q ( ∞ ) (cid:105) from the non-improved stress energy tensor. We find that this results in: (cid:104)O Q (0) T µν ( x ) J ρ ( y ) O − Q ( ∞ ) (cid:105)(cid:51) iQ [ H µρ ( x − y ) F ν ( x ) + H νρ ( x − y ) F µ ( x ) − g µν H αρ ( x − y ) F α ( x )] . (5.6)Here the functions F and H are given by: F µ ( x ) ≡ x µ s d − | x | d , H µν ( x ) ≡ s d − | x | d (cid:18) η µν − dx µ x ν x (cid:19) . (5.7)We can further evaluate the contribution to the correlator coming from T imp (cid:104)O Q (0) T µν imp ( x ) J ρ ( y ) O − Q ( ∞ ) (cid:105) = iχQ (cid:0) g µν ∂ − ∂ µ ∂ ν (cid:1) [ F ρ ( x − y ) ( G ( y ) − G ( x )) + F ρ ( y ) G ( x − y ) + ( Q − F ρ ( y ) G ( x )] . (5.8)Here G is given by (5.1). This correlator has a finite macroscopic limit, but at the end of theday this does not contribute to the sum rules. The reason is that the improvement terms dropfrom the expressions for the charges and the sum rules stem from looking at the correlatorsof the charge generator and an operator that plays the role of an order parameter. In whatfollows, we will thus work with the non-improved T µν (unless otherwise stated) in order toshow the saturation of sum rules. The same conclusion holds for the case of the free scalarfield even when considering the sum rule associated with the broken dilatations (2.17).Finally, the right hand side of the sum rule requires us to find (cid:104)O Q (0) J d ( z = 1 , ¯ z = 1) O − Q ( ∞ ) (cid:105) = − iQs d − . (5.9)Once we are equipped with the expression for the free space correlator, we take the macro-scopic limit, which is the subject of the next subsection. By the state/operator correspondence, the operator O Q in equation (5.2) describes a statewith charge density ρ on the cylinder S d − × R . This can be seen by noting that (cid:104) Ω | J | Ω (cid:105) cyl = R − d (cid:104)O Q (0) J (1) O − Q ( ∞ ) (cid:105)(cid:104)O Q (0) O − Q ( ∞ ) (cid:105) = ρ . (5.10)– 26 –n the macroscopic limit, the charge density is kept finite as we take R → ∞ limit. Thescaling dimension associated with the operator O Q satisfy ∆ Q ∼ Q ∼ R d − . The energydensity reads: (cid:15) = ∆ H s d − R d → . (5.11)In the macroscopic limit, we take: x µ = y µ + u µ R , where R → ∞ , and u µ is fixed . (5.12)We also set y µ = δ µ . In terms of z, ¯ z we have: z = 1 + uR , ¯ z = 1 + ¯ uR . (5.13)We note that the convention of taking the macroscopic limit is different compared to [11]. Wemade this choice in order to ensure that there is no parity transformation implemented whiletaking the macroscopic limit. Altogether, the macroscopic limit of the correlation functionswhich are of interest for the saturation of the sum rules is given by: (cid:104) Ω | J i (0) T ττ ( u µ ) | Ω (cid:105) mac ≡ lim R →∞ (cid:32) R − d (cid:104) Ω(0) T dd ( x ) J i (1)Ω( ∞ ) (cid:105)(cid:104) Ω(0)Ω( ∞ ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) x µ = δ µd + u µ /R (cid:33) , (5.14)and (cid:104) Ω | J τ | Ω (cid:105) mac ≡ lim R →∞ (cid:32) R − d (cid:104) Ω(0) J d (1)Ω( ∞ ) (cid:105)(cid:104) Ω(0)Ω( ∞ ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) x µ = δ µd + u µ /R (cid:33) . (5.15)The correlators associated with the numerator on the right-hand side in both equations abovecorrespond to the correlators given by equations (5.6) and (5.9). Both limits are well-behaved.Setting y µ = δ µd , the resulting expressions (in Euclidean signature) in the macroscopic limitafter covariantizing are: (cid:104) Ω | J i (0) T ττ ( u µ ) | Ω (cid:105) mac = − i ρds d − τ u i u d +2 (5.16)and (cid:104) Ω | J τ | Ω (cid:105) mac = − iρ . (5.17)One can see from the last equation that the macroscopic limit has been taken in a way sothat the charge density ρ is constant. In this section we use the somewhat unfortunate notation u = τ + iu , and when we refer to a spacetimepoint, we always write u µ . (In previous sections we instead used u = τ + ix , but here x µ is already reservedfor coordinates on the plane before mapping to the cylinder and taking the macro limit. – 27 – .2.3 Saturation of the Sum Rules The sum rules are inherently statements in Minkowski signature. For the purpose of evaluat-ing the sum rules, we evaluate the following Wightman correlation function by doing properanalytic continuation of (5.16) and recalling T = − T ττ : G T ,J i ( u , (cid:126)u ) ≡ (cid:104) Ω | T ( u , (cid:126)u ) J i (0) | Ω (cid:105) . (5.18)Here we have defined ( u , (cid:126)u ) to denote the Minkowski coordinates. The proper analyticcontinuation is achieved by letting τ = iu + (cid:15) and then take (cid:15) → + . We find that G T ,J i ( u , (cid:126)u ) = − ρds d − u u i u d +2 . (5.19)We have already computed the spectral density from (a very close analog of) this Green’sfunction in section 4, see (4.17) and (4.18). Plugging into those formulas we get ρ T ,J i ( ω, (cid:126)p ) = 12 ρ p i [Θ( ω ) δ ( ω − p ) + Θ( − ω ) δ ( ω + p )] (cid:39) p → ρp i δ ( ω ) , (5.20)Thus, we find that the sum rule (2.15) associated with the broken boosts symmetry is sat-isfied. As for the sum rule associated with the broken scale invariance, we note that in themacroscopic limit, without including the improvement term in the stress-tensor (5.3), thefollowing happens to be true for the free scalar (cid:104) Ω | T i J | Ω (cid:105) = (cid:104) Ω | T J i | Ω (cid:105) , where T µν above refers to the stress-tensor (5.3) without the improvement part T imp . Togetherwith the analysis shown at the beginning of subsection 5.2.1, this immediately tells us thatthe sum rule associated with the broken scale symmetry (2.17) is satisfied. The extra factorof ( d −
1) in the sum rule comes from the contraction of spatial indices in the expression for x i T i . In this subsection, we identify the states that are responsible for saturation of the sum rulesdiscussed in the previous subsection. To this effect, we would like to study a sum rule inanalogy to the T T sum rule that was studied in the superfluid case in section 4. In thefree scalar case, however, the energy density vanishes in the macroscopic limit and the T T spectral density hence vanishes in the p → T J correlator, which reads: ρ T J ( ω, p ) (cid:39) p → − ρp δ (cid:48) ( ω ) . (5.21)– 28 –he above equation follows from combining the conservation of J with the T J i sum rulejust as in section 2.2. In the large Q limit, we can write for τ > (cid:104) Ω | T ( τ ) J (0) | Ω (cid:105) = Q s d − R d − + Qs d − R d − (cid:34)(cid:88) (cid:96) =1 (cid:96) ( d − e − (cid:96) | τ | /R C ( d/ − (cid:96) (cos θ ) (cid:35)(cid:124) (cid:123)(cid:122) (cid:125) terms for the sum rule + · · · . (5.22)The RHS of the above expression gives (cid:104) Ω | J (0) T ( τ ) | Ω (cid:105) for τ <
0. This equation should bethought of as an analogue of (4.8) valid in the superfluid for the free scalar case. Here, thedots indicate terms that are not important for reproducing the sum rule in the macroscopiclimit, i.e to reproduce the p → ω (cid:96) = (cid:96) . The calculationproceeds exactly the same way as in the superfluid case, with the only difference being thathere ω (cid:96) = (cid:96) for all (cid:96) . Once again, the states (cid:96) ∼ pR become important in the infinite volumelimit.Now let us understand in detail how this single Regge trajectory on the cylinder comesabout from the previous calculation of the four-point correlator on the plane via Wick con-traction. Schematically, we have the following type of contractions in the correlator (cid:104) φ Q − φφ | ∂ ¯ φ ( x ) ∂φ ( x ) ¯ φ ( y ) ∂φ ( y ) | ¯ φ ¯ φ ¯ φ Q − (cid:105) ∼ Q , which gives the first line of (5.22), and (cid:104) φ Q − φ | ∂ ¯ φ ( x ) ∂φ ( x ) ∂ ¯ φ ( y ) φ ( y ) | ¯ φ ¯ φ Q − (cid:105) ∼ Q , which gives the second line of (5.22). The contractions that yield result proportional to Q survive in the macroscopic limit and eventually a part of it QR d − ∂ τ G ( τ, θ ) (5.23)is responsible for saturating the sum rules. G ( τ, θ ) is the free scalar propagator on the cylinder: G ( τ, θ ) = 1( d − s d − (cid:16) e − | τ | /R + 2 e −| τ | /R cos θ (cid:17) − d − . (5.24)The underbraced term in (5.22) comes precisely from the expansion of ∂ τ G ( τ, θ ) in terms ofGegenbauer polynomials.As a final remark, let us understand the single Regge trajectory in terms of single or– 29 –ultiparticle states. Of course, it is clear that the relevant states would be labelled by asingle number p ∼ (cid:96)R . The question is whether this corresponds to single or multiparticlestates. To proceed, recall that the relevant contribution arises from contracting one leg of T with the bra (cid:104) Ω | and one leg of J with ket | Ω (cid:105) , and then contracting the leftover leg of T withleft over leg of J . We can think of breaking apart the T J
Wick contraction and inserting acomplete set of states. To make this notion more precise, we start by noting that the state | Ω (cid:105) defines a Bose-Einstein condensate on the cylinder. One can view it as created by thezero angular momentum modes ( a † ) of the scalar field φ , i.e. | Ω (cid:105) ∼ ( a † ) Q | (cid:105) , where | (cid:105) is thetrue vacuum. On the cylinder, we can annihilate states with zero angular momentum andcharge 1 from this condensate, and create a particle with angular momentum (cid:96) and charge 1on top of it. This resembles a bit the case of a particle-hole pair in the theory of Fermi surface(studied in section 6), albeit one important difference: the particle-hole excitations on top ofthe Fermi surface are labelled by two numbers, the angular momentum of the particle andthe angular momentum of the hole, both of which can take non-zero values. In the case of thefree scalar field, however, the hole carries zero angular momentum. Thus, the particle-holepairs are labelled by a single number and form the Regge trajectory as discussed above. Notethat we need a particle-hole pair as opposed to a single particle excitation on the cylindersince (cid:104) Ω | T a † (cid:96) a | Ω (cid:105) is non-zero whereas (cid:104) Ω | T a † (cid:96) | Ω (cid:105) = 0 due to the charge conservation on thecylinder. Nonetheless, in the macroscopic limit, both | Ω (cid:105) and a | Ω (cid:105) define the same statewith equal and finite charge density. Thus, as we take R → ∞ limit, the aforementionedparticle-hole pair on the cylinder (that consists of a hole carrying zero angular momentum)behaves like a single particle excitation, labelled by a single momentum vector (cid:126)p in the infinitevolume theory. This is similar to the behavior described in section 5 of [6], in the context ofthe free massive particle. We study systems of free fermionic field theories at finite charge density ρ and energy density (cid:15) .In subsection 6.1 we address the large charge sector of these models in three and four spacetimedimensions. In subsection 6.2 we consider the relativistic Fermi gas in four dimensions,calculate the spectral density, and show the saturation of the sum rules as described insection 2 for the broken symmetries. In this subsection we consider free fermionic field theories with a global U (1) symmetry in d = 3 and d = 4 dimensions. We denote the lightest operator of charge Q by O Q , its dimensionbeing ∆ Q , and focus on the large Q limit. In addition, we restrict the discussion to cases inwhich the ground state is homogeneous and isotropic. Under this assumption, and in the limitof large charge, the lightest operator of charge Q can be constructed using simple countingarguments. Similar arguments can be found in [37, 38], where the leading order term in theexpansion of ∆ Q was calculated for the d = 3 case. Our main findings in this subsection are:– 30 –. There is no Q term in the expansion for ∆ Q .2. The energy difference between the first excited state and the ground state is O (1):∆ +1 ,Q − ∆ Q = O (1) . (6.1)We emphasize that the analysis described in this subsection holds only under the assumptionof a homogeneous and isotropic ground state. Towards the end of this subsection, we make acomment regarding cases in which this is not the situation.We start with the d = 3 case. The fermionic field ψ is a two component complexGrassmann spinor. Under global U (1) symmetry it transforms as: ψ → e iα ψ. (6.2)As a result of the Fermi statistics, operators of the form ψ n vanish for all n > Therefore, inorder to construct operators of charge Q under the transformation (6.2), one necessarily has toinclude fermions dressed with derivatives, in order to construct a product with more than twofermions. The resulting operator O Q is therefore expected to be of the form ψ ( ∂ψ ) ( ∂ ψ ) ··· .The equation of motion reduces the number of independent physical degrees of freedom.Hence, without loss of generality, we can eliminate ∂ ψ , as it is linearly dependent on theother derivatives of ψ . We define by D n an operator which consists of n spacetime derivativesof the following form: D n ≡ ∂ n ∂ n , where n + n = n. (6.3)Note that n therefore represents the total number of derivatives of type ∂ and ∂ . The oper-ator O Q will consist of multiplication of all the possible terms of the form (cid:81) nk =0 ( ∂ n − k ∂ k ψ ) ,where n takes all integer values between 0 to a maximal value that is determined by therequirement of having a charge Q . Note that for each value of n , there are n + 1 differentterms that contain n derivatives (from either ∂ type, ∂ type, or a mixed combination). Thefermions consist of two-complex components Grassmann fermions, hence each such term canbe taken with a power of two at most. Altogether, for large Q the operator O Q takes thefollowing form: O Q ∝ ( ψ ) ( D ψ ) · · · ( D n max ψ ) n max +2 , (6.4)where n max is determined by the condition that the total number of fermions in the operator O Q is equal to Q : Q = ( n max + 1)( n max + 2) . (6.5)It is important to note that we are not obtaining all integer values of Q through this con-struction, since n max is an integer. We discuss the operators for Q ’s that cannot be producedthrough (6.5) later in this section: they have spin and hence do not correspond to homoge-neous states on the cylinder. We use the notation ψ = (cid:15) ab ψ a ψ b . – 31 –he total number of derivatives n der that appear in the operator (6.4) is given by: n der = 23 n max ( n max + 1)( n max + 2) = 23 Q n max . (6.6)Solving the quadratic equation (6.5) for n max and plugging it into (6.6), one finds the totalnumber of derivatives associated with the operator O Q to be given by: n der = 23 Q − Q + √ Q
12 + O ( Q − ) . (6.7)The dimension ∆ Q associated with the operator O Q is given by ∆ Q = Q ∆ ψ + 1 · n der , where∆ ψ = 1 is the dimension of the fermionic field in d = 3. Thus, we get:∆ Q = 23 Q + 112 (cid:112) Q + O ( Q − ) . (6.8)Next, we turn to consider the case of a Weyl fermion in d = 4 dimensions. The operator O Q will consist of multiplication of all the possible terms of the form (cid:81) nk,l =0 ( ∂ n − k − l ∂ k ∂ l ψ ) ,where again n takes all possible integer values between 0 to a maximal value that dependson Q . For each value of n , there are ( n +2)( n +1)2 different terms, each can be taken with apower of 2 at most. This yields the following expression for the large charge operator O Q : O Q ∝ ( ψ ) ( D ψ ) ( D ψ ) · · · ( D n max ψ ) ( n max +1)( n max +2) . (6.9)From the condition that the operator carries a charge Q under the global U (1) symmetry,one finds the following relation for n max : Q = 13 ( n max + 3) ( n max + 2) ( n max + 1) . (6.10)The total number of derivatives in the operator (6.9) reads: n der = 14 ( n max + 3) ( n max + 2) ( n max + 1) n max = 34 n max Q. (6.11)Using equation (6.10), we get: n der = 3 Q − Q + 14 · Q + O ( Q − ) . (6.12)The dimension ∆ Q is given by ∆ Q = Q ∆ ψ + n der = Q + n der . Using equation (6.12), we The same analysis can be automatically extended to the case of a Dirac fermion. Similar to the d = 3 case, without loss of generality, we can set ∂ ψ as the term which linearly dependson the others using the equations of motion. – 32 –nd the following expression for the scaling dimension ∆ Q :∆ Q = 3 Q + 14 · Q + O ( Q − ) , (6.13)Note that as in the d = 3 case, there is no Q term in the expansion for ∆ Q .Excited states correspond to particle-hole excitations. The lowest order excitation corre-sponds to removing a single fermion from the Fermi surface and replacing it with an excitedfermion, with an energy slightly above the Fermi energy. In the language of operators, thisproblem translates to removing a single fermion with n max derivatives from the operator (6.4)(or (6.9) in the d = 4 case), and replacing it with a fermion that carries n max + 1 derivatives.The resulting operator, which we denote by O +1 ,Q , corresponds to the next-to-lightest oper-ator that carries the same charge Q . Following (6.6) (or (6.11) in the d = 4 case), the totalnumber of derivatives such an operator contains is shifted by +1 compared to the number ofderivatives associated with the operator O Q . Hence, the energy difference between the lowestenergy excitation to the ground state energy satisfies:∆ +1 ,Q − ∆ Q = O (1) . (6.14)Let us make a comment regarding cases in which the ground state is not homogeneousand isotropic. In terms of energy levels on the cylinder S d − × R , this corresponds to casesin which the outermost energy shell is not fully occupied. For simplicity, we focus on d = 3dimensions. ∆ Q is then given by:∆ Q = j max − (cid:88) j = (2 j + 1) ε j + δQε j max , (6.15)where ε j ≡ j + are the energy eigenvalues on the sphere, and the (2 j + 1) factor aboverepresents the degeneracy. δQ represents the particles in the outermost, not necessarily filledenergy shell (as described in figure 1) and it is related to the charge Q by: Q = j max − (cid:88) j = (2 j + 1) + δQ. (6.16)In general, δQ can take any integer value in the range between 0 to 2 j max + 1, wherethe latter corresponds to the case in which the outermost shell is filled and is associated with j max , while the former corresponds to the case in which the outermost shell is the j max − δQ = 0 or δQ = 2 j max + 1 we simplyrecover the homogeneous and isotropic ground state scenario described above: Q is then suchthat equation (6.5) is satisfied with an integer value of n max , as defined above, and using the– 33 – igure 1 : An illustration describing the energy shells. For each j < j max , there are 2 j + 1occupied states in each shell. At the outermost shell, which corresponds to the j max shell,there are δQ occupied states, where 0 ≤ δQ ≤ j max + 1. For δQ = 0 or δQ = 2 j max + 1, theoutermost shell is filled and we recover the homogenous and isotropic ground state with anassociated ∆ Q that is given by equation (6.8).two equations (6.15), (6.16) one can reproduce the result (6.8) for ∆ Q based on the countingarguments. (a) (b) Figure 2 : An illustration describing the behavior of the scaling dimension for a general valueof Q . Figure 2a describes ∆ Q as a function of Q . In figure 2b, the blue dashed line shows∆ Q − Q / as a function of Q , while the orange line describes √ Q . The points in which theblue dashed line meets the orange line correspond to cases in which Q is such that equation(6.5) is satisfied with integer values of n max .In figure 2, we refer to the general Q case. One can see, from figure 2b, that ∆ Q meetsthe value (6.8) that is associated with a homogeneous and isotropic ground state only forspecific values of Q . These values correspond to the cases in which the outermost shell isfilled, as described above. It is also interesting to notice that the fluctuations which appearin the graph of the difference between the ∆ Q to the leading order term in equation (6.8),∆ Q − Q / (described in figure 2b), possess an amplitude of order O ( √ Q ). We learn that– 34 – Q is not an analytic function of Q , if we try to define it for by using all integer Q ’s. Itis however analytic, if we only consider Q ’s that correspond to completely filled shells, andanalytically continue from these points.The same analysis can be extended to other dimensions as well. In appendix B, we discussthe d = 4 case. We find that similar to the d = 3 case, outside the scope of a homogeneousand isotropic ground state there are fluctuations in the difference between ∆ Q to the leadingorder term of (6.13), these fluctuations are of order O ( Q / ) (which is the same order in Q as the next to leading order term in (6.13)), hence ∆ Q is not analytic in Q . In addition, weshow that for a general number of spacetime dimensions d the leading behavior of ∆ Q in thelarge charge limit is given by:∆ Q (cid:39) Q →∞ −(cid:100) d/ (cid:101) d − d Γ( d −
1) [Γ( d ) Q ] d/ ( d − , (6.17)where by (cid:100) d/ (cid:101) we refer to the ceiling function of d/ In [6], non-relativistic Fermi-liquid theories were studied and were shown to satisfy the sumrule associated with the broken boost symmetry by a particle-hole continuum. We extendthis analysis to the case of a relativistic Fermi gas, a state of matter that consists of manynon-interacting fermions. We show that similar to the non-relativistic case, the sum rulesassociated with the broken symmetries are satisfied by particle-hole states.As in [6], we are interested in cases in which the ground state of the theory itself breaksboosts, while preserving spacetime translational and spatial rotational invariance. The groundstate of the theory consists of fermionic particles that occupy all momentum states withmomenta | (cid:126)p | ≤ p F , where p F is the corresponding Fermi momentum. It is therefore taken tobe a tensor product of single-particle momentum eigenstates: | GS (cid:105) ≡ N (cid:89) s (cid:89) | (cid:126)p ≤ p F | (cid:126)p, s (cid:105) , | (cid:126)p, s (cid:105) ≡ c s † (cid:126)p | (cid:105) , (6.18)where c s † (cid:126)p is a fermionic creation operator creating a single particle state of momentum (cid:126)p andspin s . The constant N is a normalization constant and chosen such that (cid:104) GS | GS (cid:105) = 1.From the anti-commutation relations, { c r(cid:126)p , c s † (cid:126)q } = (2 π ) δ (3) ( (cid:126)p − (cid:126)q ) δ rs , it is clear that thefollowing properties of the ground state (6.18) hold: c s † (cid:126)p | GS (cid:105) = 0 , | (cid:126)p | ≤ p F ,c s(cid:126)p | GS (cid:105) = 0 , | (cid:126)p | > p F . (6.19)The first line above simply represents Pauli’s exclusion principle, while the second line statesthat one cannot annihilate a state which is not already contained in the Fermi ground state(6.18). – 35 –article-hole states are defined defined by [6]: | χ (cid:105) = c r † (cid:126)p c r (cid:48) (cid:126)p | FL (cid:105) , (6.20)where p ≤ p F and p > p F . The annihilation operator c r (cid:48) (cid:126)p creates a hole with momentum (cid:126)p and spin r (cid:48) , while the creation operator c s † (cid:126)p creates a fermionic particle with momentum (cid:126)p and spin r . The total momentum associated with the particle-hole state | χ (cid:105) is given by thedifference (cid:126)p = (cid:126)p − (cid:126)p . The energy associated with such a state reads E ( (cid:126)p, (cid:126)p ) = E p − E p ,where E p i = | (cid:126)p i | ≡ p i is the energy of a single particle state with momentum (cid:126)p i .In this subsection, we study the saturation of the sum rules associated with the brokenboosts and dilatation for the case of relativistic free Dirac fermions in d = 4 dimensions inflat spacetime. The action of a free, massless Dirac fermion in d = 4 dimensions is given by: S = (cid:90) d x i ¯ ψ /∂ψ , (6.21)where ¯ ψ = ψ † γ , /∂ = γ µ ∂ µ . The fermion can be written in terms of modes expansion: ψ ( x ) = (cid:90) d p (2 π ) (cid:112) ω p (cid:88) s (cid:16) c s(cid:126)p u s(cid:126)p e − ipx + d s † (cid:126)p v s(cid:126)p e ipx (cid:17) , ¯ ψ ( x ) = (cid:90) d p (2 π ) (cid:112) ω p (cid:88) s (cid:16) d s(cid:126)p ¯ v s(cid:126)p e − ipx + c s † (cid:126)p ¯ u s(cid:126)p e ipx (cid:17) , (6.22)where c s(cid:126)p , c s † (cid:126)p and d s(cid:126)p , d s † (cid:126)p are the creation and annihilation operators of fermionic particlesand anti-particles (respectively). They satisfy the following anti-commutation relations: { c r(cid:126)p , c s † (cid:126)q } = { d r(cid:126)p , d s † (cid:126)q } = (2 π ) δ (3) ( (cid:126)p − (cid:126)q ) δ rs . (6.23)The spinors u s(cid:126)p and v s(cid:126)p represent the solutions of the massless Dirac equation. They satisfy: (cid:88) s u s(cid:126)p ¯ u s(cid:126)p = (cid:88) s v s(cid:126)p ¯ v s(cid:126)p = γ · p , (6.24) (cid:88) s ¯ u s(cid:126)p γ µ u s(cid:126)p = 4 p µ , (6.25) (cid:88) r,r (cid:48) ¯ u r (cid:48) (cid:126)p γ µ u r(cid:126)p ¯ u r(cid:126)p γ ν u r (cid:48) (cid:126)p = 4 ( p µ p ν + p ν p µ − g µν p · p ) . (6.26)– 36 –he stress-tensor is given by: T µν ( x ) = i (cid:2) ¯ ψγ µ ∂ ν ψ − ∂ ν ¯ ψγ µ ψ + ( µ ↔ ν ) (cid:3) − η µν L . (6.27)We consider the system (6.21) in the ground state described by (6.18). The energy density (cid:15) and pressure P are defined as the vacuum expectation values of T and T ij (respectively)with respect to the ground state (6.18) using: (cid:15) ≡ (cid:104) GS | T | GS (cid:105) , P δ ij ≡ (cid:104) GS | T ij | GS (cid:105) . (6.28)Here we have secretly used the fact that ground state is isotropic to pull out the factor δ ij indefining the pressure P . Using (cid:104) GS | c s † (cid:126)p c s (cid:48) (cid:126)q | GS (cid:105) = (2 π ) δ (3) ( (cid:126)p − (cid:126)q ), we find: (cid:15) = − (cid:90) d q (2 π ) q i E q (cid:88) s (cid:0) ¯ u s(cid:126)q γ i u s(cid:126)q (cid:1) = p F π , (6.29) P δ ij = 12 (cid:90) d q (2 π ) E q (cid:88) s (cid:0) q i ¯ u s(cid:126)q γ j u s(cid:126)q + q j ¯ u s(cid:126)q γ i u s(cid:126)q (cid:1) = ⇒ P = 13 p F π . (6.30)In the last step, we have used (6.25) and integrated over a sphere of radius p F . Using theanti-commutation relations (6.23), as well as the definitions of the ground state (6.18) and theparticle-hole state (6.20), one can show that the only nontrivial identity involving creation-anihilation operator is given by: (cid:104) FL | c s † (cid:126)p c s (cid:48) (cid:126)q | χ (cid:105) = (2 π ) δ s (cid:48) r δ r (cid:48) s δ (3) ( (cid:126)q − (cid:126)p ) δ (3) ( (cid:126)p − (cid:126)p ) . (6.31)We define the following matrix elements: T µν ( x ) ≡ (cid:104) FL | T µν ( x ) | χ (cid:105) . (6.32)A straightforward calculation yields: T (0) = 14 1 (cid:112) E p E p ( E p + E p ) ¯ u r (cid:48) (cid:126)p γ u r(cid:126)p , T i (0) = 18 1 (cid:112) E p E p (cid:104) ( E p + E p ) ¯ u r (cid:48) (cid:126)p γ i u r(cid:126)p + (cid:0) p i + p i (cid:1) ¯ u r (cid:48) (cid:126)p γ u r(cid:126)p (cid:105) . (6.33)– 37 –sing the relation (6.26), we get: T T i ∗ ( (cid:126)p , (cid:126)p ) = 18 1 E p E p ( E p + E p ) (cid:0) E p p i + E p p i (cid:1) + 18 1 E p E p ( E p + E p ) (cid:0) p i + p i (cid:1) ( E p E p + (cid:126)p · (cid:126)p ) , T T ∗ ( (cid:126)p , (cid:126)p ) = 14 ( E p + E p ) E p E p ( E p E p + (cid:126)p · (cid:126)p ) , (6.34)where we have defined T µν T ρσ ∗ ( (cid:126)p , (cid:126)p ) ≡ (cid:80) r,r (cid:48) T µν (0) T ρσ ∗ (0). Expanding ( ?? ), (6.34) in small (cid:126)p (where (cid:126)p ≡ (cid:126)p − (cid:126)p ), we get: T T i ∗ ( (cid:126)p, (cid:126)p ) = p p i + ( (cid:126)p · (cid:126)p ) p i p + 2 p p i + O ( p ) , (6.35)and: T T ∗ ( (cid:126)p, (cid:126)p ) = 2 p + 2 (cid:126)p · (cid:126)p + O ( p ) . (6.36)In order to evaluate the spectral density we need to integrate over (cid:126)p . Note that in the limitof small (cid:126)p , the energy associated with the state of momentum (cid:126)p is given by: E ( (cid:126)p, (cid:126)p ) = E p − E p = p cos( θ ) + O ( p ) , (6.37)where θ is the angle between the vectors (cid:126)p and (cid:126)p . (a) (b) Figure 3 : An illustration describing the momentum space notation corresponding to theintegral in the expression for the spectral density (6.38) for a specific configuration of (cid:126)p , (cid:126)p :as p → p → p + F , and p + p cos θ ≈ p F . Thus the variable δp ≡ p − p F ranges from − p cos θ to 0 in p → p F in this specific configuration in momentum space, while figure 3bzooms in on the triangle AOB . – 38 –he spectral function ρ T T µ to the leading order is given by the following : ρ T T µ ( ω, (cid:126)p ) = G T T µ ( ω, (cid:126)p ) ∓ G ∗ T T µ ( − ω, − (cid:126)p ) , (6.38)where the ∓ sign takes the values of − for µ = 0 and + for spatial indices µ = i , and G T T µ ( ω, (cid:126)p ) involves the following integral (see figure 3): G T T µ ( ω, (cid:126)p ) ≡ π ) (cid:90) d p δ ( ω − E ( (cid:126)p, (cid:126)p )) T T µ ∗ ( (cid:126)p, (cid:126)p )= (cid:34) p F (2 π ) (cid:90) π/ dθ sin θ δ ( p cos θ − ω ) T T µ ∗ ( p, p F , cos θ ) (cid:90) − p cos θ dδp (cid:35) + · · · = χ [0 ,p ] ( ω )4 π ωp F p T T µ ∗ ( p, p F , ω/p ) . (6.39)Note that T T µ ∗ ( (cid:126)p, (cid:126)p ) is a function of p, p and the angle between the vectors, i.e. of cos θ .At this point we change the integration variable from p to δp = p − p F and from fig. 3,we read off the limit of integral in p → δp of T T µ ∗ ( (cid:126)p, (cid:126)p ) in the second line, which amounts to replacing p with p F and we made the angle dependence explicit. We also use (6.37) inside the deltafunction, which subsequently sets cos θ = ω/p leading to the term T T µ ∗ ( p, p F , ω/p ) ≡T T µ ∗ ( p, p , cos θ ) | p = p F , cos θ = ω/p . The function χ [0 ,p ] ( ω ) is the characteristic function ofthe interval [0 , p ].We start with the calculation of ρ T T ( ω, p ). From (6.36), we read: T T µ ∗ ( p, p F , ω/p ) =2 p F + 2 p F ω , then: ρ T T ( ω, (cid:126)p ) = χ [ − p,p ] ( ω )2 π ωp F p + · · · (cid:39) p → − ( (cid:15) + p ) p δ (cid:48) ( ω ) , (6.40)as it should be, in accordance with (2.12).Next, we turn to calculate ρ T T i . For this purpose, it is convenient to define (cid:126)p = p ˆ z , andkeep (cid:126)p arbitrary. From (6.35), we read: T T z ∗ ( p, p F , ω/p ) = p F p (1 + ω /p ) + 2 p F ω/p .Plugging it into the expression for the spectral density (6.38), after covariantizing the resultfor ρ T T z , we find the following expression for the spectral density ρ T T i : ρ T T i ( ω, (cid:126)p ) (cid:39) p → χ [ − p,p ] ( ω )2 π ω p F p p i = χ [ − p,p ] ( ω ) 3( (cid:15) + P )2 ω p p i . (6.41)Using the above result, it is straightforward to check the saturation of the sum rules. Onefinds: ∂ρ T T i ∂p j ( ω, (cid:126)p ) = (cid:39) p → ( (cid:15) + P ) δ ( ω ) δ ij , (6.42)– 39 –here we have used χ [ − p,p ] ( ω ) (cid:18) ω p (cid:19) = [Θ( ω + p ) − Θ( ω − p )] (cid:18) ω p (cid:19) (cid:39) p → δ ( ω ) . Therefore, the sum rule associated with the broken boosts (2.4) is satisfied. Using (6.42) oneeasily finds: ∂ρ T T i ∂p i ( ω, (cid:126)p ) (cid:39) p → (cid:15) + P ) δ ( ω ) = 4 (cid:15)δ ( ω ) , (6.43)thus, the sum rule (2.17) associated with the broken dilatations is satisfied (with d = 4).While in this section we have been studying the CFT of a free fermion, the analysis is infact applicable to any (possibly interacting) CFT state around which the effective theory is afree Fermi surface. To our knowledge it is not presently known if a such a free Fermi surfaceis a natural end point under the RG evolution around heavy states. It would be interestingto investigate it along the lines of [39, 40]. We will consider 2d CFTs on the cylinder with circle of radius R : ds = dτ + R ( dθ ) , with θ (cid:39) θ +2 π . The advantage of the 2d setup is that we can construct the correlation functions ofthe EM tensor explicitly and verify the existence of the large volume (macroscopic) limit. Thelow-energy states responsible for the boost Nambu-Goldstone theorem can be also identified.Remarkably, many of the things we find are similar to the superfluid discussion in section 4. TJ correlator
We take an arbitrary state | Ω (cid:105) which corresponds to a spinless highest weight state in theVerma module with dimension ∆. Let Φ be some primary and consider first the four-pointfunction in flat space ( h Φ , ¯ h Φ , h Ω , ¯ h Ω stand for the obvious scaling dimensions and we assume h Ω = ¯ h Ω .) (cid:104) Ω(0) T ( z )Φ(1)Ω( ∞ ) (cid:105) = Ch Ω z + Ch Φ z ( z − , (7.1)where (cid:104) Ω(0)Φ(1)Ω( ∞ ) (cid:105) = C . (7.2)In order to transform this to the cylinder with a circle of radius R we need to plug z = e u/R and take T ( z ) → R z − T ( u ) and Φ(1) → R ∆ Φ Φ(0). Therefore the following Euclidean The transformation of the EM tensor T ( z ) → R z − T ( u ) is missing a constant – the famous ground stateenergy from the Schwartzian. This is unimportant for us because we are considering the correlation functionsin heavy states and hence we drop this constant. – 40 –orrelation function is found on the cylinder with coordinate u = τ + iRθ : (cid:104) Ω | T ( u )Φ(0) | Ω (cid:105) = Ch Ω R Φ + Ch Φ R Φ (cosh( u/R ) − . The constant piece is necessary to account for the propagation of the state | Ω (cid:105) .Let us now investigate the macroscopic limit. For concreteness, we take ∆ Φ = 1 andΦ to be a conserved current (1 ,
0) operator. C/ ( πR ) = ρ is the charge density in the state | Ω (cid:105) while h Ω / ( πR ) = (cid:15) is the energy density. Evidently, to achieve a nontrivial macroscopiclimit, we need to take C to scale as a positive power of R – i.e., we change the state Ω as afunction of R such that CR is finite in the R → ∞ limit. This is the same as having a constantcharge density. Secondly, the constant Ch Ω R is proportional to ρ(cid:15) which we should also holdfixed. Then the macroscopic limit becomes (cid:104) Ω | T ( u ) J (0) | Ω (cid:105) = π ρ(cid:15) + πρu . Similarly, (cid:104) Ω | T ( u ) ¯ J (0) | Ω (cid:105) = π ρ(cid:15) . In components this becomes (assuming C is real) at separated points in Euclidean signa-ture (where T = π ( T tt − T tx ), ¯ T = π ( T tt + T tx ), similary J = π ( J t − J x ), ¯ J = π ( J t + J x )): (cid:104) Ω | T tt ( τ, x ) J t (0) | Ω (cid:105) = ρ(cid:15) + ρ π ( τ − x )( τ + x ) , (cid:104) Ω | T xt ( τ, x ) J x (0) | Ω (cid:105) = ρ π τ − x ( τ + x ) , (cid:104) Ω | T tt ( τ, x ) J x (0) | Ω (cid:105) = − i ρπ τ x ( τ + x ) , (cid:104) Ω | T xt ( τ, x ) J t (0) | Ω (cid:105) = − i ρπ τ x ( τ + x ) . (7.3)We do not write the matrix elements containing T xx as they are the same up to a sign asthose containing T tt .Analytic continuation to Minkowski signature is implemented in the equations above bysetting τ = it ± (cid:15) with (cid:15) → + . The ± correspond to different (Minkowski) time ordering ofthe operators: (cid:104) Ω | T tt ( t, x ) J x (0) | Ω (cid:105) = − ρ π txx + t (cid:18) ∂∂x t + x − i(cid:15) − ∂∂x t − x − i(cid:15) (cid:19) (cid:104) Ω | J x (0) T tt ( t, x ) | Ω (cid:105) = − ρ π txx + t (cid:18) ∂∂x t + x + i(cid:15) − ∂∂x t − x + i(cid:15) (cid:19) (7.4) To conform with both the conventions of the 2d CFT literature and the usual notion of T µν in higherdimensions, we use the definitions T ≡ − π √ g δSδg uu and T µν ≡ √ g δSδg µν . – 41 –ow we use the familiar identity (with (cid:15) → + assumed): u + i(cid:15) − u − i(cid:15) = − πiδ ( u ) to findan expression for the commutator in position space: (cid:104) Ω | [ T tt ( t, x ) , J x (0)] | Ω (cid:105) = − iρ txx + t (cid:0) δ (cid:48) ( t + x ) + δ (cid:48) ( t − x ) (cid:1) . (7.5)The support of the commutators on the light-cone is of course due to the non-dissipativenature of the excitations pertinent to this problem. In frequency and momentum space wefind (cid:104) Ω | [ T tt ( ω, k ) , J x (0)] | Ω (cid:105) = (cid:90) ∞−∞ dt dx e iωt − ikx (cid:104) Ω | [ T tt ( t, x ) , J x (0)] | Ω (cid:105) = − πρk ( δ ( ω + k ) + δ ( ω − k )) . (7.6)The spectral density is given by ρ T tt J x ( ω, k ) = − ρk ( δ ( ω + k ) + δ ( ω − k )) (cid:39) k → − ρkδ ( ω ) , (7.7)as required by the sum rule (2.15) for the boost. (We remind the reader that in our conventions g xx = −
1, hence k x = − k x = − k , which explains the sign in (7.7).)Similarly, one can show that ρ T xt J t ( ω, k ) = − ρk ( δ ( ω + k ) + δ ( ω − k )) (cid:39) k → − ρkδ ( ω ) , (7.8)and verify the sum rule (2.17) for the dilatation. TT correlator
The correlator involving the energy momentum tensor is given by (cid:104)
Ω(0) T ( z ) T (1)Ω( ∞ ) (cid:105) = h z + 2 h Ω z ( z − + c/ z − . (7.9)This leads to an amplitude on the circle of radius R : (cid:104) Ω | T ( u ) T (0) | Ω (cid:105) = h R + h Ω R (cosh( u/R ) −
1) + c R (cosh( u/R ) − . (7.10)The macroscopic limit requires to keep h Ω /R ≡ π(cid:15) fixed and c fixed. Then we obtain in themacroscopic limit: (cid:104) Ω | T ( u ) T (0) | Ω (cid:105) = π (cid:15) + 2 π(cid:15)u + c u . (7.11)Similarly (cid:104) Ω | T ( u ) ¯ T (0) | Ω (cid:105) = π (cid:15) . – 42 –e can now extract the energy-density correlator with itself as before by inserting T = − πT uu = − π ( T ττ − iT τx ) = π ( T tt − T tx ) and expanding to find: (cid:104) Ω | T tt ( τ, x ) T tt (0) | Ω (cid:105) = (cid:15) + 2 (cid:15) ( τ − x ) π ( x + τ ) + c π ( x + τ − x τ )4 π ( x + τ ) (7.12)The commutator is then given by (similar to the calculation of T J commutator) (cid:104) Ω | [ T tt ( t, x ) , T tt (0)] | Ω (cid:105) = i(cid:15) (cid:0) δ (cid:48) ( x + t ) − δ (cid:48) ( x − t ) (cid:1) − i c π (cid:0) δ (cid:48)(cid:48)(cid:48) ( x + t ) − δ (cid:48)(cid:48)(cid:48) ( x − t ) (cid:1) . (7.13)This can be now transformed to frequency and momentum space to find (cid:104) Ω | [ T tt ( ω, k ) , T tt (0)] | Ω (cid:105) = (cid:90) ∞−∞ dt dx e iωt − ikx (cid:104) Ω | [ T tt ( t, x ) , T tt (0)] | Ω (cid:105) = 2 π(cid:15)k ( δ ( ω − k ) − δ ( ω + k )) + cπk / δ ( ω − k ) − δ ( ω + k )) . (7.14)The spectral density is given by ρ T tt T tt (cid:39) k → − (cid:15)k δ (cid:48) ( ω ) − k (cid:2) c δ (cid:48) ( ω ) + 2 (cid:15) δ (cid:48)(cid:48)(cid:48) ( ω ) (cid:3) + · · · . (7.15)The sum rule (2.13) is saturated by the first term, as (cid:15) + P = 2 (cid:15) in 2 dimension.To understand which terms contribute to the sum rule we must study in detail theintermediate states by inserting a complete set of states. It is obvious that the only states (cid:104) Ω (cid:48) | for which (cid:104) Ω (cid:48) | T tt (0) | Ω (cid:105) (cid:54) = 0 are in the Verma module of | Ω (cid:105) (left or right descendantsbut not both). The usual basis of states L n N − N · · · L n − | Ω (cid:105) is inconvenient to use since it is notorthonormal and the matrix elements are difficult to compute. Instead we use the oscillatorbasis of [41], nicely reviewed in [42] (we are using their notations) and we start by computingthe wave function (cid:104) U | T ( u ) | Ω (cid:105) = 1 R (cid:88) n ≤ e − nu/R (cid:104) U | L n | Ω (cid:105) = 1 R (cid:88) n ≤ e − nu/R L − n ·
1= 1 R h − ∞ (cid:88) k =2 e − ku/R k − (cid:88) p =1 p ( k − p ) u p u k − p + 2 ∞ (cid:88) k =1 e − ku/R k ( µk − iλ ) u k . (7.16)where u is our usual coordinate on the cylinder of radius R and u k is an infinite set of variablescollectively denoted U . µ, λ are related to h Ω and c via c = 1 + 24 µ , h Ω = λ + µ . In thisbasis the monomials are orthogonal with norm(1 ,
1) = 1 , ( u k , u l ) = δ k,l S k, , ( u k , u l ) = δ k,l S k, , ( u k u p , u q u l ) = δ k,q δ p,l S k, S l, + δ k,l δ p,q S k, S l, . In the last term we assumed that the four indices are not all the same. We used S k,j =– 43 –2 k ) − j Γ( j + 1). We can unify the third and fourth formulas and write:(1 ,
1) = 1 , ( u k , u l ) = δ k,l k , ( u k u p , u q u l ) = ( δ k,q δ p,l + + δ k,l δ p,q ) 14 kl . Inserting a complete set of states, we find that only need to insert “single particle” and “twoparticle” states: (cid:104) Ω | T ( u ) T (0) | Ω (cid:105) = (cid:104) Ω | T ( u ) | Ω (cid:105)(cid:104) Ω | T (0) | Ω (cid:105) + (cid:88) k =1 k (cid:104) Ω | T ( u ) | u k (cid:105)(cid:104) u k | T (0) | Ω (cid:105) +4 (cid:88) k 1) + 3 R µ u/R ) − . This accounts for almost the whole answer in (7.10), except that the coefficient of the secondterm above is off. (In the full answer it is c/ (8 R ) = (3 µ + 1 / /R instead of 3 µ /R that we obtained from the one particle exchange.) One can check that the difference ismade up by the two-particle states. As we have seen above in (7.11)–(7.15), the first term, R ( λ + µ ) u/R ) − is enough to saturate the boost Nambu-Goldstone theorem in themacroscopic limit. Therefore, the boost Nambu-Goldstone theorem is saturated by one-particle states | Ω (cid:105) and | u k (cid:105) . Note that of these states, only | u (cid:105) is an ordinary conformaldescendant of | Ω (cid:105) (indeed, it is proportional to the action of L − on | Ω (cid:105) ).In frequency space the commutator directly on the cylinder as a result of these one-particle exchanges is (we denote by p = n/R , with n an integer, the momentum on the circleof radius R ) (cid:104) Ω | [ T ( ω, p ) , T (0)] | Ω (cid:105) = 2 nR ( µ n + λ ) δ (cid:16) ω − nR (cid:17) + · · · (7.17)All the states that appear in the intermediate channel have energy and momentum that arerelated by ω n = p n = n/R . This is because they are excitations of the ground state given bythe action of a holomorphic EM tensor.As always, the spectral density, being a sum of delta functions, needs some smearingbefore it can be written in the infinite volume limit. By contrast, correlation functionsfor u (cid:28) R land themselves to a nice macroscopic limit more directly. From the spectraldensity (7.17) we see that any one individual state, even if it has n ∼ R and λ ∼ R asrequired in the macroscopic limit, leads to a vanishing spectral density in the macroscopic– 44 –imit (the contribution is suppressed as 1 /R ). As was discussed in detail in section 3.4, wehave to smear over a band of states centered around n ∼ R and λ ∼ R to recover the correctresult.Note the nice analogy to the superfluid effective theory and mean field theory: in (4.8)and (5.22) we have contributions from what is analogous to one-particle states in the Vermamodule | u n (cid:105) , of which | u (cid:105) is a conformal descendant of the ground state. These are sufficientto reproduce the boost sum rule. The contribution comes from the states with energy n ∼ R while the ground state has energy ∼ R . It is tempting to try and reproduce the above results with an effective theory. One candidateis the superfluid EFT of section 4 specialized to d = 2, which could apply to the lowestenergy state at fixed (large) U (1) charge. In CFTs with a discrete operator spectrum it isa fundamental result [43] that the U (1) symmetry is enhanced to a u (1) × u (1) Kaˇc-Moodyalgebra, and the Energy-Momentum tensor of the full theory decomposes to two separatelyconserved Energy-Momentum tensors.Below we present an effective theory in d = 2 with the following properties: • The U (1) symmetry is of course not spontaneously broken – it is in the usual “log-ordered” phase which is common in d = 2. • It has one compact boson but it allows for an energy-momentum tensor with arbitrary central charge. • The U (1) symmetry can be promoted to a Kaˇc-Moody symmetry only if the centralcharge is c = 1.Besides describing the compact boson large charge limit, which is somewhat trivial, thetheory we present is potentially interesting for situations where the U (1) symmetry of a CFTdoes not enhance to Kaˇc-Moody. Such CFTs must have a continuous spectrum and onemight worry that the large charge limit would be necessarily more complicated.A more realistic and interesting application for our EFT is to describe superfluids witha boundary, or equivalently, the large charge limit of 3d boundary CFT (BCFT). Indeed,the aforementioned shift in the central charge of a single compact boson will be crucial formatching the boundary trace anomaly of the superfluid. For some literature on boundarytrace anomalies see [44–50]. We leave the development of this direction to future work. The simplest such example is a noncompact complex scalar Φ, with the U (1) symmetry rotating aroundthe origin of the target space C and the associated current J µ = i (cid:16) Φ † ∂ µ Φ − Φ ∂ µ Φ † (cid:17) (7.18)which does not get enhanced to a Kaˇc-Moody symmetry. – 45 – .2.1 First Look at the Large Charge Effective Theory Let us start with the EFT (4.1) specialized to d = 2: S = κπ (cid:90) d x ∂ϕ ¯ ∂ϕ , (7.19)where ϕ is a compact scalar ϕ ∼ ϕ + 2 π . We expand the theory around ϕ = − iµτ + π suchthat κµ/ (2 π ) = ρ and κµ / (4 π ) = (cid:15) and try to match the correlators we found from thetheory for the fluctuations: κ/π (cid:82) d x ∂π ¯ ∂π . The stress tensor of the theory S in (7.19) is T = − κ ( ∂ϕ ) , which in terms of the fluctuations takes the form T = κµ iκµ∂π − κ ( ∂π ) . (7.20)The stress tensor two point function can be computed using the propagator (cid:104) ∂π ( u ) ∂π (0) (cid:105) = − κu , (7.21)and we get (cid:104) T ( u ) T (0) (cid:105) = π (cid:15) + 2 π(cid:15)u + 12 u . (7.22)The first two pieces are exactly right but the third one is not. This is because our effectivetheory has central charge 1 instead of c . We have so far merely reproduced the known resultthat the compact boson has central charge 1.If we want to reproduce (7.22) exactly, we face the question of how to make a singlecompact boson ϕ have c (cid:54) = 1, which seems at first sight impossible. A similar in spiritproblem arises in the quantization of the effective string [12] and the solution here is similar.We are allowed to add singular terms to the effective action since we are anyhow expandingaround a nontrivial background: S = βπ (cid:90) d x ∂ ϕ ¯ ∂ ϕ∂ϕ ¯ ∂ϕ . (7.23) β will turn out to be proportional to the shift in central charge. Expanded about the superfluidsolution we find that this leads to a contribution to the effective action of the fluctuations S = − βπµ (cid:90) d x ∂ π ¯ ∂ π + O (1 /µ ) . (7.24)On the one hand, as will be discussed in the next section, S is somewhat trivial; onemanifestation of this is that it does not lead to a modification of the propagator (7.21) up to O (1 /µ ) (this is true since S vanishes on shell). On the other hand, the conformal symmetry By d x we mean dτ dx = dud ¯ u . – 46 –f the action S = S + S is modified: with precision O (1 /µ ) the deformed action S isinvariant under the corrected conformal transformations δϕ ( u, ¯ u ) = λ ( u ) ∂ϕ − β κ ∂ λ ( u ) ∂ϕ , (7.25)(there is an independent antiholomorphic copy of the symmetry) and the stress tensor thatgenerates this symmetry is T = − κ ( ∂ϕ ) + β ∂ϕ ∂ ϕ − ( ∂ ϕ ) ( ∂ϕ ) + O (1 /µ )= κµ iκµ∂π − κ ( ∂π ) + 2 iβµ ∂ π + O (1 /µ ) . (7.26)The stress tensor two-point function can now be straightforwardly computed using Wickcontractions using the propagator (7.21). We get (cid:104) T ( u ) T (0) (cid:105) = (7.22) + 12 βu + O (1 /µ ) . (7.27)If we set β = ( c − / 24, we recover (7.11) to O (1 /µ ). To work out the predictions of theEFT to higher orders, we need a more systematic approach, which we turn to next.We will also see below that there is no holomorphic current, i.e. a weight (1 , 0) primaryoperator of the Virasoro symmetry generated by the deformed T of (7.26) unless β = 0. There is a systematic procedure to construct all terms allowed by symmetry in the effectiveaction. We define the Weyl invariant metric ˆ g µν ≡ g µν | ∂ϕ | , where | ∂ϕ | ≡ − g µν ∂ µ ϕ∂ ν ϕ .Then in the derivative expansion we can write the following terms: S deriv = κπ (cid:90) d x (cid:112) ˆ g (cid:104) α , ˆ R + α , ˆ ∇ µ ˆ ∇ ν ˆ R ∂ µ ϕ∂ ν ϕ + . . . (cid:105) , (7.28)where α k,i is the coefficient of the i th term at k th derivative order. We used the leadingorder equation of motion ∇ ϕ = 0, that in 2 d the Riemann tensor has only one independentcomponent, ˆ R , and that (cid:82) d x √ ˆ g ˆ R is a topological invariant, the Euler characteristic of themanifold to reduce the number of terms in (7.28). There is one famous term that is missingfrom S deriv , since it is not a local Weyl invariant in itself, but transforms with a shift that isa total derivative. The Wess-Zumino term takes the form [51, 52]: S WZ = α (cid:90) d x √ g (cid:2) ( ∂ µ τ ) + τ R (cid:3) ,τ ≡ − log | ∂ϕ | . (7.29)– 47 –ote that τ here is a composite dynamical field (as opposed to a background field, which isthe more common case in the literature).Let us now take g µν to be flat. In complex coordinates the leading order equation ofmotion is ∂ ¯ ∂ϕ = 0, and by dropping terms proportional to it, we realize that S WZ = S from(7.23) with β = πα / 4. So we make contact with the considerations in the previous section.The equation of motion also implies that on shell ϕ ( u, ¯ u ) = χ ( u ) + ¯ χ (¯ u ) and d ˆ s = ˆ g µν dx µ dx ν = − ∂χ ¯ ∂ ¯ χ g µν dx µ dx ν = − ∂χ ¯ ∂ ¯ χ dzd ¯ z = − dχd ¯ χ , (7.30)i.e. ˆ g µν is flat. This then implies that on-shell all curvature invariants that we used to build S deriv in (7.28) vanish. In addition, as we have seen before, S WZ is also a total derivativemodulo the equations of motion. The EFT is nontrivial despite the action having no termswhich are nonzero on-shell. This is because the EM tensor could (and should) receive variouscorrections.Since all the higher terms in the effective action vanish on-shell, we can use a powerfulgeneral result in EFTs that there exists a field redefinition ϕ → ˜ ϕ , that makes the actionquadratic. S = κπ (cid:90) d x ∂ ˜ ϕ ¯ ∂ ˜ ϕ . (7.31)We can then expand around the superfluid background by taking ˜ ϕ = − iµτ + ˜ π . The relationbetween π and ˜ π is (see also [53]): π = ˜ π − βκµ ∂ ¯ ∂ ˜ π + O (1 /µ ) . (7.32)The symmetry transformation of (7.25) and the corresponding stress tensor (7.26) become δ ˜ π ( u, ¯ u ) = − iµ λ ( u ) − iβ κµ ∂ λ ( u )+ λ ( u ) ∂ ˜ π + O (1 /µ ) ,T ( u ) = κµ iκµ∂ ˜ π + 2 iβµ ∂ ˜ π − κ ( ∂ ˜ π ) + O (1 /µ ) , (7.33)where in the first lines we collected terms that shift ˜ π and hence their generators are linearin ˜ π , while the second lines correspond to conformal transformations. (In the expressionof T we dropped terms proportional to the equation of motion.) To O ( µ ) the symmetryis just a combination of the shift and conformal symmetry of the free compact boson, and– 48 –orrespondingly T is just a sum of the conventional current and stress tensor. At higherorders the symmetry and its generator becomes more exotic. To the order we wrote downformulas the computation of the stress tensor correlator is identical in the ˜ π and π variables,but the introduction of ˜ π streamlines the computation at higher orders.In summary, we are faced with the problem of constructing a tensor in a derivativeexpansion from a free scalar governed by the action S = κ/π (cid:82) d x ∂ ˜ π ¯ ∂ ˜ π . The stress tensoris supposed to obey the OPE T ( u ) T (0) = c u + 2 u T (0) + 1 u ∂T (0) + regular , (7.34)which can be achieved order by order in 1 /µ . The remarkable fact is that c is tunable. Wecould have started from this formulation of the problem, but for physical intuition and tomake contact with the literature, we took a detour.Using the Mathematica package OPEdefs [54], by imposing the stress tensor OPE, wefound that the first few orders of the stress tensor are: T ( u ) = κµ iκµ∂ ˜ π − κ ( ∂ ˜ π ) + 1 µ (cid:20) i ( c − ∂ ˜ π + γ ∂ ˜ π∂ ˜ π (cid:21) + 1 µ (cid:20) ( c − (cid:0) ( ∂ ˜ π ) + ∂ ˜ π∂ ˜ π (cid:1) + γ ∂ ˜ π + γ ( ∂ ˜ π ) ∂ ˜ π (cid:21) + O (1 /µ ) , (7.35)where γ i are arbitrary coefficients (they are not quite Wilson coefficients, since, as we re-marked, the action does not admit terms beyond the free kinetic term). We now attempt toconstruct a (1 , 0) holomorphic primary in the 1 /µ expansion by imposing the OPEs: T ( u ) j (0) = 1 u j (0) + 1 u j (cid:48) (0) + regular ,j ( u ) j (0) = κ u + regular . (7.36)We succeed to O ( µ ), but at O (1 /µ ) the most general Ansatz j = κµ iκ∂ ˜ π + 1 µ (cid:2) λ ∂ ˜ π + γ ∂ ˜ π∂ ˜ π (cid:3) + O (1 /µ ) (7.37)leads to a contradiction with (7.36): the OPE with T wants to set λ = − i ( c − / 12, whilethe OPE with j to λ = 0. (Of course these are consistent for c = 1.) This is how the generaltheorem of [43] manifests itself in our concrete computation. The absence of a holomorphiccurrent in effective string theory is due to similar reasons [12]. There is another stress tensor T = κ ( ∂φ ) + V ∂ ˜ φ (that of the linear dilaton CFT) that produces atunable central charge c = 1 + 6 V /κ from a free scalar action for the noncompact scalar φ ; our setup with acompact ϕ is different as a linear dilaton term is forbidden. – 49 – .2.3 Ground State Energy While there are undetermined coefficients in the stress tensor (7.35), it turns out that thevacuum energy is universal in this 2d EFT to all orders in the large charge expansion.The argument consists of two simple steps. First we note that while the conformal trans-formations implemented by T in (7.35) are exotic (as displayed in (7.33)), L + ¯ L generatesordinary time translations on the cylinder. This can be seen either from the transformationlaw it generates for constant λ ( u ) in (7.33), or by noticing that all higher order terms in 1 /µ are total derivatives, T ( u ) = κµ iκµ∂ ˜ π − κ ( ∂ ˜ π ) + ∂ψ ,ψ ≡ µ (cid:20) i ( c − ∂ ˜ π + γ ∂ ˜ π ) (cid:21) + 1 µ (cid:20) ( c − ∂ ˜ π∂ ˜ π + γ ∂ ˜ π + γ ∂ ˜ π ) (cid:21) + O (1 /µ ) . (7.38)The cylinder partition function in the fixed Q sector can be written as Z Q [ β ] = Tr Q fixed (cid:104) e − βR ( L +¯ L − c ) (cid:105) = (cid:90) Q fixed D ˜ ϕ e − S free [ ˜ ϕ ] . (7.39)While our manipulations above were in classical field theory, we can argue that keeping trackof the Jacobian of the field redefinition ϕ → ˜ ϕ would not change the conclusion that thereexists a field redefinition that makes the theory free. The Jacobian is a local Weyl invariantfunctional of ϕ , hence can be exponentiated and written in terms of ˆ g µν . Since we have writtenall these terms in the action (7.28), keeping track of the Jacobian only changes coefficientsin the action. Then there must exist a field redefinition that makes the action free. Thisargument is reminiscent of the classic argument of [55, 56].In the β → ∞ limit from the representation as a trace, we see that we pick up the groundstate energy exp (cid:104) − βR (cid:0) ∆ Q − c (cid:1)(cid:105) . Evaluating the path integral at fixed charge gives Z Q [ β ] → exp (cid:20) − β (cid:18) κµ R − R (cid:19)(cid:21) , (7.40)where the second term is the Casimir energy of the free real scalar. Using the relation (4.5)with c = κ/ (4 π ), we conclude that∆ Q = Q κ + c − 112 + O ( e − Q ) . (7.41)We hope that this prediction can be tested in a situation where our effective theory wouldapply (e.g. in the context of BCFT). – 50 –e note that there exists another method for computing the dimension of the lowestdimension large charge scalar, analogous to the approach of [53]. This method gives the sameresult as (7.41). Acknowledgements We thank Gabriel Cuomo, Bruno Le Floch, Petr Kravchuk and Cumrun Vafa for very usefuldiscussions. ZK, MM, and ARM are supported in part by the Simons Foundation grant488657 (Simons Collaboration on the Non-Perturbative Bootstrap) and the BSF grant no.2018204. The work of ARM was also supported in part by the Zuckerman-CHE STEMLeadership Program. SP acknowledges the support from DOE grant DE-SC0009988. A The Zoo of Correlation Functions Consider the Euclidean time-ordered two point function of two (bosonic) operators A, B : G AB ( τ ) = − (cid:104) T τ A ( τ ) B (0) (cid:105) , (A.1)where the symbol T τ denotes the time-ordering operator, defined such that T τ A ( τ ) B (0) = A ( τ ) B (0) if τ > B (0) A ( τ ) if τ < 0. We define the Fourier transform of this function as G AB ( iν ) = (cid:82) ∞−∞ dτ G AB ( τ ) e iντ for real ν . It is a fundamental result, derived from inserting acomplete set of states, that this can be analytically continued to give a function G AB ( ω ) onthe complex ω plane (first sheet) except for the real ω axis, where it has a branch cut. Thefunction has the following spectral representation: G AB ( ω ) = (cid:90) ∞−∞ dω (cid:48) ρ AB ( ω (cid:48) ) ω − ω (cid:48) . (A.2) ρ AB ( ω ) is called the spectral density associated with the operators A and B .All other Green’s functions can be computed from G AB ( ω ) using an analytic continuation.We are interested in the following Green’s functions: G ( R ) AB ( t ) ≡ − i Θ( t ) (cid:104) [ A ( t ) , B (0)] (cid:105) ,G ( A ) AB ( t ) ≡ i Θ( − t ) (cid:104) [ A ( t ) , B (0)] (cid:105) ,G ( comm ) AB ( t ) ≡ (cid:104) [ A ( t ) , B (0)] (cid:105) = i (cid:16) G ( R ) AB ( t ) − G ( A ) AB ( t ) (cid:17) , (A.3)where G ( R ) AB ( t ) is the retarded function, G ( A ) AB ( t ) is the advanced function and G ( comm ) AB ( t ) isdefined by the third line above as the correlator of the commutator [ A ( t ) , B (0)]. Its Fouriertransform is proportional (up to normalization) to the spectral function ρ AB ( ω ). From theabove expressions, it is clear that the Fourier transformed function G ( R ) AB ( ω ) is analytic in the For A, B fermionic operators, one needs to replace the commutator in (A.3) with anti-commutators. – 51 –pper half plane of complex ω , while G ( A ) AB ( ω ) is analytic in the lower half plane of complex ω . The following relations hold in momentum space: G ( R ) AB ( ω ) = G AB ( ω + i(cid:15) ) G ( A ) AB ( ω ) = G AB ( ω − i(cid:15) ) G ( comm ) AB ( ω ) = i (cid:104) G ( R ) AB ( ω ) − G ( A ) AB ( ω ) (cid:105) = i [ G AB ( ω + i(cid:15) ) − G AB ( ω − i(cid:15) )]= i (cid:90) ∞−∞ dω (cid:48) ρ AB ( ω (cid:48) ) (cid:20) ω − ω (cid:48) + i(cid:15) − ω − ω (cid:48) − i(cid:15) (cid:21) = i (cid:90) ∞−∞ dω (cid:48) ρ AB ( ω (cid:48) ) (cid:2) − πiδ ( ω − ω (cid:48) ) (cid:3) = 2 πρ AB ( ω ) . (A.4)As an example, let us see how these identities work out for the case of the complex freescalar field. For the free scalar, the Euclidean propagator is given by G ¯ φφ ( iν, (cid:126)p ) = − ν + | (cid:126)p | (A.5)One analytically continues to to obtain a function defined on the complex ω plane except ofthe real axis: G ¯ φφ ( ω, (cid:126)p ) = 1 ω − | (cid:126)p | , ω / ∈ R (A.6)Now we can use (A.4) by choosing A = ¯ φ and B = φ : G ( comm )¯ φφ ( ω ) = G ( R )¯ φφ ( ω ) − G ( A )¯ φφ ( ω )= i (cid:2) G ¯ φφ ( ω + i(cid:15), (cid:126)p ) − G ¯ φφ ( ω − i(cid:15), (cid:126)p ) (cid:3) = i (cid:20) ω + i(cid:15) ) − | (cid:126)p | − ω − i(cid:15) ) − | (cid:126)p | (cid:21) = i | (cid:126)p | (cid:20) ω + i(cid:15) ) − | (cid:126)p | − ω + i(cid:15) ) + | (cid:126)p | − ω − i(cid:15) ) − | (cid:126)p | + 1( ω − i(cid:15) ) + | (cid:126)p | (cid:21) = 2 π | (cid:126)p | [ δ ( ω − | (cid:126)p | ) − δ ( ω + | (cid:126)p | )]= (2 π )sgn( ω ) δ ( p µ p µ ) , (A.7)and the spectral function reads: ρ ¯ φφ ( ω ) = sgn( ω ) δ ( p µ p µ ) . (A.8)(A.8) can also be obtained using an equivalent description of analytically continuing thespace-time coordinates. While the above method is used in sec. 2, the following method is– 52 –sed in the rest of the paper.We start with the Wightman correlation function for free scalar in Minkowski spacetime: (cid:104) | ¯ φ ( t, (cid:126)x ) φ (0) | (cid:105) = 1( d − s d − (cid:2) ( (cid:15) + it ) + (cid:126)x · (cid:126)x (cid:3) − ∆ φ , (cid:15) → + , (A.9) (cid:104) | φ (0) ¯ φ ( t, (cid:126)x ) | (cid:105) = 1( d − s d − (cid:2) ( − (cid:15) + it ) + (cid:126)x · (cid:126)x (cid:3) − ∆ φ , (cid:15) → + . (A.10)The Fourier transformed Wightman function G W ¯ φφ ( ω, (cid:126)p ) reads: G W ¯ φφ ( ω, (cid:126)p ) = (cid:90) d d x e i ( ωt − (cid:126)p · (cid:126)x ) (cid:104) | ¯ φ ( t, (cid:126)x ) φ (0) | (cid:105) = 2 π Θ( ω )Θ( p µ ) 4 ∆ ∗ − ∆ φ Γ(∆ ∗ )Γ(∆ φ ) ( p µ ) ∆ φ − ∆ ∗ − Γ(∆ φ − ∆ ∗ ) , (A.11)where we used the notation p µ ≡ p µ p µ = ω − (cid:126)p · (cid:126)p and ∆ ∗ = d − is the unitarity bound, i.e.∆ φ ≥ ∆ ∗ is saturated by the free scalar field. In the limit where ∆ φ → ∆ + ∗ , one gets:Θ( p µ ) ( p µ ) ∆ φ − ∆ ∗ − Γ(∆ φ − ∆ ∗ ) → δ ( p µ ) . Thus, for the free scalar field we find: G W ¯ φφ ( ω, (cid:126)p ) = 2 π Θ( ω ) δ ( p µ ) . (A.12)The commutator correlator is given by: G ( comm )¯ φφ ( ω, (cid:126)p ) = (cid:90) d d xe i ( ωt − (cid:126)p · (cid:126)x ) (cid:104) | (cid:2) ¯ φ ( t, (cid:126)x ) , φ (0) (cid:3) | (cid:105) = G W ¯ φφ ( ω, (cid:126)p ) − [ G W ¯ φφ ( − ω, − (cid:126)p )] ∗ = 2 π sgn( ω ) δ ( p µ ) . (A.13)From above, one can rederive the spectral function (A.8). B ∆ Q in the Free Fermionic Phase In this appendix, we extend the analysis described at the end of subsection 6.1 for the caseof d = 4 dimensions. Considering the d = 4 theory of free fermions on the cylinder S × R ,we have: ∆ Q = j max − (cid:88) j =1 / g ( d =4) j ε ( d =4) j + δQε ( d =4) j max , (B.1)– 53 –here g ( d =4) j represents the degeneracy and it is given by: g ( d =4) j = (cid:18) j + 32 (cid:19) (cid:18) j + 12 (cid:19) , (B.2)and ε ( d =4) j = j + 1 are the energy eigenvalues. The charge Q is given by: Q = j max − (cid:88) j =1 / g ( d =4) j + δQ, (B.3)where δQ represents the amount of occupied states in the outermost energy shell and it cantake values in the range 0 ≤ δQ ≤ (cid:0) j max + (cid:1) (cid:0) j max + (cid:1) . (a) (b) Figure 4 : An illustration describing the behavior of the scaling dimension for a general valueof Q in the d = 4 case. Figure 4a describes ∆ Q as a function of Q . In figure 4b, the bluedashed line corresponds to ∆ Q − / Q / as a function of Q , while the orange line describes Q / · / . The points in which the blue dashed line meets the orange line correspond to cases inwhich Q is such that equation (6.10) is satisfied with integer values of n max .In figure 4, we describe the general Q case. One can see, from figure 4b, that ∆ Q meetsthe value (6.13) that is associated with a homogeneous and isotropic ground state only forspecific values of Q . These values again correspond to cases in which the outermost shell isfilled. Note that similar to the d = 3 case, there are fluctuations in the difference between∆ Q to the leading order term of (6.13), these fluctuations are of order O ( Q / ) (which is thesame order in Q as the next to leading order term in (6.13)), hence ∆ Q is not analytic in Q .Let us make a comment regarding the leading order term in the expansion for ∆ Q for ageneral number of spacetime dimensions d . The energy eigenvalues for a general d are givenby: ε ( d ) j = j + d − . (B.4)In the Q → ∞ limit, the behavior of g ( d ) j is governed by large j , and can be obtained for any– 54 –imension (one can systematically study this using Hilbert series [57, 58]): g ( d ) j (cid:39) j →∞ d − j d − dim / , / , · · · (cid:124) (cid:123)(cid:122) (cid:125) r − = j d − Γ( d − 1) 2 (cid:100) d (cid:101)− , (B.5)where r is the rank of SO ( d ). Since we know that j max ( Q ) is an increasing function of Q and g ( d ) j is increasing with j , the large Q behavior is controlled by large j behavior of g ( d ) j . Atthis point, we can use (B.5) and find: Q (cid:39) j max (cid:88) j =1 / j d − Γ( d − 1) 2 (cid:100) d (cid:101)− (cid:39) j max j d − Γ( d ) 2 (cid:100) d (cid:101)− . (B.6)This can be solved in the leading order to obtain: j max ( Q ) = (cid:104) Γ( d )2 −(cid:100) d (cid:101) Q (cid:105) / ( d − . (B.7)This leads to:∆ Q = j max (cid:88) j (cid:15) ( d ) j g ( d ) j (cid:39) Q →∞ j max ( Q ) (cid:88) j j d − Γ( d − 1) 2 (cid:100) d (cid:101)− = j max ( Q ) d d Γ( d − 1) 2 (cid:100) d (cid:101)− . (B.8)Substituting back the value of j max ( Q ) using (B.7) in the above equation, we find the result(6.17) for the leading order term in ∆ Q in a general number of spacetime dimension. C Contact Terms in Energy-Momentum Correlators In the paper we have studied some general constraints on the spectrum of the theory in thesituation that the boost symmetry is spontaneously broken. This essentially boiled down tostudying the commutators of the conformal charges with the energy-momentum operator.For a conformal Killing vector ξ µ , ∂ µ ξ ν + ∂ ν ξ µ = 2 d η µν ∂ · ξ , (C.1)we can define a corresponding conserved conformal charge Q ξ = (cid:82) d d − x ξ µ T µ , where theintegral is over a space-like slice.The general transformation rule of the Energy-Momentum tensor (assuming d ≥ 3) is[ Q ξ , T ρσ ] = ξ λ ∂ λ T ρσ + ∂ξ µ ∂x ρ T µσ + ∂ξ ν ∂x σ T νρ + d − d ∂ · ξT ρσ . (C.2)The above equation is equivalent to assigning some operator contact terms in the products T µµ ( x ) T ρσ ( x (cid:48) ) and ∂ µ T µν ( x ) T ρσ ( x (cid:48) ). Indeed, since ∂ ν ( ξ µ T µν ) = ξ µ ∂ ν T µν + d ∂ · ξT µµ (where we– 55 –sed the equation satisfied by conformal Killing vectors (C.1)), using the Stokes theorem andthe fact that Q ξ is invariant under small deformations we can compute [ Q ξ , T ρσ ] from thecontact terms in the products T µµ ( x ) T ρσ ( x (cid:48) ) and ∂ µ T µν ( x ) T ρσ ( x (cid:48) ).A very important fact is that the contact terms in the products T µµ ( x ) T ρσ ( x (cid:48) ) and ∂ µ T µν ( x ) T ρσ ( x (cid:48) ) are not fixed uniquely by the commutators [ Q ξ , T ρσ ]. To understand whythis is so, note that we can add contact terms T µν ( x ) T ρσ ( x (cid:48) ) ∼ Aδ d ( x − x (cid:48) ) ( η µν T ρσ + η ρσ T µν ) + Bδ d ( x − x (cid:48) ) ( η µρ T νσ + η µσ T νρ + η νρ T µσ + η νσ T µρ )(C.3)with two arbitrary coefficients A, B . While these contact terms clearly modify the contactterms in the products T µµ ( x ) T ρσ ( x (cid:48) ) and ∂ µ T µν ( x ) T ρσ ( x (cid:48) ), they do not modify the commutators[ Q ξ , T ρσ ]. The latter statement is true as can be verified by a direct computation or byobserving that [ Q ξ , T ρσ ] is a separated-points observable and hence it can only be sensitiveto the contact terms that truly originate from separated points physics.Nevertheless the contact terms in the products T µµ ( x ) T ρσ ( x (cid:48) ) and ∂ µ T µν ( x ) T ρσ ( x (cid:48) ) can becompletely fixed once the Energy-Momentum tensor correlators are precisely defined.A local QFT can be coupled to a space-time metric g and hence we can define thegenerating functional of connected correlators W [ g ]. The functional W [ g ] is not entirelydetermined by the underlying theory. It suffers from an ambiguity by local terms, for instance,in 3+1 dimensions we may add (cid:82) d x √ gR to W [ g ]. ( R is the Ricci scalar.) We can alsoadd higher powers of R suppressed by a cutoff if we are dealing with an effective theory.For reasons that will become clear shortly, such ambiguities should be called ultra-local.These ambiguities will not affect our discussion below or the coefficients A, B in (C.3). TheEnergy-Momentum correlators can be extracted, by definition, via functional derivatives of W [ g ]: (cid:104) T µ ν ( x ) · · · T µ n ν n ( x n ) (cid:105) = ( − n (cid:112) g ( x ) · · · (cid:112) g ( x n ) δδg µ ν ( x ) · · · δδg µ n ν n ( x n ) W . (C.4)Equivalently, the Energy-Momentum correlators can be worked out from the expansion of thegenerating functional around some given background metric g as follows. We define g (cid:48) ρσ = g ρσ + δg ρσ . The first few terms in the expansion are W [ g (cid:48) ] = W [ g ] − (cid:90) d d x (cid:112) g ( x ) δg µν (cid:104) T µν (cid:105) + 18 (cid:90) d d x (cid:90) d d y (cid:112) g ( x ) (cid:112) g ( y ) δg µν δg ρσ (cid:104) T µν T ρσ (cid:105)− · (cid:90) d d x (cid:90) d d y (cid:90) d d z (cid:112) g ( x ) (cid:112) g ( y ) (cid:112) g ( z ) δg µν δg ρσ δg φχ (cid:104) T µν T ρσ T φχ (cid:105) + ... . (C.5)– 56 –he functional derivative is defined in the obvious way δδg µν ( x ) g αβ ( y ) = 12 (cid:16) δ αµ δ βν + δ αν δ βµ (cid:17) δ d ( x − y ) . (C.6)The definition (C.4) is not unique, however, it has the advantage that Bose symmetry ismanifestly obeyed both at separated and coincident points. The definition (C.4) is standardin the literature. See for instance [59, 60] and references therein.We now see why scheme ambiguities such as (cid:82) d x √ gR do not matter for the coefficients A, B in (C.3). Indeed, scheme ambiguities lead to c -number contact terms in T T and notoperator ones. Sometimes operator contact terms are referred to as semi-local while c -numbercontact terms are referred to as ultra-local. (Operator contact terms are referred to as semi-local since in the presence of additional operators there would be delta functions over somepositions but not all positions.)To understand the consequences of diffeomorphism invariance we start with some metricand perform a change of variables ds = g µν ( x ) dx µ dx ν = ∂x µ ∂x (cid:48) ρ ∂x ν ∂x (cid:48) σ g µν ( x ) dx (cid:48) ρ dx (cid:48) σ As usual now we can view the new metric as a function of x (cid:48) , g (cid:48) ρσ ( x (cid:48) ) = ∂x µ ∂x (cid:48) ρ ∂x ν ∂x (cid:48) σ g µν ( x ( x (cid:48) )) . In the absence of gravitational anomalies (which we will assume throughout for simplicity)the metrics g, g (cid:48) give rise to the same W [ g ] = W [ g (cid:48) ]. The invariance of the effective actionunder diffeomorphisms implies a differential equation (cid:90) d d x ( ∇ ρ η σ + ∇ σ η ρ ) δδg ρσ ( x ) W = 0 (C.7)that is valid for any vector field η . (Note that there is no √ g in this integral because thefunctional derivative of W behaves as a density.) Equation (C.7) is of course equivalent tothe conservation equation ∇ ρ (cid:104) T ρσ (cid:105) = 0. Next we must vary this equation δ (cid:90) d d x ( ∇ ρ η σ + ∇ σ η ρ ) δδg ρσ ( x ) W = 0 . A somewhat tedious computation of the variation above at the end of which we set themetric to be flat again, reduces to the precise prescription of contact terms in the prod-– 57 –ct ∂ µ T µν ( x ) T ρσ ( x (cid:48) ): ∂ µ T µν ( x ) T ρσ ( x (cid:48) )= − δ d ( x − x (cid:48) ) ∂ ν T ρσ ( x (cid:48) ) + ∂ xν δ d ( x − x (cid:48) ) T ρσ ( x (cid:48) ) + ∂ xρ δ d ( x − x (cid:48) ) T νσ ( x (cid:48) ) + ∂ xσ δ d ( x − x (cid:48) ) T νρ ( x (cid:48) ) . 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