Ward Identity and Homes’ Law in a Holographic Superconductor with Momentum Relaxation

# Ward Identity and Homes’ Law in a Holographic Superconductor with Momentum Relaxation

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###### Abstract

We study three properties of a holographic superconductor related to conductivities, where momentum relaxation plays an important role. First, we find that there are constraints between electric, thermoelectric and thermal conductivities. The constraints are analytically derived by the Ward identities regarding diffeomorphism from field theory perspective. We confirm them by numerically computing all two-point functions from holographic perspective. Second, we investigate Homes’ law and Uemura’s law for various high-temperature and conventional superconductors. They are empirical and (material independent) universal relations between the superfluid density at zero temperature, the transition temperature, and the electric DC conductivity right above the transition temperature. In our model, it turns out that the Homes’ law does not hold but the Uemura’s law holds at small momentum relaxation related to coherent metal regime. Third, we explicitly show that the DC electric conductivity is finite for a neutral scalar instability while it is infinite for a complex scalar instability. This shows that the neutral scalar instability has nothing to do with superconductivity as expected.

a]Keun-Young Kim, a,b]Kyung Kiu Kim, c]and Miok Park

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Ward Identity and Homes’ Law in a Holographic Superconductor with Momentum Relaxation

• School of Physics and Chemistry, Gwangju Institute of Science and Technology, Gwangju 61005, Korea

• Department of Physics, College of Science, Yonsei University, Seoul 120-749, Korea

• School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea

Keywords: Gauge/Gravity duality, Holographic superconductor, Homes’ law

## 1 Introduction

Holographic methods have provided novel and effective tools to study strongly correlated systems [1, 2, 3, 4] and they have been applied to many condensed matter problems. In particular, holographic understanding of high superconductor is one of the important issues. After the first holographic superconductor model proposed by Hartnoll, Herzog, and Horowitz (HHH)111The HHH model is a class of Einstein-Maxwell-complex scalar action with negative cosmological constant. [5, 6], there have been extensive development and extension of the model. For reviews and references, we refer to [2, 3, 7, 8].

The HHH model is a translationally invariant system with finite charge density. Therefore, it cannot relax momentum and exhibits an infinite electric DC conductivity even in normal phase not only in superconducting phase. To construct more realistic superconductor models, a few methods incorporating momentum relaxation were proposed. One way of including momentum relaxation is to break translational invariance explicitly by imposing inhomogeneous (spatially modulated) boundary conditions on a bulk field [9, 10, 11, 12, 13]. Massive gravity models [14, 15, 16, 17, 18, 19, 20] give some gravitons mass terms, which breaks bulk diffeomorphism and translation invariance in the boundary field theory. Holographic Q-lattice models [21, 22, 23, 24, 25] take advantage of a global symmetry of the bulk theory. For example, a global phase of a complex scalar plays a role of breaking translational invariance. Models with massless scalar fields linear in spatial coordinate [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] utilize the shift symmetry. Some models with a Bianchi VII symmetry are dual to helical lattices [37, 38, 39]. Based on these models, holographic superconductor incorporating momentum relaxation have been developed [40, 41, 42, 43, 44, 45, 46, 47, 48, 49].

In this paper, we study the HHH holographic superconductor model with massless scalar fields linear in spatial coordinate [44], where the strength of momentum relaxation is identified with the proportionality constant to spatial coordinate. The property of the normal phase of this model such as thermodynamics and transport coefficients were studied in [30, 26, 50, 28, 29, 51, 52]. The superconducting phase was analysed in [43, 44]. In particular, optical electric, thermoelectric and thermal conductivities of the model have been extensively studied in [29, 44, 51, 52]. Building on them, we further investigate interesting properties related to conductivities and momentum relaxation. There are three issues that we want to address in this paper: (1) conductivities with a neutral scalar hair instability, (2) Ward identities: constraints between conductivities, (3) Homes’ law and Uemura’s law. We explain each issue in the following.

(1) In a holographic superconductor model of a Einstein-Maxwell-scalar action [6, 7], a superconducting state is characterized by the formation of a complex scalar hair below some critical temperature. In essence, the complex scalar is turned on by coupling between the maxwell field and complex scalar through the covariant derivative. Interestingly, it was also observed [6, 7] that a different mechanism for the instability forming neutral scalar hair222A neutral scalar may arise from a top-down setting [53, 54]. is possible. This instability was not associated with superconductivity because it does not break a symmetry, but at most breaks a symmetry . Therefore, in this system with a neutral scalar hair, it is natural to expect that DC electric conductivity will be finite contrary to the case with a complex scalar hair (superconductor). However, to our knowledge, it has not been checked yet. In the early models without momentum relaxation, this question is not well posed since electric DC conductivity is always infinite due to translation invariance and finite density. In this paper, in a model with momentum relaxation, we show that DC electric conductivity is indeed finite with a neutral scalar hair.

(2) It was shown [3, 55], in normal phase without momentum relaxation, there are two constraints relating three transport coefficients: electric conductivity(), thermoelectric conductivity() and thermal conductivity(). The constraints can be derived by the Ward identity regarding diffeomorphism. Thanks to these two constraints, and can be obtained algebraically once is computed numerically. This is why only is presented in the literature [2]. In our model, there is an extra field, a massless scalar for momentum relaxation, and it turns out there are three Ward identities of six two-point functions: , and and three more two-point functions related to the operator dual to a scalar field. Therefore, the information of alone cannot determine and . If we know three two-point functions then the Ward identities enable us to compute the other three two-point functions. In this paper, following the method in [3], we first derive the Ward identities for two-point functions analytically from field theory perspective. Next, we confirm them numerically from holographic perspective. This confirmation of the Ward identities also demonstrates the faithfulness of our numerical method.

(3) Homes’ law and Uemura’s law are empirical and material independent universal laws for high-temperature and some conventional superconductors [56, 57]. The law states that, for various superconductors, there is a universal material independent relation between the superfluid density () at near zero temperature and the transition temperature () multiplied by the electric DC conductivity () in the normal state right above the transition temperature .

 ~ρs(~T=0)=CσDC(~T% c)~Tc, (1.1)

where , and are scaled to be dimensionless, and is a dimensionless universal constant: or . They are computed in [45] from the experimental data in [56, 57]. For in-plane high superconductors and clean BCS superconductors . For c-axis high superconductors and BCS superconductors in the dirty limit . Notice that momentum relaxation is essential here because without momentum relaxation is infinite. There is another similar universal relation, Uemura’s law, which holds only for underdoped cuprates [56, 57]:

 ~ρs(~T=0)=B~Tc. (1.2)

where is another universal constant. In the context of holography Homes’ law was studied in [58, 45]. It was motivated [58] by holographic bound of the ratio of shear viscosity to entropy density () in strongly correlated plasma [1] and its understanding in terms of quantum criticality [59] or Planckian dissipation [60],where the time scale of dissipation is shortest possible. Since Homes’ law also may arise in systems of Planckian dissipation [60] there is a good chance to find universal physics in condensed matter system as well as in quark-gluon plasma. In [45] Homes’ law was observed in a holographic superconductor model in a helical lattice for some restricted parameter regime of momentum relaxation, while Uemura’s law did not hold in that model. However, physic behind Homes’ law in this model has not been clearly understood yet. For further understanding on Homes’ law, in this paper, we have checked Homes’ law and Uemuras’ law in our holographic superconductor model. We find that Homes’ law does not hold but Uemura’s law holds at small momentum relaxation region, related to coherent metal regime.

This paper is organised as follows. In section 2, we introduce our holographic superconductor model incorporating momentum relaxation by massless real scalar fields. The equilibrium state solutions and the method to compute AC conductivities are briefly reviewed. In section 3, the conductivities with a neutral scalar instability are computed and compared with the ones with a complex hair instability. In section 4, we first derived Ward identities giving constraints between conductivities analytically from field theory perspective. These identities are confirmed numerically by holographic method. In section 5, after analysing conductivities at small frequency, we discuss the Home’s law and Uemura’s law in our model. In section 6 we conclude.

## 2 AC conductivities: holographic model and method

### 2.1 Equilibrium state

In this section we briefly review the holographic superconductor model we study, referring to [26, 29, 51, 52, 61] for more complete and detailed analysis. We consider the action333The complete action includes also the Gibbons Hawking term and some boundary terms for holographic renormalization, which are explained in [26, 29, 51, 52, 61] in more detail.

 S=∫d4x√−g[R−2Λ−14F2−|DΦ|2−m2|Φ|2−122∑I=1(∂ψI)2], (2.1)

where and is the holographic direction. is the Ricci scalar and is the cosmological constant with the AdS radius . We have included the field strength for a gauge field , the complex scalar field with mass , two massless scalar fields, . The covariant derivative is defined by with the charge of the complex scalar field. The action (2.1) yields equations of motion

 RMN−12gMN(R+6−14F2−|DΦ|2−m2|Φ|2−122∑I=1(∂ψI)2) =12FMQFNQ+12(DMΦDNΦ∗+DNΦDMΦ∗)+122∑I=1∂MψI∂NψI, (2.2) ∇MFMN=−iq(Φ∗DNΦ−ΦDNΦ∗), (2.3) (D2−m2)Φ=0,∇2ψI=0, (2.4)

for which we make the following ansatz:

 A=At(r)dt+12B(xdy−ydx),Φ=Φ(r),ψI=(βx,βy), (2.5) ds2=−U(r)e−χ(r)dt2+dr2U(r)+r2(dx2+dy2). (2.6)

In the gauge field, encodes a finite chemical potential or charge density and plays a role of an external magnetic field. is dual to a superconducting phase order parameter, condensate. Near boundary , with two undetermined coefficients and , which are identified with the source and condensate respectively. The dimension of the condensate is related to the bulk mass of the complex scalar by . In this paper, we take and to perform numerical analysis. is introduced to give momentum relaxation effect where is the parameter for the strength of momentum relaxation. For , the model becomes the original holographic superconductor proposed by Hartnoll, Herzog, and Horowitz (HHH) [5, 6].

First, if (no condensate), the solution corresponds to a normal state and its analytic formula is given by

 U(r)=r2−β22−m0r+n2+B24r2,χ(r)=0,At=n(1rh−1r), (2.7)

where is the location of the black brane horizon defined by , , and is interpreted as charge density. It is the dyonic black brane [62] modified by due to  [52]. The thermodynamics and transport coefficients(electric, thermoelectric, and thermal conductivity) of this system was analysed in detail in [52]. In the case without magnetic field, see [29]. Next, if , the solution corresponds to a superconducting state with finite condensate and its analytic formula is not available444A nonzero induces a nonzero , which changes the definition of ‘time’ at the boundary so field theory quantities should be defined accordingly.. For , the solutions are numerically obtained in [6] for and in [44] for . For example we display numerical solutions for some cases in Figure 1, where we set and plot dimensionless quantities scaled by : , , and .

For , due to the generation of vortex our ansatz (2.5) should be modified. In this paper we will not consider this case and refer to [2, 63, 64].

### 2.2 AC conductivities

The purpose of this subsection is to briefly describe the essential points of a method to compute the AC thermo-electric conductivities. For more details and clarification regarding our model at , see [52, 51] for normal phase and [44] for superconducting phase. At see [29] for normal phase.

In order to study transport phenomena holographically, we introduce small bulk fluctuations around the background obtained in the previous subsection. For example, to compute electric, thermoelectric, and thermal conductivities it is enough to consider

 δAi(t,r)=∫∞−∞dω2πe−iωtai(ω,r),δgti(t,r)=∫∞−∞dω2πe−iωtr2hti(ω,r),δψi(t,r)=∫∞−∞dω2πe−iωtξi(ω,r), (2.8)

where for and is enough for thanks to a rotational symmetry in space. For the sake of illustration of our method, we consider the case for  [44] and refer to [52] for . In momentum space, the linearized equations of motion around the background are555For case, the bulk fluctuations to direction should be turned on so the number of equations of motion are doubled too.

 a′′x+(U′U−χ′2)a′x+(ω2U2eχ−2q2Φ2U)ax+r2eχA′tUh′tx=0,h′tx+A′tr2ax+iβUe−χr2ωξ′=0,ξ′′+(U′U−χ′2+2r)ξ′−iβωeχU2htx+ω2eχU2ξ=0. (2.9)

Near boundary () the asymptotic solutions are

 htx=h(0)tx+1r2h(2)tx+1r3h(3)tx+⋯,ax=a(0)x+1ra(1)x+⋯,ξ=ξ(0)+1r2ξ(2)+1r3ξ(3)+⋯, (2.10)

 S(2)ren=12∫dω2π[Ja−ωAab(ω)Jbω+Ja−ωBab(ω)Rbω], (2.11)

where

 Ja=⎛⎜ ⎜⎝a(0)xh(0)txξ(0)⎞⎟ ⎟⎠,Ra=⎛⎜ ⎜⎝a(1)xh(3)txξ(3)⎞⎟ ⎟⎠,A=⎛⎜⎝0 −n00 2U(1)0000⎞⎟⎠,B=⎛⎜⎝1000−30003⎞⎟⎠. (2.12)

Here is the coefficient of when is expanded near boundary and is charge density. The index in and are suppressed.

The remaining task for reading off the retarded Green’s function is to express in terms of . It can be done by the following procedure. First let us denote small fluctuations in momentum space by collectively. i.e.

 Φa=(δai,δhti,δξi). (2.13)

Near black brane horizon (), solutions may be expanded as

 Φa(r)=(r−1)−iω4πT+na(φa+~φa(r−1)+⋯), (2.14)

which corresponds to incoming boundary conditions for the retarded Green’s function [65] and is some integer depending on specific fields, . The leading terms are only free parameters and the higher order coefficients such as are determined by the equations of motion. A general choice of can be written as a linear combination of independent basis , (), i.e. . For example, can be chosen as

 (2.15)

Every yields a solution , which is expanded near boundary as

 Φai(r)  →  Sai+⋯+Oairδa+⋯, (2.16)

where and the leading terms are the sources of -th solutions and are the corresponding operator expectation values. and can be regarded as regular matrices of order , where is for row index and is for column index. A general solution may be constructed from a basis solution set :

 Φa(r)=Φai(r)ci →  Saici+⋯+Oaicirδa+⋯ (2.17) ≡    Ja +⋯+Rarδa  +⋯, (2.18)

with arbitrary constants ’s. For a given , we always can find 666There is one subtlety in our procedure. The matrix of solutions with incoming boundary condition are not invertible and we need to add some constant solutions, which is related to a residual gauge fixing  [51].

 ci=(S−1)iaJa, (2.19)

so the corresponding response may be expressed in terms of the sources

 Ra=Oaici=Oai(S−1)ibJb. (2.20)

With (2.20), the action (2.11) becomes

 S(2)ren=12∫ω≥0dω2π[Ja−ω[Aab(ω)+BacOci(S−1)ib(ω)]Jbω]≡12∫ω≥0dω2π[Ja−ωGabJbω], (2.21)

where the range of is chosen to be positive following the prescription in [65] and the retarded Green’s functions are explicitly denoted as

 Gab≡⎛⎜ ⎜ ⎜⎝GJJGJTGJSGTJGTTGT%$S$GSJGSTGS%$S$⎞⎟ ⎟ ⎟⎠. (2.22)

Finally, the thermo-electric conductivities are related to the retarded Green’s functions as

 (σαT¯αT¯κT)=−iω(GJ%$J$−μGJJ+GJT−μGJJ+GTJ   GTT−GTT(ω=0)−μ(GJT+GTJ−μGJJ)). (2.23)

## 3 Conductivities with a neutral scalar hair instability

By the numerical method reviewed in the previous subsection, the electric, thermoelectric and thermal conductivities of the model (2.1) have been computed in various cases [29, 52, 44]. As an example, in Figure 2, we show the results for [44], which is reproduced here for easy comparison with new results in this paper.

Figure 2 shows AC electric conductivity (), thermoelectric conductivity (), and thermal conductivity () for and at different temperatures. The colors of curves represent the temperature ratio, , where is the critical temperature of metal/superconductor phase transition. for dotted, red, orange,green, and blue curves respectively. In particular, the dotted curve is the case above and the red curve corresponds to the critical temperature. The first row is the real part and the second row is the imaginary part of conductivities.

One feature we want to focus on in Figure 2 is pole in Im[] below the critical temperature. There is no pole above the critical temperature. By the Kramers-Kronig relation, the pole in Im[] implies the existence of the delta function at in Re[]. It means that in superconducting phase the DC conductivity is infinite while in normal phase the DC conductivity is finite due to momentum relaxation.

Unlike the studies in [44], here we set . Between finite and zero , there is a qualitative difference in the instability of a Reissner-Nordstrom AdS black hole [6]. The origin of the superconductor (or superfluidity) instability responsible for the complex scalar hair may be understood as the coupling of the charged scalar to the charge of the black hole through the covariant derivative . In other words, the effective mass of defined by can be compared with the Breitenlohner-Freedman (BF) bound. The BF bound for AdS is . The effective mass may be sufficiently negative near the horizon to destabilize the scalar field since becomes bigger at low temperature777As the temperature of a charged black hole is decreased, develops a double zero at the horizon.. Based on this argument one may expect that when the instability would turn off. However, it turns out that a Reissner-Nordstrom AdS black hole may still be unstable to forming neutral scalar hair, if is a little bit bigger than the BF bound for AdS. It can be understood by the near horizon geometry of an extremal Reissner-Nordstrom AdS black hole. It is AdS R so scalars above the BF bound for AdS may be below the bound for AdS. These two instability conditions can be summrized by one ineqaulity [44]

 m2eff=⎡⎢ ⎢⎣m2−2q21+β2μ2⎤⎥ ⎥⎦⎡⎢ ⎢⎣16⎛⎜ ⎜⎝1+β2μ21+β2μ2⎞⎟ ⎟⎠⎤⎥ ⎥⎦<−14=m2BF, (3.1)

which reproduces the result for in [2]

 m2eff=(m2−2q2)(16)<−14=m2BF. (3.2)

Here, we see can be below the BF bound when .

However, it was discussed in [6, 7] that the instability to forming neutral scalar hair for is not associated with superconductivity because it does not break a symmetry, but at most breaks a symmetry . Therefore, it would be interesting to see if the DC conductivity is infinite or not in the background with a neutral scalar hair.888We thank Sang-Jin Sin for suggesting this. Without momentum relaxation () this question is not well posed since the DC conductivity is always infinite with or without a neutral scalar hair due to translation invariance and finite density. Now we have a model with momentum relaxation (), we can address this issue properly.

To have an instability at we choose the same parameters as Figure 2: and . For , , which is below the BF bound (3.1). Figure 3 shows our numerical results of conductivites, where all temperatures are below : for red, orange,green, and blue curves respectively. A main difference of Figure 3 from Figure 2 is the disappearance of pole in Im[] below . It confirms that the neutral scalar hair has nothing to do with superconductivity as expected.

In Figure 3 it is not easy to see the conductivities in small regime, so we zoom in there in Figure 4. Contrary to the conductivity of normal component in superconducting phase, the DC electric conductivity is not so sensitive to temperature and increases as temperature decreases, which is the property of metal. The thermoelectric and thermal conductivities decrease as temperature increases except a small increase of thermoelectric conductivity near the critical temperature. As a cross check, we have also computed these DC conductivities analytically by using the black hole horizon data according to the method developed in [50]. Since there is no singular behavior in the conductivities as we may regard the real scalar field here as the dilaton in [50] and the conductivities read

 σ=1+4πQ2β2s,α=4πQβ2,¯κ=4πsTβ2, (3.3)

where , and are the entropy density, charge density and temperature in the dual field theory. They are given by , and . The analytic values are designated by the red dots in Figure 4 and they agree to the numerical values very well. For a special case with , in Figure 5, we see that , different from superconducting case ( shown in [44]), but , same as superconducting case.

### 3.1 Superfluid density with a complex scalar hair

We have found that for there is no pole in Im[], of which strength corresponds to superfluid density. To understand it better, we derive an expression for superfluid density for . Let us start with the Maxwell equation,

 ∇MFMN=−iq(Φ∗DNΦ−ΦDNΦ∗)  . (3.4)

Once we assume that all fields depend on and and the fluctuations are allowed only for the -direction, the -component of the Maxwell equation reads

 ∂r√−gFxr=∂t√−gFtx+iq√−g(Φ∗DxΦ−ΦDxΦ∗)  . (3.5)

The integration of (3.5) from horizon to boundary gives the boundary current

 (3.6)

By hydrodynamic expansion for small , it turns out that the first term and the second term goes to zero as

 limr→rh√−gFxr∼O(ω)e−iωt,∫∞rhdr∂t√−gFtx∼O(ω2)e−iωt, (3.7)

while the last term goes to constant. Here we used the expansions of the fileds near horizon

 δAx=(U(r)r2)−iω4πT^ax(ω,r)e−iωt  ,δgtx=r2(U(r)r2)−iω4πT^hx(ω,r)e−iωt+iωr2ζe−iωt  ,δψx=(U(r)r2)−iω4πT^χx(ω,r)e−iωt+βζe−iωt  , (3.8)

where is a constant residual gauge parameter fixing [51], and , and can be expanded as

 ^ax(ω,r)∼A0+A1(r−rh)+⋯,^hx(ω,r)∼H1(r−rh)+⋯,^χx(ω,r)∼~χ0+~χ1(r−rh)+⋯. (3.9)

With the following source-vanishing-boundary conditions

 limr→∞1r2δgtx=limr→∞(^hx−iωζ)e−iωt=0,limr→∞δψx=limr→∞(^χx+βζ)e−iωt=0, (3.10)

except , the current (3.6) can be interpreted as

 Jx=−iωσxx(ω)δAx|r=∞  . (3.11)

Because only the last term of (3.6) contribute to for , as discussed in (3.7), the superfluid density (the strength of the pole of Im[]) is given by

 Ks≡limω→0ωIm[σ]=limω→0JxδAx(r=∞)=limω→0iqδAx(r=∞)∫∞rhdr√−g(Φ∗DxΦ−ΦDxΦ∗)=limω→02q2δAx(r=∞)∫∞rhdre−χ/2|Φ|2(δAx+AtU(r)eχδgtx). (3.12)

This shows how the hairy configuration contributes to . If , vanishes, which confirms our numerical analysis.

## 4 Ward identities: constraints between conductivities

In this section, we first analytically derive the Ward identities regarding diffeomorphism from field theory perspective. It gives constraints between conductivities() and two-point functions related to the operator dual to the real scalar field. Next, these identities are confirmed by computing all two-point functions numerically from holographic perspective.

### 4.1 Analytic derivation: field theory

To derive the Ward identities, we closely follow the procedure in [3]999See [66] for a holographic derivation. and extend the results therein to the case with real and complex scalar fields, which are and in (4.1). Our final results are (4.44)-(4.45) and (4.56)-(4.58).

Let us start with a generating functional for Euclidean time ordered correlation functions:

 eW[¯hαβ,¯Aμ,¯ψI,¯Φ,¯Φ∗]=∫DXe−S[X,¯hαβ,¯Aμ,¯ψI,¯Φ,¯Φ∗], (4.1)

where , , , , and are the non-dynamical external sources of the stress-energy tensor , current , real scalar operators , and complex operators respectively. We define the one-point functions by functional derivatives of :

 (4.2)

where these expectation values are not tensors but tensor densities under diffeomorphism. One more functional derivatives acting on one-pint functions give us Euclidean time ordered two-point functions:

 Gμν,αβE(x,y)≡⟨Pt(Tμν(x)Tαβ(y))⟩=4δ2Wδ¯hμν(x)δ¯hαβ(y), (4.3) Gμν,αE(x,y)≡⟨Pt(Tμν(x)Jα(y))⟩=2δ2Wδ¯hμν(x)δ¯Aα(y), (4.4) Gμν,IE(x,y)≡⟨Pt(Tμν(x)OI(y))⟩=2δ2Wδ¯hμν(x)δ¯ψI(y), (4.5) Gμν,ΦE(x,y)≡⟨Pt(Tμν(x)OΦ(y))⟩=2δ2Wδ¯hμν(x)δ¯Φ∗(y), (4.6) Gμ,αE(x,y)≡⟨Pt(Jμ(x)Jα(y))⟩=δ2Wδ¯Aμ(x)δ¯Aα(y), (4.7) (4.8) Gμ,ΦE(x,y)≡⟨Pt(Jμ(x)OΦ(y))⟩=δ2Wδ¯Aμ(x)δ¯Φ∗(y), (4.9) GJ,IE(x,y)≡⟨Pt(OJ(x)OI(y))⟩=δ2Wδ¯ψJ(x)δ¯ψI(y), (4.10) GJ,ΦE(x,y)≡⟨Pt(OJ(x)OΦ(y))⟩=δ2Wδ¯ψJ(x)δ¯Φ∗(y), (4.11) GΦ,Φ∗E(x,y)≡⟨Pt(OΦ(x)OΦ†(y))⟩=δ2Wδ¯Φ∗(x)δ¯Φ(y). (4.12)

We consider the generating functional invariant under diffeomorphism, , and the variation of the fields can be expressed in terms of a Lie derivative with respect to the vector field

 δ¯hμν=(Lζ¯h)μν=∇μζν+∇νζμ  , (4.13) δ¯Aμ=(Lζ¯A)μ=ζλ∇λ¯Aμ+(∇μζν)¯Aν  , (4.14) δ¯ψI=(Lζ¯ψI)=ζλ∇λψI  , (4.15) δ¯Φ=(Lζ¯Φ)=ζλ∇λ¯Φ  . (4.16)

For diffeomorphism invariance, the variation of should vanish:

 δW=∫d3x(δWδ¯hμν(x)(Lζ¯h)μν+δWδ¯Aμ(x)(LζA)μ+δWδ¯ψI(x)(Lζ¯ψI)+δWδ¯Φ(x)(Lζ¯Φ)+δWδ¯Φ∗(x)(Lζ¯Φ∗))=0, (4.17)

which, after integration by parts, yields the Ward identity for one-point functions regarding diffeomorphism.

 Dμ⟨Tμν⟩+¯Fλ ν⟨Jλ⟩+⟨OI⟩¯hνλ∂λ¯ψI+⟨OΦ⟩¯hνλ∂λ¯Φ∗+⟨OΦ†⟩¯hνλ∂λ¯Φ=0, (4.18)

where .

By taking a derivative of (4.18) with respect to either , , or , we obtain the Ward identities for the two-point functions:

 Dμ⟨Pt(Jα(y)Tμν(x))⟩+¯Fμ ν⟨Pt(Jα(y)Jμ(x))⟩−¯hνβ⟨Jα(x)⟩∂∂xβδ(x−y)+¯hνα⟨Jμ(x)⟩∂∂xμδ(x−y)+¯hμν∂μ¯ψ⟨Pt(Jα(y)OI(x))⟩+2Re{¯hμν∂μ¯Φ∗⟨Pt(Jα(y)OΦ(x))⟩}=0, (4.19)
 Dμ⟨Pt(Tαβ(y)Tμν(x))⟩+δ(x−y