Violation of Thermal Conductivity Bound in Horndeski Theory
aa r X i v : . [ h e p - t h ] O c t Violation of Thermal Conductivity Bound in Horndeski Theory
Hai-Shan Liu
Institute for Advanced Physics & Mathematics,Zhejiang University of Technology, Hangzhou 310023, China
ABSTRACTWe consider charged AdS planar black holes in the four-dimensional Einstein-Maxwell-Horndeski theory with two free axions and analyse the black hole thermodynamics. Wecalculate the holographic thermoelectric conductivities of the dual field theory and deter-mine the ratio of thermal conductivity over the temperature. At low temperature andwith zero electric current, we find that the ratio is proportional to temperature squaredand hence can be arbitrarily small, providing the first example that violates the previouslyconjectured thermal conductivity bound.
Emails: [email protected]
Introduction
The guage/gravity duality has brought many remarkable insights to the dynamics of somestrongly coupled condensed matter systems [1–4]. Several universal bounds of transport co-efficients, including the well-known bound of the ratio of the shear viscosity to the entropydensity [5, 6], have been conjectured based on the holographic “bottom-up” models. How-ever, many of these bounds were later violated through various ways [7, 8]. One bound thatstands out is that for the thermal conductivity κ DC , at zero electric current. This thermalbound was formulated by using a simple holographic model of Einstein-Maxwell-Dilatontheory, together with some momentum dissipation mechanism, which states that [9]: κ DC T ≥ C , (1.1)where C is a non-zero finite constant, as long as the dilaton potential is bounded from below . Remarkbly, this bound has been tested positive against a variety of holographic models,and up until now, there is no counterexample.Motivated by searching for an counterexample, we study the holographic thermoelectricproperties of black holes in Horndeski models. The Horndeski theories were constructed in1970s [11], and they were rediscovered and have received much attention in cosmology [12].A particular property of Horndeski theories is that although their Lagrangians containterms which have more than two derivatives, the equations of motion involve at most twoderivatives on each field. This is analogous to Lovelock gravities [13], and the theories canbe ghost free.Black hole solutions that are asymptotic to (locally) AdS spacetime have been con-structed in Horndeski gravity theories in [14, 15] and the thermodynamics of these AdSblack holes were analyzed in [16, 17]. The stability and causality were studied in [18–20].Holographic properties in Horndeski theories were deeply investigated in [21–25]. Furtherapplications and properties were discussed in [26–28]. It turns out that for general cou-pling constants, there is no holographic a -theorem for Horndeski gravity; however, thereexists a critical point, holographic a -theorem can be established [25]. This implies thatholographic application for Horndenski gravity is only sensible at the critical point of thecoupling constant, where the dual field theory is scale invariant, rather than fully conformalinvariant [25]. It is worth pointing out that for unbounded potential the constant C approaches 1 / ( − V min ), where V min is the minimum of the dilaton potential and approaches −∞ , and thus κ DC /T can be zero at lowtemperatures as is mentioned in [10].
2n this paper, we shall consider D = 4 Einstein-Maxwell-Horndeski gravity with a barecosmological constant, together with two free axions for momentum dissipation. In section 2,we obtain charged AdS black holes at the critical point of the coupling constants. In section3, we analyse the black hole thermodynamics. We study the holographic DC electrothermalconductivities in section 4 and find a counterexample of the bound (1.1). We conclude thepaper in section 5. The Lagrangian of Einstein-Maxwell-Horndeski theory with bare cosmological constant Λand two free axions φ i is L = √ g h κ ( R − − F ) − ( αg µν − γG µν ) ∂ µ χ∂ ν χ −
12 2 X i =1 ( ∂φ i ) i , (2.1)where F = dA and G µν is the Einstein tensor, and γ is the Horndeski coupling. The theoryadmits an AdS planar black hole solution [21]: ds = − h ( r ) dt + dr f ( r ) + r dx i dx i , A = a ( r ) dt , χ = χ ( r ) , φ , = λ x , ,h = g r − κλ βγ + 4 κ − m r + κ (cid:0) g q ( βγ + 4 κ ) − κλ (cid:1) g r ( βγ + 4 κ ) − κ q g r ( βγ + 4 κ ) − κ λ q g r ( βγ + 4 κ ) ,f = 36 g r ( βγ + 4 κ ) (6 g r ( βγ + 4 κ ) − κ ( q + 2 λ r )) h , χ ′ = s βγg r − κ ( q + 2 λ r )6 γg r f ,a = a − qr + qκλ g r ( βγ + 4 κ ) + κq g r ( βγ + 4 κ ) , (2.2)at the critical point of the coupling constantsΛ = − g ( βγ + 2 κ )2 κ , α = 3 γg . (2.3)The solution with λ = 0 was constructed in [15]. The parameters a , q , m are integrationconstants. When the parameters ( m , q, λ ) all vanish, the solution becomes the planar AdSvacuum but with non-vanishing χ , namely χ = χ + p βg − log r . (2.4)3hus the special conformal transformation of the AdS vacuum is not preserved by χ , but thePoincar´e symmetry and the scale invariance survive, leading to a scale invariant quantumfield theory [25]. It is important to note that at the critical condition (2.3) for the couplingconstants, the Horndenski coupling γ does not have zero limit; it is part of the vacuumconstruction analogous to the bare cosmological constant and should not be viewed asa perturbative parameter. In fact for large AdS radius ℓ = 1 /g where the AdS/CFTcorrespondence is applicable, the critical condition (2.3) implies that the Horndeski couplingconstant must be large, which is significant different from the Lovelock theories, or the non-critical cases where γ is fixed and can be arbitrarily small.There is a curvature singularity located at r = r ∗ , where f diverges, and it is determinedby F ( r ∗ ) ≡ g r ∗ ( βγ + 4 κ ) − κ ( q + 2 λ r ∗ ) = 0 . (2.5)In order to describe a black hole, we require that the largest root r ∗ should be inside theevent horizon, that is r ∗ < r , where the radius of the event horizon r is the largest rootof f ( r ) = 0. The Hawking temperature can be obtained through standard method T = 6 g r ( βγ + 4 κ ) − κq − r κλ πr ( βγ + 4 κ ) . (2.6)It is worthwhile pointing out that the requirement r ∗ < r guarantees that the temperatureis positive definite. In particular, the temperature can be arbitrarily close to zero, butcannot reach zero.Though the linearized equations of motion for Horndeski theory involve only two deriva-tives, it is still necessary to insure that the kinetic term of the Horndeski scalar is positiveto avoid any possible ghostlike behavior. The kinetic term for the axion perturbation δχ is P δ ˙ χ δ ˙ χ , where P = 144 γg κr ( βγ + 4 κ ) (cid:0) q + λ r (cid:1) F ( r ) . (2.7)Here F ( r ) is defined in (2.5). In order to avoid any ghostlike excitation, P should benon-negative from horizon to asymptotic infinity. As was discussed above, the requirement r ∗ < r implies that F ( r ) is always positive on and outside the horizon. It follows that thepositivity of P requires that γ >
0, which is consistent with the critical condition (2.3).4
Black hole thermodynamics
The thermodynamics of an AdS black hole in Horndeski theory have been studied exten-sively in [16, 17] with the aid of Wald formalism [29, 30]. Here, we begin by reviewingthe main results. The variation of Lagrangian L gives the equations of motion and totalderivative terms δ L = e.o.m + √ g ∇ µ J µ , (3.1)from which we can define a 1- form J (1) = J µ dx µ and its Hodge dual Θ (3) = − ∗ J (1) .Specializing the variation to be induced by an infinitesimal diffeomorphism δx µ = ξ µ , wecan define a 3-form and show that J (3) ≡ Θ (3) − i ξ ∗ L = e.o.m − d ∗ J (2) , (3.2)where i ξ represents a contraction of ξ µ with the 3-form ∗L and J (2) = dξ . Then, onecan define a 2-form Q (2) ≡ ∗ J (2) , such that J (3) = dQ (2) on shell. The variation of theHamiltonian is given by δH = δQ − i ξ Θ = − r s hf (cid:16) κ + γ f χ ′ (cid:17) δf Ω (2) − r s hf κ (cid:16) fh aδa ′ + aa ′ δfh − f δhh ) (cid:17) Ω (2) . (3.3)Note that here we consider λ and g = 1 /ℓ as thermodynamical constants. We also choose agauge that the electric potential vanishes on the horizon, then the variation of the Hamil-tonian on the horizon is δH + = 16 πT ( κ + γ f χ ′ ) δ ( r , (3.4)where T is given in (2.6), while at the infinity, it is δH ∞ = κµδq − (4 κ + βγ ) δm , (3.5)And Wald showed that the variation of the Hamiltonian vanishes on the Cauchy surface. Fora black hole it implies that δH + + δH ∞ = 0, which gives the first law of the thermodynamics, dM = T dS + Φ e dQ e + Φ + χ dQ + χ . (3.6)5ith M = (4 κ + βγ ) m , Φ e = µ , Q e = κq ,S = (cid:16) κ + γ f χ ′ ) | r (cid:17) πr = 16 πr ( βγ + 4 κ )3 g T , Φ + χ = − γ r T p f χ ′ | r ,Q + χ = 16 π Z r = r q ( ∂χ ) = 16 π p f χ ′ (cid:12)(cid:12)(cid:12) r = r . (3.7)Φ + χ and Q + χ are the scalar potential and charge respectively, (see [17] for more details). Thetemperature is given in (2.6). It is important to note that owing to our parametrization weappear to be able to set γ = 0 in the above thermodynamical quantities; however, as wehave remarked earlier, there is no smooth limit of γ = 0. There are various ways to obtain the holographic DC thermoelectric conductivities [31–34].The key step is to construct the relevant radially conserved current, which serves as abridge connecting the boundary physical properties to the black hole horizon information.We consider the following perturbations around the background solution, δg tx = tU ( r ) + Ψ tx , δg rx = Ψ rx , δA x = tU ( r ) + a x , δφ = Φ( r ) λ . (4.1)The radially conserved electric current can be easily obtained with the help of Maxwellequation ∂ r ( √ gF rx ) = 0, J = κ √ gF rx . (4.2)The radially conserved holographic heat current is more difficult to construct, since itsconservation involves both Einstein and Maxwell equations. Fortunately, a general formulaof deriving the holographic heat current was proposed in [24] by using Noether symmetryfor general classes of gravity theories. Applying this formula in our theory, we obtain theholographic heat current Q = √ g (cid:16) κ (2 ∇ r ξ x + aF rx ) + γ g rr ( ∂ r χ ) ∇ r ξ x (cid:17) , (4.3)6here ξ is the time-like Killing vector ∂ t . We find that the electric and heat current can betime independent by choosing U = − ζh , U = − E + ζa , (4.4)where E and ζ are constants which parameterize the sources for the electric and heatcurrents, respectively. Near the black hole horizon, we impose the ingoing wave condition a ′ x = − E + ζa √ hf + . . . , Ψ tx = Ψ (0) tx − ζh Z √ hf + . . . , (4.5)where Ψ (0) tx is a regular function whose value on the horizon can be determined by thelinearized perturbative equation of motionΨ (0) tx ( r ) = − Eg κqr ( βγ + 4 κ ) + ζ (cid:0) κ (cid:0) q + 2 λ r (cid:1) − g r ( βγ + 4 κ ) (cid:1) g κλ r ( βγ + 4 κ ) . (4.6)Now, we are in a position to evaluate the radially conserved currents on the horizon J = (cid:0) κ + κq λ r (cid:1) E + 4 π q ( βγ + 4 κ )3 g λ r T ζ , Q = 4 π q ( βγ + 4 κ )3 g λ r T E + 16 π ( βγ + 4 κ ) g κλ T ζ . (4.7)The DC conductivity matrix is then given by σ DC = ∂ J ∂E = κ (1 + q r λ ) , α DC = 1 T ∂ J ∂ζ = 4 π q ( βγ + 4 κ )3 g r λ T , ¯ α DC = 1 T ∂ Q ∂E = 4 π q ( βγ + 4 κ )3 g r λ T , ¯ κ DC = 1 T ∂ Q ∂ζ = 16 π ( βγ + 4 κ ) κg λ T . (4.8)It is thus clear that the electric bound, which was proposed in [36], is satisfied σ DC = κ (1 + q r λ ) ≥ . (4.9)The form of the electric conductivity σ DC is the same as that of Einstein-Maxwell case[35] while it is expressed in terms of black hole horizon radius r , as is pointed out in [21].It is easy to check that α DC = ¯ α DC , which means the Onsager relation holds. Furthermore,we also find that the thermal relation ST α DC − Q e ¯ κ DC = 0 holds for this system.7he thermal conductivity at zero electric current is κ DC = 16 π ( βγ + 4 κ ) κg (cid:16) λ + q r (cid:17) T . (4.10)There are two more quantities of interest, the Lorentz ratios of the thermal conductivitiesover the electric conductivities, which are given by¯ L = ¯ κ DC σ DC T = S κ (cid:0) q + λ r (cid:1) , L = κ DC σ DC T = λ r S κ (cid:0) q + λ r (cid:1) . (4.11)Usually, the Lorentz ratio L is a constant, due to the fact that the heat transport and theelectric transport both involve the charge carriers, like free electrons in metal, which is wellknown as the Wiedemann-Franz law. As we can see, this law is violated in our case, whichmay be explained in terms of independent transportation of charge and heat in a stronglycoupled system.It was observed that there is a bound for Lorentz ratio ¯ L in [34]¯ L ≤ S Q e . (4.12)From (4.11), we can see this bound is indeed satisfied in our case¯ L = S κ (cid:0) q + λ r (cid:1) ≤ S Q e . (4.13)Having established that both the electric bound (4.9) and the Lorentz ratio bound (4.13)are satisfied, we now examine the thermal conductivity bound (1.1). It follows from (4.10),we find κ DC T = 16 π ( βγ + 4 κ ) κg (cid:16) λ + q r (cid:17) T . (4.14)As discussed in section 2, although the temperature of the black hole cannot be zero, it canbe arbitrarily close to zero. For sufficiently low temperature, the ratio becomes κ DC T ∼ π ( βγ + 4 κ ) g √ κ p g q ( βγ + 4 κ ) + κλ T + O ( T ) . (4.15)Thus, the ratio can be arbitrarily small at low temperature and it is obvious that thescalar potential of Horndeski theory is bounded, hence our holographic model violates theproposed thermal conductivity bound (1.1). 8 Conclusions
In this paper, we considered the Einstein-Maxwell-Hordeski theory with bare cosmologicalconstant and two free axions. The theory admits analytical charged AdS planar black holes,where the axions span over the two-dimensional plane. It is important to note that the AdSspacetime where the black holes are immersed in is the vacuum solution at the critical pointof the coupling constants. The special conformal transformation of the AdS is broken bythe Horndeski scalar χ , but the Poincar´e and scale invariance survive, giving rise to scaleinvariant quantum field theory at the boundary. We analysed the thermodynamics of theblack hole and calculated the holographic thermoelectric conductivities of the dual fieldtheory. The focus of the paper is to examine various related universal holographic boundsproposed in literature.The Horndeski term doesn’t contribute directly to the electric conductivity, which takesthe same form as that of Einstein-Maxwell theory. Thus the electric conductivity boundis preserved. Furthermore, the Onsager relation α DC = ¯ α DC and the thermal relation ST α DC − Q e ¯ κ DC = 0 are both satisfied. The situation for the thermal conductivity, onthe other hand, is quite different. Although the Lorentz ratio bound is satisfied, the ratioof thermal conductivity at zero electric current over temperature turns out to be, at lowtemperature, constantly proportional to the square of the temperature; therefore, it can bearbitrarily small as the temperature is low, violating the conductivity bound (1.1). Thisrare counterexample indicates that an underlying principal is needed to understand thecondition when the bound is valid. It is of interest to investigate whether the breaking ofthe conformal symmetry to sale invariance is the culprit for the bound violation. Acknowledgement
We are grateful to anonymous referees for their useful suggestions which improve the paper alot. H-S.L. is supported in part by NSFC grants No. 11305140, No. 11375153, No. 11475148and No. 11675144.
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