DC Conductivities with Momentum Dissipation in Horndeski Theories
MMI-TH-1744
DC Conductivities with Momentum Dissipation in Horndeski Theories
Wei-Jian Jiang , Hai-Shan Liu , , H. L¨u and C.N. Pope , Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310058, China Institute for Advanced Physics & Mathematics,Zhejiang University of Technology, Hangzhou 310023, China George P. & Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,Texas A&M University, College Station, TX 77843, USA Department of Physics, Beijing Normal University, Beijing 100875, China DAMTP, Centre for Mathematical Sciences, Cambridge University,Wilberforce Road, Cambridge CB3 OWA, UK
ABSTRACTIn this paper, we consider two four-dimensional Horndeski-type gravity theories withscalar fields that give rise to solutions with momentum dissipation in the dual boundarytheories. Firstly, we study Einstein-Maxwell theory with a Horndeski axion term and twoadditional free axions which are responsible for momentum dissipation. We construct staticelectrically charged AdS planar black hole solutions in this theory and calculate analyticallythe holographic DC conductivity of the dual field theory. We then generalize the resultsto include magnetic charge in the black hole solution. Secondly, we analyze Einstein-Maxwell theory with two Horndeski axions which are used for momentum dissipation. Weobtain AdS planar black hole solutions in the theory and we calculate the holographic DCconductivity of the dual field theory. The theory has a critical point α + γ Λ = 0, beyondwhich the kinetic terms of the Horndeski axions become ghost-like. The conductivity asa function of temperature behaves qualitatively like that of a conductor below the criticalpoint, becoming semiconductor-like at the critical point. Beyond the critical point, theghost-like nature of the Horndeski fields is associated with the onset of unphysical singularor negative conductivities. Some further generalisations of the above theories are consideredalso.
Emails: [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] A p r ontents Gauge/Gravity duality has served as a powerful tool in understanding the phenomena ofstrongly coupled systems in condensed matter physics [1–3]. Especially, much attention hasbeen paid to the holographic description of systems with momentum relaxation. Such sys-tems with broken translational symmetry are needed in order to give a realistic descriptionof materials in many condensed matter systems.Since momentum is conserved in a system with translational symmetry, a constantelectric field can generate a charge current without current dissipation in the presence ofnon-zero charge density. Thus, the conductivity of the system would become divergentat zero frequency. In more realistic condensed matter materials, the momentum is notconserved due to impurities or a lattice structure, thus leading to a finite DC conductivity.In the context of holography, there are various ways to achieve momentum dissipation,such as periodic potentials, lattices and breaking diffeomorphism invariance [4–18]. Amongthese, the model in [12] is particular simple. It comprises an Einstein-Maxwell theorytogether with a set of minimally-coupled massless scalar fields that have linear dependenceon the boundary coordinates. These axionic scalars preserve the homogeneity of the bulk2tress tensor, since they have no mass terms or interactions that would break translationalinvariance.In this paper, we shall generalise the models with momentum dissipation that wereconstructed in [12] by introducing non-minimal Horndeski type couplings of some of thescalar fields to gravity. The Horndeski theories were first constructed in the 1970s [19],and they have received much attention recently through their application to cosmology inGalileon theories (see, for example, [20]). A characteristic feature of Horndeski theoriesis that although terms in their Lagrangians involve more than two derivatives, the fieldequations and the energy-momentum tensor involve no higher than second derivatives ofthe fields. This is analogous to the situation in Lovelock gravities [21].Specifically, we shall generalise the model in [12] in two parallel ways. Firstly, in section2, we shall consider a Horndeski extension of an Einstein-Maxwell plus scalar theory in whichtwo minimally-coupled axions that provide the momentum dissipation are supplemented bya third axion with a non-minimal Horndeski coupling. Although this axion has a significanteffect in terms of modifying the geometrical structure of the black hole background, we findthat the DC conductivity in the boundary theory is essentially unaltered, at least if oneexpresses the result as a function of the black hole horizon radius. In section 3, we shallconsider instead an Einstein-Maxwell theory with Horndeski couplings to the two axionsthat provide the momentum dissipation. Here, we find that the effects of the non-minimalHorndeski couplings are much more substantial, and in fact as the strength of the non-minimal term is increased to a critical value, the qualitative behaviour of the conductivitiesas a function of temperature changes. Below the critical coupling the high-temperaturebehaviour is similar to that of a metal, whilst at the critical coupling the behaviour becomesmore like that of a semiconductor. We summarize our results in section 4. In appendix, weextend the theories and solutions that we studied in the main text to arbitrary spacetimedimensions.
In this section, we consider AdS planar black holes of Horndeski theory in four dimensions.The solutions have been constructed in [22, 23], and the thermodynamics have been studiedin [24, 25]. In these solutions, the Horndeski axion χ depends on the radial coordinate. In3rder to achieve momentum dissipation, we include two additional free axions φ i as in [12]: I = 116 π (cid:90) d x √− g L ,L = κ (cid:16) R − − F −
12 2 (cid:88) i =1 ( ∂φ i ) (cid:17) − ( αg µν − γG µν ) ∂ µ χ ∂ ν χ , (2.1)where κ , α , γ are coupling constants, G µν ≡ R µν − Rg µν is the Einstein tensor, and F = dA is the electromagnetic field strength. The equations of motion with respect to the metric g µν , the Maxwell potential A µ , the Horndeski scalar χ and the axions φ i are given by κ ( G µν + Λ g µν − F µν + F g µν ) − κ ∂ µ φ ∂ ν φ + ∂ µ φ ∂ ν φ ) + κ (cid:0) ( ∂φ ) + ( ∂φ ) (cid:1) g µν − α (cid:16) ∂ µ χ∂ ν χ − g µν ( ∂χ ) (cid:17) − γ (cid:16) ∂ µ χ∂ ν χR − ∂ ρ χ ∂ ( µ χ R ν ) ρ − ∂ ρ χ∂ σ χ R µρν σ − ( ∇ µ ∇ ρ χ )( ∇ ν ∇ ρ χ ) + ( ∇ µ ∇ ν χ ) (cid:3) χ + G µν ( ∂χ ) − g µν (cid:2) − ( ∇ ρ ∇ σ χ )( ∇ ρ ∇ σ χ ) + ( (cid:3) χ ) − ∂ ρ χ∂ σ χ R ρσ (cid:3)(cid:17) = 0 , ∇ µ (cid:0) ( αg µν − γG µν ) ∇ ν χ (cid:1) = 0 , ∇ ν F νµ = 0 , (cid:3) φ i = 0 . (2.2)One of the remarkable properties of a Horndeski theory is that each field has no higher thansecond-derivative terms in the equations of motion, even though the Lagrangian involveslarger numbers of derivatives (up to four derivatives, in our case). Although terms quadraticin second-derivatives are present, linearised perturbations around a background will involveat most second-order linear differential equations, and thus can be ghost free.We are interested in static planar black hole solutions in this paper. In this section, weshall take the Horndeski axion χ to depend only on the radial coordinate, whilst the twoadditional axions φ i span the planar directions: ds = − h ( r ) dt + dr f ( r ) + r dx i dx i ,χ = χ ( r ) , A = a ( r ) dt , φ = λx , φ = λx , (2.3)where λ is a constant. The Maxwell equation can be used to express the electrostaticpotential in terms of the metric functions, as a (cid:48) = qr (cid:115) hf , (2.4)where q is an integration constant, parameterising the electric charge, and a prime denotes4 derivative with respect to r . The equation of motion for the Horndeski scalar χ can thenbe written as (cid:16)(cid:114) fh (cid:16) γ (cid:0) rf h (cid:48) + f h (cid:1) − αr h (cid:17) χ (cid:48) (cid:17) (cid:48) = 0 . (2.5)Following [22, 23], we focus on the special class of solutions obtained by taking γf (cid:0) rh (cid:48) + h (cid:1) − αr h = 0 . (2.6)With this, we can solve the Einstein equations and obtain the black hole solution a = a − qr + κq g (4 κ + βγ ) r + κqλ κ + βγ ) g r ,χ (cid:48) = (cid:115) β − κ ( q + 2 λ r )6 γg r √ f , f = 36(4 κ + βγ ) g r (cid:0) κ ( q + 2 λ r ) − κ + βγ ) g r (cid:1) h ,h = g r − µr + κq (4 κ + βγ ) r − κ q κ + βγ ) g r − κλ κ + βγ − κ λ g r (4 κ + βγ ) − κ q λ κ + βγ ) g r , (2.7)where the parameters are such that α = 3 g γ , Λ = − g (cid:0) βγ κ (cid:1) . (2.8)The solution has non-trivial integration constants µ , q and λ , together with a pure gaugeparameter a . The Hawking temperature can be calculated by standard methods, and isgiven by T = 6 g r ( βγ + 4 κ ) − κ ( q + 2 λ r )8 πr ( βγ + 4 κ ) , (2.9)where r is the radius of event horizon, which is the largest root of h ( r ) = 0. There are many ways to compute the holographic conductivities. For the DC conductivity,a simple method makes us of the “membrane paradigm” [17, 26–31]. The key point isto construct a radially conserved current, which allows one to read off the holographicboundary properties in terms of the black hole horizon data. Here, we shall follow theprocedure described in [28]. 5e consider perturbations around the black hole solutions, of the form δg tx = r ψ tx , δg rx = r ψ rx , δA x = − Et + a x , δφ = Φ λ . (2.10)The equation of motion for the vector field ∂ r ( √ gF rx ) = 0 implies that we can define aradially-conserved current J = κ √ gF rx . (2.11)Explicitly, this current is given by J = κ (cid:16) f a (cid:48) x (cid:0) − g r ( βγ + 4 κ ) + κq + 2 κλ r (cid:1) − g qr ψ tx ( βγ + 4 κ ) (cid:17) g r ( βγ + 4 κ ) , (2.12)and it obeys ∂J/∂r = 0.The Einstein equations imply f (cid:0) Φ (cid:48) − λ ψ rx (cid:1) (cid:0) g r ( βγ + 4 κ ) − κ ( q + 2 λ r ) (cid:1) g r ( βγ + 4 κ ) = Eq ,f (cid:0) κqa (cid:48) x + r ( βγ + 4 κ ) (cid:0) rψ (cid:48)(cid:48) tx + 4 ψ (cid:48) tx (cid:1)(cid:1) = 24 κg λ r ( βγ + 4 κ ) ψ tx (6 g r ( βγ + 4 κ ) − κ ( q + 2 λ r )) . (2.13)Regularity on the horizon requires that a (cid:48) x = − E √ hf + O (1) . (2.14)The last equation in (2.13) shows that near horizon, ψ tx = − Eqλ r + O ( r − r ) . (2.15)With these, we can evaluate the current on the horizon, finding J = κ (1 + q λ r ) E , (2.16)and hence the conductivity is given by σ = ∂J∂E = κ (1 + q λ r ) . (2.17) Note that the perturbation ψ rx is non-dynamical, and could in fact be removed by a coordinate trans-formation. We choose to keep it here in order to make the presentation parallel with the one we shall givebelow when a magnetic field is turned on, since in that case one cannot remove the analogous perturbationsby means of a coordinate transformation. χ does not explicitly contribute to the conductivitywhen σ is expressed in terms of r , and hence the result (2.17) is the same as in [12]. Ofcourse, the Horndeski term modifies the relation between the temperature and r , and soin the σ ( T ) relation the Horndeski term has non-trivial effects. However at large T (corre-sponding to large r , with T ∼ g r / (4 π )), the σ ( T ) dependence approaches that obtainedin [12]. We can obtain a more general class of dyonic black hole solutions, by extending the ansatzfor the vector potential in (2.3) to include a magnetic term: A = adt + B x dx − x dx ) . (2.18)We find the dyonic black hole solution is given by φ = λx , φ = λx .a = a − qr + κq ( B + q )30 g (4 κ + βγ ) r + κqλ κ + βγ ) g r ,χ (cid:48) = (cid:115) β − κ ( B + q + 2 λ r )6 γg r √ f ,f = 36 g r ( βγ + 4 κ ) (cid:0) κ ( B + q + 2 λ r ) − g r ( βγ + 4 κ ) (cid:1) h ,h = g r − µr + κ ( B + q )(4 κ + βγ ) r − κ ( B + q ) κ + βγ ) g r − κλ κ + βγ − κ λ g r (4 κ + βγ ) − κ λ ( B + q )9(4 κ + βγ ) g r . (2.19)It is interesting to note that this dyonic solution is rather simply related to the previouspurely electric solution by means of a replacement in which the quadratic powers of q in(2.7) are sent to q + B , while the linear powers of q are left unchanged, in the sense thatone makes the formal replacements q → q , q → q + B , q → q ( q + B ) . (2.20)7he Hawking temperature for the dyonic black hole is given by T = 6 g r ( βγ + 4 κ ) − κ ( B + q + 2 λ r )8 πr ( βγ + 4 κ ) . (2.21)We are now in a position to calculate the DC conductivity in the dyonic black holebackground. In this case, we turn on perturbations in both the spatial boundary directions x i , δg tx = r ψ t , δg rx = r ψ r , δg tx = r ψ t , δg rx = r ψ r ,δA x = − E t + a , δA x = − E t + a , δφ = Φ λ , δφ = Φ λ . (2.22)Following similar methods to those we used in the previous subsection, we construct aradially-conserved 2-component current J i = κ √ gF rx i . (2.23)The regularity conditions on the horizon are a (cid:48) = − E √ hf + O (1) , a (cid:48) = − E √ hf + O (1) . (2.24)The currents can be evaluated on the horizon, and we define the conductivity matrix by σ ij = ∂J i ∂E j , with , { i, j = 1 , } . (2.25)Explicitly, the conductivity matrix elements are given by σ = σ = λ r (cid:0) B + q + λ r (cid:1) B + B (cid:0) q + 2 λ r (cid:1) + λ r ,σ = − σ = Bq (cid:0) B + q + 2 λ r (cid:1) B + B (cid:0) q + 2 λ r (cid:1) + λ r . (2.26)The Hall angle is defined (for small angles) by θ H = σ σ = Bq (cid:0) B + q + 2 λ r (cid:1) λ r (cid:0) B + q + λ r (cid:1) . (2.27)As in the purely electrically-charged black holes, the inclusion of the Horndeski scalar χ doesnot modify these transport quantities when they are expressed in terms of the r variable.In particular the Hall angle goes to zero at high temperature, as θ H ∼ /T .8 Momentum dissipation using Horndeski axions
In this section, we consider a system in which the axionic scalars that provide the momentumdissipation are themselves taken to have Horndeski couplings rather than minimal couplingsto gravity. The Lagrangian describing the theory is given by L = κ ( R − − F ) − ( αg µν − γG µν ) (cid:88) i =1 ∂ µ χ i ∂ ν χ i . (3.1)We shall assume that α is positive, and so for γ = 0 we recover the Einstein-Maxwell theorywith two free axions, proposed in [12]. The equations of motion are κ ( G µν + Λ g µν − F µν + 18 F g µν ) − (cid:88) i α (cid:16) ∂ µ χ i ∂ ν χ i − g µν ( ∂χ i ) (cid:17) − (cid:88) i γ (cid:16) ∂ µ χ i ∂ ν χ i R − ∂ ρ χ i ∂ ( µ χ i R ν ) ρ − ∂ ρ χ i ∂ σ χ i R µρν σ − ( ∇ µ ∇ ρ χ i )( ∇ ν ∇ ρ χ i ) + ( ∇ µ ∇ ν χ i ) (cid:3) χ i + G µν ( ∂χ i ) − g µν (cid:2) − ( ∇ ρ ∇ σ χ i )( ∇ ρ ∇ σ χ i ) + ( (cid:3) χ i ) − ∂ ρ χ i ∂ σ χ i R ρσ (cid:3)(cid:17) = 0 , ∇ µ (cid:0) ( αg µν − γG µν ) ∇ ν χ i (cid:1) = 0 , ∇ ν F νµ = 0 . (3.2)It is clear that these equations admit a pure AdS vacuum solution where R µν = Λ g µν andthe electromagnetic and scalar fields vanish. In this vacuum, the effective kinetic term forthe Horndeski axions χ i becomes L ( χ i , kin) = − ( α + γ Λ) (cid:88) i ( ∂χ i ) . (3.3)This will be of the standard sign, signifying ghost-freedom, if ( α + γ Λ) >
0. In this paperwe shall consider only solutions for which Λ is negative. Stability requires that ( α + γ Λ)should be non-negative, but novel features can arise at the critical point where ( α + γ Λ)vanishes. (An analogous situation can also arise in Einstein-Gauss-Bonnet theories, see,e.g., [32].) Thus γ can lies in the range − ∞ < γ ≤ α ( − Λ) . (3.4)Typically, the cosmological constant is viewed as a fixed parameter that is part of the9pecification of a theory, but it can alternatively arise as an integration constant for an n -form field strength in n dimensions. Thus here we may replace the cosmological constantterm in (3.1) by a term L F (4) = 14! F (4) . (3.5)The equation of motion for F (4) can be solved by taking F µνρσ = √− (cid:15) µνρσ , where Λis an arbitrary non-positive constant that acquires an interpretation as the cosmologicalconstant. In this new theory, one may treat the “cosmological constant” as a thermodynamicvariable, which has an interpretation as a pressure (see, for example, [33, 34]). Changingthe cosmological constant, i.e. the pressure, can lead to a phase transition from a stable toan unstable regime as the sign of ( α + γ Λ) turns negative. The critical point where ( α + γ Λ)vanishes gives, as we shall see, some interesting features in the boundary theory.We now construct dyonic AdS planar black holes where the two Horndeski axions arelinear functions of the spatial boundary coordinates x i , i.e., ds = − h ( r ) dt + dr f ( r ) + r dx i dx i ,A = a ( r ) dt + B x dx − x dx ) , χ i = λx i . (3.6)The equations of motion for the axions are trivially satisfied. The Maxwell equation impliesthat a (cid:48) = q (cid:115) hf r − , (3.7)where q is an integration constant. With this, the Einstein equations give4 κr f h (cid:48) + h (cid:0) f (cid:0) γλ + 2 κr (cid:1) + κ ( q + B ) + 4 κ Λ r + 2 αλ r (cid:1) = 04 κr f (cid:48) + f (cid:0) κr − γλ (cid:1) + κ ( q + B ) + 4 κ Λ r + 2 αλ r = 0 h (cid:0) κr f (cid:48) h (cid:48) + f (cid:0) κr h (cid:48)(cid:48) + h (cid:48) (cid:0) κr + γλ r (cid:1)(cid:1)(cid:1) + h (cid:0) f (cid:48) (cid:0) κr + γλ r (cid:1) − γλ f − κ (cid:0) q + B − r (cid:1)(cid:1) − κr f h (cid:48) = 0 . (3.8)These equations can be easily solved, leading to the black hole solutions h = U f , U = e γλ κr a = a − √ πκq √ γλ erfi( √ γλ √ κr ) ,f = − λ κ (3 α + γλ ) − µe − γλ κr r − Λ r √ πe − γλ κr erfi (cid:16) √ γλ √ κr (cid:17) (cid:0) γλ (3 α + γ Λ) + 3 κ ( q + B ) (cid:1) √ γκ / λr , (3.9)where erfi( x ) is the imaginary error function, defined by erfi( x ) = 2 π − / (cid:82) x e z dz . Theasymptotic forms of the metric functions near infinity are given by − g tt = h ( r ) ∼ − Λ3 r − λ (3 α + 2 γ Λ)6 κ − µr + O ( 1 r ) ,g rr = f ( r ) ∼ − Λ3 r − λ (3 α + γ Λ)6 κ − µr + O ( 1 r ) , (3.10)which shows that the solution is asymptotic to dS or AdS for Λ > < < T = (cid:0) − κ Λ r − κ ( q + B ) − αλ r (cid:1) πκr exp (cid:16) γλ κr (cid:17) . (3.11)Although the linearised equations of motion for the Horndeski terms are of two deriva-tives, it is still necessary to check the sign of the kinetic terms for possible ghost-likebehaviour. The kinetic terms for the perturbative axions δχ i are given by (cid:88) i P δ ˙ χ i δ ˙ χ i , with P µν = − ( αg µν − γG µν ) . (3.12)In order to avoid ghosts, the P component of P µν , which is given by P = α − γ ( f r ) (cid:48) hr = γκ (cid:0) B + q (cid:1) − γ λ f + 4 κr ( α + γ Λ) + 2 αγλ r κr h , (3.13)should be non-negative, both on and outside the horizon. The asymptotic form of P nearinfinity is given by P ∼ − α + γ Λ)2Λ r + λ (cid:0) α + 12 αγ Λ + 5 γ Λ (cid:1) κ Λ r + O ( 1 r ) . (3.14)The positivity of P therefore implies, as a necessary condition, that α + γ Λ ≥ < α + γ Λ = 0, the leading term of P vanishes11nd the asymptotic form of P becomes simpler, with P ∼ γ λ κr + O ( 1 r ) , (3.15)which is still greater than zero. It can then be checked from (3.13) that P is indeed alwayspositive in the region from the horizon to infinity when α and γ are both positive. Now, we turn to the calculation of the DC conductivity of this system. We follow a similarprocedure to the one described in the previous section. Here we shall omit the details ofthe calculation, and just present the final results. We begin with the simpler case where B = 0, for which we find the conductivity is given by σ = κ + 4 κ q r λ (cid:0) κr ( α + γ Λ) + 2 αγλ r + γκq (cid:1) . (3.16)When γ = 0, this result reduces to (2.17). This demonstrates that the couplings of theaxions for dissipative momenta plays a crucial role in shaping the conductivity. Although σ contains the same “charge-conjugation symmetric” term κ , as one would expect, it has avery different “dissipative” term associated with λ that has a richer structure. At large T ,however, it has the same qualitative behaviour as that of the Einstein-Maxwell case in thehigh-temperature limit for generic parameters σ ∼ κ + κ q λ r ( α + γ Λ) ∼ κ + κ q Λ π λ ( α + γ Λ) 1 T . (3.17)On the other hand, at the critical point α + γ Λ = 0, the temperature dependence is character-istically different. The denominator of the dissipative term in (3.16) has three contributions,with the leading-order power of r being proportional to ( α + γ Λ). When α + γ Λ is positive,the conductivity rises from a positive in initial value at zero temperature, rises to a peak,and then decreases to a constant value at high temperature. Especially, when γ = 0, theconductivity decreases monotonically from its initial value as the temperature increases,behaving much like a normal conductor. If on the other hand α + γ Λ = 0, the conductivitymonotonically increases with temperature, approaching a constant in the high-temperaturelimit. This behaviour is closer to that of a semiconductor. We illustrate the various be-haviours in Fig.1, where the parameters are fixed such that κ = α = q = 1 and λ = 1 /
2. In This phenomenon was observed in [18], where massive gravity was used to achieve momentum dissipation. − σ versus T for four repre-sentative values of γ . The top curve corresponds to the critical case ( α + γ Λ) = 0, while thelower curves correspond cases with ( α + γ Λ) >
0. In the right-hand diagram we instead fix γ = 1 / α + γ Λ) = 0being the curve at the top, with the lower curves having ( α + γ Λ) >
0. The critical casecan be thought of as representing a phase transition where the high-temperature behaviourof the material changes from that of a metal ( σ falls to a small constant κ as T increases)to a semiconductor ( σ rises to a limiting value as T increases) in the critical case. Fromthe bulk point of view, the transition can be viewed as being induced when the pressure( ∼ ( − Λ)) becomes sufficiently large.It is interesting to note that in the left-hand diagram in Fig. 1, all the conductivitycurves originate from the same value when T = 0. The reason for this can be seen from theexpressions for the temperature and the conductivity, namely T = − e γλ κr (cid:0) κ Λ r + 2 αλ r + κq (cid:1) πκr , (3.18) σ = κ + 4 κ q r γ (cid:0) κ Λ r + 2 αλ r + κq (cid:1) + 4 ακr . (3.19)The temperature becomes zero when the factor in parentheses in (3.18) vanishes, and then(3.19) implies that the corresponding zero-temperature conductivity is given by σ (0) = κ + κ q αλ r , (3.20)with r being given by r = αλ + (cid:112) α λ − κ q Λ4 κ ( − Λ) . (3.21)Thus at fixed Λ, with κ , α , q and λ also fixed as in left-hand diagram, the zero-temperatureconductivity is independent of γ . By contrast, if γ is fixed instead of Λ, as in the right-handdiagram, the zero-temperature conductivity does depend on Λ.We have not included plots for values of the parameters for which α + γ Λ is negative.Here, the dissipative part of the conductivity can be negative, and for a range of temper-atures the full expression for the conductivity can be negative or divergent. This suggestsan unphysical instability, and is in fact consistent with our previous observation that theHorndeski axions acquire ghost-like kinetic terms when α + γ Λ is negative.The case when B (cid:54) = 0 is considerably more complicated, and we shall not present the13 = (cid:144) g= (cid:144) g= (cid:144) g= T s L=- L=-
L=-
L=- T s Figure 1:
Plots of the conductivity σ versus temperature, for various parameter choices. In eachdiagram we have set κ = α = q = 1 and λ = 1 / . In the left-hand diagram we fix Λ = − andtake various choices for the parameter γ . The top line has the critical value γ = 1 / , for which ( α + γ Λ) = 0 . In the right-hand diagram we instead fix γ = 1 / and take various choices for theparameter Λ . Again, the top line corresponds to the critical value. In both diagrams, the lower linesall correspond to ( α + γ Λ) > , and they approach κ (which we have set equal to 1 for the purposesof these plots) at large T . general expression for the conductivity matrix here. However, in the high temperature limitwe find that it takes the form for α + γ Λ > σ = σ ∼ κ + κ Λ (cid:0) q − B (cid:1) π λ ( α + γ Λ) T ,σ = − σ ∼ κ Λ qB π λ ( α + γ Λ) T + κ Λ qB (cid:0) α λ + 2 αγλ Λ + Λ (cid:0) − B κ + γ λ Λ + κ q (cid:1)(cid:1) π λ ( α + γ Λ) T , (3.22)and the Hall angle is given by θ H ∼ κ Λ qB π λ ( α + γ Λ) T + Λ qB (cid:0) α λ + 2 αγλ Λ + Λ (cid:0) − B κ + γ λ Λ − κ q (cid:1)(cid:1) π λ ( α + γ Λ) T . (3.23)At the critical point α + γ Λ = 0, the conductivity and Hall angle at high temperaturebecome σ = σ ∼ γ κλ Λ (cid:0) γ λ Λ − κ ( q + B ) (cid:1) B κ + 4 B ( κ q − γ κ λ Λ) + γ λ Λ + O ( 1 T ) ,σ = − σ ∼ Bκ q (cid:0) κ ( q + B ) − γ λ Λ (cid:1) B κ + 4 B ( κ q − γ κ λ Λ) + γ λ Λ + O ( 1 T ) ,θ H ∼ Bκ q (cid:0) κ ( B + q ) − γ λ Λ (cid:1) γ λ Λ ( − B κ + γ λ Λ − κ q ) + O ( 1 T ) . (3.24)In particular, the Hall angle approaches a constant at large T . At T = 0, on the other14and, which occurs when the factor in parentheses in the numerator in (3.11) vanishes, theconductivity and Hall angle become σ = σ = ακλ r (cid:0) B κ + κq + αλ r (cid:1) B κ + B κ (cid:0) κq + 2 αλ r (cid:1) + α λ r ,σ = − σ = Bκ q (cid:0) B κ + κq + 2 αλ r (cid:1) B κ + B κ (cid:0) κq + 2 αλ r (cid:1) + α λ r ,θ H (0) = κqB (cid:0) B κ + κq + 2 αλ r (cid:1) αλ r (cid:0) B κ + κq + αλ r (cid:1) , (3.25)where r = αλ + (cid:112) α λ − κ Λ ( B + q ) − κ Λ . (3.26)It is of interesting to note that at T = 0, both σ ’s and θ H are independent of γ . This impliesthat in the zero temperature limit the DC conductivities and Hall angle are the same asthose in the Einstein-Maxwell case (2.26,2.27), if the results are expressed in terms of thehorizon radius r and we set α → κ . In this paper, we studied two four-dimensional gravity theories involving scalar fields withnon-minimal Horndeski-type couplings to gravity. We first considered Einstein-Maxwellgravity with one non-minimally coupled Horndeski axion and two minimally coupled ax-ions. The two minimally coupled axions have linear dependence on the spatial boundarycoordinates, and they generate momentum dissipation in the standard way. We constructeda charged AdS planar black hole in the theory, and calculated the holographic DC conduc-tivity in the dual field theory. Interestingly, although the Horndeski scalar in these solutionsplays a role in determining the geometry of the black hole background, it does not contributedirectly to the conductivity. To be precise, if written in terms of the horizon radius r theconductivity is the same as that in Maxwell-Einstein gravity.In the second model, we used two Horndeski axions, non-minimally coupled to Einstein-Maxwell gravity, to drive the momentum dissipation. We obtained a static AdS blackhole solution in the theory. We analyzed the kinetic terms of the axion perturbations,and showed that the theory has a critical point at α + γ Λ = 0. When α + γ Λ <
0, thekinetic terms of the axion perturbations become negative, implying that the excitationsbecome ghost-like. We then obtained the conductivity in the dual boundary theory, andfound that the conductivity has two terms, a “charge-conjugation symmetric” term and15dissipative” term as usual. However, the dissipative term has richer features than in astandard minimally-coupled theory. At the critical point α + γ Λ = 0, the conductivityincreases monotonically as a function of temperature, which is typical of the behaviour in asemiconductor. When α + γ Λ >
0, on the other hand, the conductivity rises to a maximumthen falls, finally approaching a constant. In the special case γ = 0, corresponding toturning off the Horndeski modification of the usual minimal coupling of the axions, theconductivity decreases monotonically with temperature, and the behavior is more like anormal conductor. We chose a set of parameters in the paper and plotted the conductivityversus temperature curves for various values of γ , in Fig. 1. We showed that from thecritical point γ = − α/ Λ to the special case γ = 0, the behavior of the conductivity asa function of temperature changes from that reminiscent of a semiconductor to that of anormal conductor.Momentum dissipation is the key for obtaining finite holographic DC conductivity.While free axions provide one of the simplest models for such a mechanism, the resultingDC conductivity generally tends to have a fairly simple structure whose qualitative featuresare independent of the parameters. Our work demonstrated that using non-minimally cou-pled axions in the momentum-dissipation mechanism can lead to a much richer pattern ofholographic DC conductivities. Acknowledgements
We are grateful to Sera Cremonini for discussions. H-S.L. is supported in part by NSFCgrants No. 11305140, 11375153, 11475148, 11675144 and CSC scholarship No. 201408330017.The work of H.L. is supported in part by NSFC grants No. 11475024, No. 11175269 andNo. 11235003. C.N.P. is supported in part by DOE grant DE-FG02-13ER42020.
A Higher dimensional case
In this section, we generalise the theory in section 2 to include
N p -form fields in arbitrarydimension, L = κ (cid:16) R − − F − N (cid:88) i =1 p ! (cid:0) F i ( p ) (cid:1) (cid:17) −
12 ( αg µν − γG µν ) ∂ µ χ ∂ ν χ , (A.1)where κ , α and γ are coupling constants, G µν ≡ R µν − Rg µν is the Einstein tensor, F = dA is the electromagnetic field strength and F i = d A i is one of the form fields which span all16pacial dimension with N p = n −
2. The equations of motion are given by κ ( G µν + Λ g µν − F µν + 18 F g µν ) + N (cid:88) i =1 (cid:2) − κ p − (cid:0) F i (cid:1) µν + κ p ! (cid:0) F i (cid:1) g µν (cid:3) − α (cid:16) ∂ µ χ∂ ν χ − g µν ( ∂χ ) (cid:17) − γ (cid:16) ∂ µ χ∂ ν χR − ∂ ρ χ ∂ ( µ χ R ν ) ρ − ∂ ρ χ∂ σ χ R µρν σ − ( ∇ µ ∇ ρ χ )( ∇ ν ∇ ρ χ ) + ( ∇ µ ∇ ν χ ) (cid:3) χ + G µν ( ∂χ ) − g µν (cid:2) − ( ∇ ρ ∇ σ χ )( ∇ ρ ∇ σ χ ) + ( (cid:3) χ ) − ∂ ρ χ∂ σ χ R ρσ (cid:3)(cid:17) = 0 , ∇ µ (cid:0) ( αg µν − γG µν ) ∇ ν χ (cid:1) = 0 , ∇ ν F νµ = 0 , ∇ ν F νµ ··· µ p − i = 0 (A.2)We consider static planar black hole ansatz ds = − h ( r ) dr + dr f ( r ) + r dx i dx i ,χ = χ ( r ) , A = a ( r ) dt , F i = λdx i ∧ · · · ∧ dx ip . (A.3)where λ is a constant. The Maxwell’s equation can be used to express the electrical potentialin terms of metric functions a (cid:48) = q (cid:115) hf r − n , (A.4)where q is an integration constant. And the equation of motion for scalar can be written as (cid:16) r n − (cid:114) fh (cid:16) γ (cid:0) ( n − rf h (cid:48) + ( n − n − f h (cid:1) − αr h (cid:17) χ (cid:48) (cid:17) (cid:48) = 0 . (A.5)We focus on a special class of solution, as what we did in section 2, by letting γ (cid:0) ( n − rf h (cid:48) + ( n − n − f h (cid:1) − αr h = 0 . (A.6)Under these setup, we can obtain the black hole solution a = a − q ( n − r n − + κq g (3 n − n − n − βγ + 4 κ ) r n − + N κλ qg ( n − n − n + 2 p − βγ + 4 κ ) r n +2 p − ,χ (cid:48) = (cid:115) β − κ ( q + N λ r n − p − ) γg ( n − n + 2) r n − √ f ,f = g ( n − ( n − ( βγ + 4 κ ) r n − ( κq − g ( n − n + 2) ( βγ + 4 κ ) r n − + N κλ r n − p − ) h , = g r − µr n − + 2 κq ( n − n − βγ + 4 κ ) r n − + κ q g (7 − n )( n − ( n − βγ + 4 κ ) r n − − N κλ ( n − n − p − βγ + 4 κ ) r p − + N κ λ g ( n − ( n − n − p − βγ + 4 κ ) r p − − N κ λ q g ( n − ( n − n + 2 p − βγ + 4 κ ) r n + p ) − , (A.7)with parameters under constraint α = ( n − n − g γ , Λ = − ( n − n − g (cid:16) βγ κ (cid:17) . (A.8) B Einstein-Maxwell-Dilaton theory with Horndeski axions
In section 3, we studied the theory of Einstein-Maxwell gravity with two non-minimallycoupled Horndeski axions. Here, we give a generalisation in which we include also a dilatonicscalar field with an exponential coupling to the Maxwell field, and exponential potentialterms. The Lagrangian is given by L = κ [ R − e δ φ − V e δ φ − ( ∂φ ) − e δ φ F ] − (cid:88) i ( αg µν − γG µν ) ∂ µ χ i ∂ ν χ i . (B.1)where δ , δ , δ , V , κ , γ , and α are constants. The second potential term, with coefficient V , is required for the case where a magnetic field is included. The equations of motion aregiven by κ ( G µν + (Λ e δ φ + V e δ φ ) g µν − ∂ µ φ∂ ν φ + ( ∂φ ) g µν − e δ φ F µν + e δ φ F g µν ) − (cid:88) i α (cid:16) ∂ µ χ i ∂ ν χ i − g µν ( ∂χ i ) (cid:17) − (cid:88) i γ (cid:16) ∂ µ χ i ∂ ν χ i R − ∂ ρ χ i ∂ ( µ χ i R ν ) ρ − ∂ ρ χ i ∂ σ χ i R µρν σ − ( ∇ µ ∇ ρ χ i )( ∇ ν ∇ ρ χ i ) + ( ∇ µ ∇ ν χ i ) (cid:3) χ i + G µν ( ∂χ i ) − g µν (cid:2) − ( ∇ ρ ∇ σ χ i )( ∇ ρ ∇ σ χ i ) + ( (cid:3) χ i ) − ∂ ρ χ i ∂ σ χ i R ρσ (cid:3)(cid:17) = 0 , ∇ µ (cid:0) ( αg µν − γG µν ) ∇ ν χ i (cid:1) = 0 , E µA ≡ ∇ ν ( e δ φ F νµ ) = 0 , (cid:3) φ − δ e δ φ − V δ e δ φ − δ e δ φ F = 0 . (B.2)18e consider the static planar black hole in four dimensions ds = − h ( r ) dr + dr f ( r ) + r dx i dx i ,χ i = λx i , A = a ( r ) dt + B x dx − x dx ) , φ = β log r , (B.3)where λ , β and B are constants. The Maxwell equation implies a (cid:48) = q (cid:115) hf r − − δ β . (B.4)We find that there are two inequivalent classes of solutions, where the parameters ( δ , δ , δ )are given by Class 1 : δ = − β , δ = β , δ = β − β , Class 2 : δ = − β , δ = − β , δ = − β . (B.5)In both classes we have h = U f , U = r β e γλ κr . For class 1 we find f = e − γλ κr (cid:104) αλ κ ( β −
4) Ei (cid:18) + β , − γλ κr (cid:19) + B β − r −
2+ 12 β Ei (cid:18) + β , − γλ κr (cid:19) − q r − − β Ei (cid:18) − β , − γλ κr (cid:19) − µ r − − β (cid:105) , (B.6)with parameters Λ = αβ λ κ (4 − β ) , V = β B − β ) . (B.7)Ei is the exponential integral, defined byEi( z, x ) = (cid:90) ∞ t − z e − xt dt = Γ(1 − z ) x z − − (cid:88) n ≥ ( − x ) n n ! ( n + 1 − z ) . (B.8)The Hawking temperature for the class 1 solutions is given by T = e γλ κr r β (cid:18) B κr β − (cid:0) β − (cid:1) κq + 8 αλ r β +20 (cid:19) π ( β − κr . (B.9)19he positivity of temperature require β >
4. In the large r limit, the temperatureapproaches T ∼ B r ( β − π ( β − . (B.10)For the class 2 solutions we find f = e − γλ κr (cid:104) ( αλ + κq ) κ ( β −
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