Thermoelectric conductivities, shear viscosity, and stability in an anisotropic linear axion model
TThermoelectric conductivities, shear viscosity, and stability in ananisotropic linear axion model
Xian-Hui Ge , ∗ Yi Ling , † Chao Niu , ‡ and Sang-Jin Sin , § Department of Physics, Shanghai University, 200444 Shanghai, China Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Department of Physics, Hanyang University, Seoul 133-791, Korea School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
Abstract
We study thermoelectric conductivities and shear viscosities in a holographically anisotropicmodel, which is dual to a spatially anisotropic N = 4 super-Yang-Mills theory at finite chemicalpotential. Momentum relaxation is realized through perturbing the linear axion field. Ac conduc-tivity exhibits a coherent/incoherent metal transition. Deviations from the Wiedemann-Franz laware also observed in our model. The longitudinal shear viscosity for prolate anisotropy violatesthe bound conjectured by Kovtun-Son-Starinets. We also find that thermodynamic and dynamicalinstabilities are not always equivalent by examining the Gubser-Mitra conjecture. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] a r X i v : . [ h e p - t h ] N ov . INTRODUCTION One of the advantages of holography is that it provides a nonperturbative method forcalculating transport coefficients of strongly coupled systems. Transport coefficients ofanisotropic plasma are of interest because the quark-gluon plasma created in the RHICand LHC is actually anisotropic and nonequilibrium during the period of time τ out after thecollision. The neutral anisotropic black brane solution at zero temperature was found origi-nally in Ref. [1], and at nonzero temperature was constructed from type-IIB supergravity byMateos and Trancanelli [2, 3]. Interestingly, the shear viscosity longitudinal to the directionof anisotropy violates the viscosity bound [4], which is up to now the first example of suchviolation in Einstein gravity [5].In Refs. [6, 7], the R-charged version of the anisotropic black brane solution was de-rived via nonlinear Kaluza-Klein reduction of type-IIB supergravity to five dimensions. Thenonlinear Kaluza-Klein reduction of type-IIB supergravity to five dimensions leads to thepresence of an Abelian field in the action. The introduction of the U (1) gauge field breaksthe SO(6) symmetry and thus leads to the excitations of the Kaluza-Klein modes. In addi-tion, we also consider the analytic continuation in which the anisotropy parameter is takento be imaginary, resulting in an oblate anisotropy.In this paper, we will calculate the optical conductivity (longitudinal to the anisotropydirection), dc thermoelectric conductivities, and shear viscosities in this anisotropic system.An accurate realization of thermoelectric conductivities in real condensed matter systemsrequires us to include mechanisms for momentum dissipation. Recently, it was suggested tointroduce momentum relaxation in holography by exploiting spatially dependent sources forscalar operators or using the massive gravity [8–25]. The reduction action for the anisotropicblack brane used in this paper, is the Einstein-Maxwell dilaton-axion theory. We will showthat momentum relaxation can be realized in our model by explicitly breaking the transla-tional invariance (for works on spontaneously symmetry breaking, see Ref. [26] ). Remark-ably, the resistivity shows its linear temperature behavior, signaling the presence of “strangemetals.”On the condensed matter theory side, there is still a lack of a satisfying explanation of thelinear temperature dependence of resistivity at sufficiently high temperatures in materialssuch as organic conductors, heavy fermions, fullerenes, vanadium dioxide, and pnitides.2he linear temperature dependence of resistivity at high temperature, a signature of abreakdown of the Boltzman theory, is expected when the quasiparticle mean free path l becomes shorter than the lattice parameter ˜ a . It is the self-consistency of the Boltzmanntheory which requires that the charge carrier transport should satisfy the Mott-Ioffe-Regel(MIR) limit σ MIR ∼ k F e / (cid:126) with l ∼ ˜ a or k F l ∼ ω = − iτ − rel + · · · , (1)where τ rel denotes the momentum relaxation time scale and the ellipsis represents themomentum-dependent terms. The linear fluctuations are purely decaying modes when τ − rel >
0, indicating that the black brane is stable under such dynamical perturbations.This is what we mean by “wall of stability.” The wall of stability requires the momentumrelaxation time scale to be τ rel ≥
0; otherwise, the dual field will absorb momentum ratherthan dissipating it. It may be interesting to relate the question of the wall of stabilitywith the GM conjecture and ask the following question: does the regime with τ rel ≥ τ rel ≥ S d − horizon topology, the phase diagram is analogous tothe phase structure of van der Waals’s liquid-gas system [41–46]. For event horizons withtopology R d − , the black brane phase structure is usually considered as dominated by theblack brane phase for all temperatures without any thermodynamic instabilities.However, in Refs. [6, 7] we found that a planar black brane does not necessarily needto be thermodynamically stable by demonstrating that an anisotropic black brane has abranch of solution with negative specific heat. One may naturally connect the thermody-namic instability uncovered in this anisotropic but translationally symmetric system withthe GM conjecture [34]. Therefore, in this paper, we will first compute the dc and opticalconductivities with momentum relaxation and then check the GM conjecture.The organization of the contents is as follows: In Sec. II, we briefly review the anisotropicblack brane solution and its thermodynamic properties. In Sec. III, we calculate the dc andac conductivities. In Sec. IV, we calculate the transverse and longitudinal shear viscositiesfor the prolate brane solutions ( and the oblate case just for completeness of our discussions).By analyzing the causality structure, we show that oblate anisotropy with a < I. R-CHARGED ANISOTROPIC BLACK BRANE SOLUTION
The five-dimensional axion-dilaton Maxwell gravity bulk action reduced from type-IIBsupergravity is written as [6, 7] S = 12 κ (cid:20) (cid:90) d x √− g (cid:16) R + 12 − ( ∂φ ) − e φ ( ∂χ ) − F µν F µν (cid:17) − (cid:90) d x √− γK (cid:21) , (2)where we have set the AdS radius L = 1, and κ = 8 πG = π N c . The counterterm takes theform S ct = 1 κ (cid:90) d x √ γ (3 − e φ ∂ i χ∂ i χ ) − log v (cid:90) d x √ γ A , (3)where A is the conformal anomaly in the axion-dilaton gravity system, v is the Fefferman-Graham coordinate, and γ is the induced metric on a v = v surface.The background solutions with anisotropy along the z direction for the equations ofmotion are ds = e − φ r (cid:16) − F B dt + dx + dy + H dz (cid:17) + e − φ dr r F (4) A = A t ( r ) dt, φ = φ ( r ) , χ = az (5)The metric functions φ , F , B , and H = e − φ are functions of the radial coordinate r only.The electric potential is given by A t ( r ) = (cid:82) rr H drQ √B e φ /r , where Q is an integral constantrelated to the charge. A dimensionless charge can be introduced by defining q ≡ Q √ r , andthe physical range of q is 0 ≤ q <
2. The horizon locates at r = r H with F ( r H ) = 0,and the boundary is at r → ∞ where F = B = H = 1. The asymptotic AdS boundarycondition requires the boundary condition φ ( ∞ ) = 0. We note that the above ansatz isinvariant under the scaling t → λt , x i → λx i , r → λ − r , and a → λ − a . The Hawkingtemperature is given by T = r H F (cid:48) ( r H ) √B H π = (cid:112) B H (cid:20) r H e − φH π (cid:18)
16 + a e φH r H (cid:19) − e φ H q r H π (cid:21) , (6)through the Euclidean method. The entropy density is given by s = N c e − φH π r H . Note thatthe prolate anisotropy a > H ( r H ) >
1, since the metrichas a z axis longer than the x and y axes. From an analytical continuation in which theanisotropy parameter a is taken to be imaginary, one can obtain an oblate metric. Thenumerical and semianalytic black brane solution was given in Refs. [6, 7]. For example, we5an plot the numerical solutions (4) in Fig. 1 for prolate and oblate anisotropy. We willprove in Sec. IV that the oblate anisotropy is unphysical in the holographic setup. HB F f - u F H B f u FIG. 1: The metric functions for a = 64 . Q = 9 .
76 (left), which corresponds to the prolateanisotropy; and a = 1 . i , Q = 1 / u H = 1, which depicts the oblate anisotropy. The Ward identity obeys ∇ i (cid:104) T ij (cid:105) = (cid:104)O φ (cid:105)∇ j φ (0) + (cid:104)O χ (cid:105)∇ j χ (0) + F (0) ij (cid:104) J i (cid:105) . (7)If we consider the background fields only, the translational symmetry is unbroken because ∇ i (cid:104) T ij (cid:105) = 0, with the facts that ∇ j φ (0) = 0, (cid:104)O χ (cid:105) = 0, and F (0) ij = 0. However, theblack brane solution is not translationally invariant in the z direction. By considering thefluctuations around the background as we will do in later sections, we can prove that (cid:104)O χ (cid:105) isfinite. Thus, the Ward identity suggests an approach to holographic momentum relaxationand finite dc response through the spatially dependent source term for the axion.Actually, the anisotropic black brane has very special thermodynamic properties, aspointed out in Refs. [6, 7]. Considered the prolate anisotropy with a >
0, the blackbrane suffers thermodynamic instabilities. This can be easily seen from (6) in the small-horizon-radius limit r H (cid:28)
1; that is to say T ∼ √B H a e φ H πr H , (8)which in turn results in negative specific heat, since ∂T /∂r H <
0. On the other hand, forthe larger horizon radii with r H (cid:29)
1, we have T ∼ √B H a e − φ H / π (cid:18) − q e φ H π (cid:19) r H . (9)6hus for the prolate anisotropy case, for a fixed temperature there are two branches ofallowed black brane solutions: a branch with larger horizon radii and one with smaller.The smaller branch solution is unstable with negative specific heat. This situation is verysimilar to the case of Schwarzschild-AdS black holes with a spherical horizon. In general,the larger black brane branch will appear and will match to the small black brane solutionat some temperature T min , below which there is only the thermal gas solution. At somehigher temperature, a first-order phase transition (from thermal gas) to the large blackbrane branch will take place. So the system will be a doped Mott-like insulator up to acritical temperature, and then there will be a first-order phase transition to a conductingphase [47–49]. In the following, we mainly focus on the prolate anisotropy case with a > III. DC AND OPTICAL CONDUCTIVITY WITH MOMENTUM RELAXATION
In what follows, we will first compute the dc conductivity with momentum dissipationand discuss its physical meaning. In the r = 1 /u coordinate, the metric can be recast as ds = − g tt ( r ) dt + g rr ( r ) dr + g xx ( r ) dx + g yy ( r ) dx + g zz ( r ) dz , (10)where g xx = g yy (cid:54) = g zz ( r ).It was argued in several papers that axions having a spatially dependent source leadsto the fact that momentum is dissipated at the linearized level [10, 50, 51]. For the scalartype of metric perturbations, the independent variables are h tt , h tz , h xx = h yy , h zz , δφ , δχ together with the t and z components of the gauge field A µ . In the zero-momentumlimit, it is easy to check that h tz , A z , and δχ decouple from other variables. Therefore, tocompute the conductivity with momentum dissipation, we only need to consider linearizedfluctuations of the form δg tz (0) = h tz ( t, r ) , δA z ( t, r ) = a z ( t, r ) , δχ = a − ¯ χ ( t, r ) , (11)and all the other metric and gauge perturbations vanished. We observe that the axiononly sources momentum relaxation along the anisotropic direction. If we instead considerthe metric perturbations h tx and δA x ( t, r ), we expect that it is metallic along the isotropicdirection without any momentum dissipation. Here we choose the gauge h rz = 0 and theelectromagnetic perturbation along the z direction at zero momentum. We shall work in the7ourier decomposition h tz ( t, r ) = (cid:90) dω π e − iωt h tz ( ω, r ) , (12) a z ( t, r ) = (cid:90) dω π e − iωt a z ( ω, r ) , (13)¯ χ ( t, r ) = (cid:90) dω π e − iωt ¯ χ ( ω, r ) . (14)The linearized equations of motion corresponding to the ( r, z ) component of Einstein’s equa-tion, the Maxwell equation, and the dilaton equation are given by0 = A (cid:48) t a z + h (cid:48) tz − g (cid:48) zz g zz h tz − g tt e φ ¯ χ (cid:48) iω , (15)0 = a (cid:48)(cid:48) z + (cid:18) g (cid:48) tt g tt − g (cid:48) zz g zz + g (cid:48) xx g xx − g (cid:48) rr g rr (cid:19) a (cid:48) z + A (cid:48) t g tt ( h (cid:48) tz − g (cid:48) zz g zz h tz ) + ω g rr g tt a z , (16)0 = ¯ χ (cid:48)(cid:48) + (cid:18) g (cid:48) zz g zz + g (cid:48) tt g tt + g (cid:48) xx g xx − g (cid:48) rr g rr + 2 φ (cid:48) (cid:19) ¯ χ (cid:48) + ω g rr g tt ¯ χ − iωa g rr g zz g tt h tz , (17)0 = h (cid:48)(cid:48) tz − (cid:18) g (cid:48) tt g tt + g (cid:48) rr g rr + g (cid:48) zz g zz − g (cid:48) xx g xx (cid:19) h (cid:48) tz + (cid:18) g (cid:48) rr g (cid:48) zz g rr g zz + g (cid:48) tt g (cid:48) zz g tt g zz − g (cid:48)(cid:48) zz g zz + g (cid:48) zz g zz − g (cid:48) xx g (cid:48) zz g xx g zz − g rr e φ a g zz (cid:19) h tz + A (cid:48) t a (cid:48) z − iωg rr e φ ¯ χ, (18)where the prime denotes a derivative with respect to r . For the sake of convenience, we caneliminate h tz by taking a radial derivative of (17) and substituting the expression for h (cid:48) tz in(16) and (17). We can introduce a new variable˜ χ = ω − g xx g zz g tt g − rr e φ ¯ χ (cid:48) , (19)and recast the equation of motion as0 = g − xx g zz (cid:18) g xx g − zz (cid:114) g tt g rr a (cid:48) z (cid:19) (cid:48) + ω (cid:114) g tt g rr a z − A (cid:48) t Qg xx √ g zz a z − iQ √ g rr g tt g xx g zz ˜ χ, (20)0 = e φ g xx g zz (cid:18) e − φ g − xx g − zz (cid:114) g tt g rr ˜ χ (cid:48) (cid:19) (cid:48) + ω (cid:114) g tt g rr ˜ χ + a e φ (cid:18) ig xx A (cid:48) t √ g zz a z − √ g rr g tt g zz ˜ χ (cid:19) . (21)Following Refs. [10, 12], we can rewrite the fluctuation equations (20) and (21) in the form L L a z ˜ χ + ω (cid:114) g tt g rr a z ˜ χ = M a z ˜ χ , (22)8here L and L are linear differential operators and M is the mass matrix, M = Q √ g rr g tt / ( g xx g zz ) iQ √ g rr g tt / ( g xx g zz ) − ia Qe φ √ g rr g tt /g zz a e φ √ g rr g tt /g zz . (23)Clearly, there exists a massless mode, because det M = 0. Let us introduce the followinglinear combinations: λ = b − ( r ) (cid:18) e φ a z + Qia g xx ˜ χ (cid:19) , (24) λ = b − ( r ) (cid:18) Q a g xx a z − Qia g xx ˜ χ (cid:19) , (25)where b ( r ) = e φ + Q a g xx . (26)Then, we obtain the master equation for the massless mode λ (cid:18) e − φ g xx (cid:114) g tt g rr g zz b ( r ) λ (cid:48) − (cid:114) g tt g rr g zz c ( r ) λ (cid:19) (cid:48) + ω b ( r ) e − φ g xx (cid:114) g rr g tt g zz λ = 0 , (27)where c ( r ) = ( g xx e φ ) (cid:48) / ( e φ g xx ). We can easily find that the following quantity is radiallyconserved at zero frequency:Π = e − φ g xx (cid:114) g tt g rr g zz b ( r ) λ (cid:48) − (cid:114) g tt g rr g zz c ( r ) λ . (28)The dc membrane conductivity to each radial slice is defined as σ DC ( r ) = lim ω → − Π iωλ (cid:12)(cid:12)(cid:12)(cid:12) r . (29)It was proved in Refs. [10, 12] that σ DC ( r ) does not evolve radially [i.e. σ DC ( ∞ ) = σ DC ( r H )].So it can be evaluated at the horizon. In order to evaluate (29), we note that the ingoingboundary conditions for the fields are given by a z = ( r − r H ) − iω/ ( F (cid:48) ( r H ) √ B ( r H )) [ a Hz + O ( r − r H )] , (30)˜ χ = ( r − r H ) − iω/ ( F (cid:48) ( r H ) √ B ( r H )) [ ˜ χ Hz + O ( r − r H )] . (31)Substituting the metric functions (4) and (30) into (29), we finally obtain σ DC = r H e φ ( r H)4 (cid:18) r H e − φ ( r H ) q a (cid:19) , (32)9here q = Q √ r . The first term in the round brackets is the conductivity due to the pairproduction in the dual field theory, The dc conductivity can also be recast as σ DC = ( g xx g yy g zz ) / (cid:12)(cid:12)(cid:12)(cid:12) r = r H + q a e φ ( g xx g yy g zz ) / (cid:12)(cid:12)(cid:12)(cid:12) r = r H = r H e φ ( r H)4 + Q a r H e − φ ( r H ) / . (33)This result is consistent with Refs. [20, 50]. We will provide an alternative calculation onthe dc conductivity later as a consistent check. The dc conductivity can be related to ascattering time τ rel by [22] σ DC = ( g xx g yy g zz ) / (cid:12)(cid:12)(cid:12)(cid:12) r = r H + Q (cid:15) + P z τ rel . (34)The scattering rate is given by Γ = τ − rel = sa π ( E + P z ) . (35)For the prolate solution, we have τ rel >
0. For the case τ rel <
0, the wall of stability will beviolated. As stated in the Introduction, the scattering rate is the characteristic time scaleof momentum relaxation in the dual field theory. When τ rel <
0, the metric fluctuations (quasinormal modes) absorb momentum, resulting in the fact that small perturbations of thestate will grow exponentially in time. Therefore, we demand that τ rel ≥ In the dual gravitational picture, the unstable quasi-normal modes are identified with unstable uniformplasma with respect to certain nonuniform perturbation [52, 53]. On the gravitational side, this instabilityseems similar in certain respects to the Gregory-Laflamme instability for black strings[52]. .06 0.07 0.08 0.09 0.10 0.11 0.12 T m s DC m T m · · · · s DC m T m m s DC - FIG. 2: Left: DC conductivity as a function of the temperature corresponding to the smaller-horizon-radius branch, where we set q = 1 .
414 and a = 0 .
1. Middle: DC conductivity as a functionof the temperature in the thermodynamically stable regime, where we set q = 1 .
414 and a = 0 . ρ ∼ ρ MIR T shows its linear temperature dependence in the small-horizonregime and that it can violate the Mott-Ioffe-Regel limit at high temperatures. To the linear order, the Ward identity (7) reduces to ∂ t δ (cid:104) T zz (cid:105) = − aδ (cid:104)O χ (cid:105) + ρ∂ t a z . (36)We have proved that the linearized contribution to the vacuum expectation value (cid:104)O χ (cid:105) isfinite, although at zeroth order (cid:104)O χ (cid:105) is vanishing. • Dc conductivity with prolate anisotropy.
We would like to comment on the temperaturedependence of the dc electric conductivity. It was proven in Ref. [2] that the IR geometry ofthe prolate black brane is asymptotic Lifshitz at zero temperature, and thus the dc electricconductivity obtained here behaves quiet differently from those with near-horizon geometry
AdS × R [10]. There are two branches in the high-temperature phase: small and large blackhole radius. As can be seen in Fig. 2 (middle), the conductivity goes up as the temperatureincreases, which corresponds to the larger-horizon-radius branch solution of the black brane.The system undergoes a first-order Hawking-Page-like confinement/deconfinement phasetransition as the temperature varies. It has been proved that the smaller-horizon-radiusbranch of the solution has a negative specific heat, so it is not physically realized. If thesmall-horizon-radius r H were stable, we could identify it as the dual to doped “bad metals,”since its resistivity is linear in temperature as shown in Fig. 2 (right). This is because inthe holographic set up, quasiparticle descriptions lose their validity and dc resistivity can beboth weaker and stronger than the MIR limit ρ MIR ∼ (cid:126) /k F e . It suggests that we need to11tabilize the small-radius branch by twisting our present model, which suggests a directionto future model building.In Refs. [47–49], the authors argued that the confinement phase is dual to the Mottinsulators. The strange metal behaviors can be obtained by doping a Mott insulator. In oursetup, the linear axion field’s parameter a might play the role of dopants. A. Optical conductivity
In this section, we try to solve Eqs. (15)-(18) numerically. In order to calculate ac thermo-electric conductivities numerically, we need to evaluate the on-shell action that gives a finiteand quadratic function of the boundary values. In general, for the variation of the action,we have δS = (cid:90) ∂ µ (cid:18) ∂L∂∂ µ ϕ i δϕ i (cid:19) dr + (cid:90) E.O.M.δϕ i dr. (37)We obtain the on-shell action in the momentum space, S = lim r →∞ V (cid:90) dω π √− g (cid:20) g zz g rr g tt h (cid:48) tz h tz − a (cid:48) z a z g rr g zz − e φ ¯ χ (cid:48) ¯ χg rr + A (cid:48) t a z g rr g tt h tz − E h tz h tz (cid:21) , (38)where E denotes the energy density. The Green function is defined via G i ; j = δ S/δϕ i δϕ j .Near the boundary ( r → ∞ ), the asymptotic solutions go as h tz = r h (0) tz + h (2) tz + 1 r h (3) tz + 1 r h (4) tz · · · , (39) a z = a (0) z + 1 r a (2) z + · · · , (40)¯ χ = ¯ χ (0) + 1 r ¯ χ (2) + 1 r ¯ χ (3) + · · · . (41)Note that a (0) z is the source of the electric current J z , and h (0) tz is dual to the source of theenergy-momentum tensor T tz . In order to compute the ac electric conductivity numerically,we need to impose the ingoing boundary condition at the horizon and adopt the numericalmethod developed in Refs. [19] and [51].The optical conductivity is given by σ ( ω ) = − iG JzJz ω . (42)The numerical computation shows that the optical conductivity takes the form of the Drudeconductivity as σ ( ω ) = σ Q + Kτ − iωτ , (43)12 /µ . . . . . . . . K/µ .
751 0 .
762 0 .
743 0 .
710 0 .
698 0 .
634 0 .
628 0 . τ µ .
715 134 .
337 64 .
016 39 .
909 27 .
456 17 .
966 14 .
924 11 . a/µ at T /µ = 0 . Ω (cid:144) Μ R e (cid:64) Σ (cid:68) & I m (cid:64) Σ (cid:68) (cid:144) Μ Ω (cid:144) Μ R e (cid:64) Σ (cid:68) & I m (cid:64) Σ (cid:68) (cid:144) Μ Ω (cid:144) Μ R e (cid:64) Σ (cid:68) & I m (cid:64) Σ (cid:68) (cid:144) Μ Ω (cid:144) Μ R e (cid:64) Σ (cid:68) & I m (cid:64) Σ (cid:68) (cid:144) Μ Ω (cid:144) Μ R e (cid:64) Σ (cid:68) & I m (cid:64) Σ (cid:68) (cid:144) Μ Ω (cid:144) Μ R e (cid:64) Σ (cid:68) & I m (cid:64) Σ (cid:68) (cid:144) Μ FIG. 3: The optical conductivity as a function of frequency for different wave numbers a/µ =0 . , . , . , . , . , . T /µ = 0 . with σ Q = r H e φ ( r H)4 , constant K , and relaxation time τ . Figure 3 shows how the opticalconductivity changes as the anisotropic parameter a (i.e. the dissipation strength) changes.Since the anisotropic parameter a is nonvanishing, the 1 /ω pole in the imaginary part dis-appears. As shown in Table I, when a becomes bigger, the maximum value of the peakin the real part decreases, which is in good agreement with the dc conductivity (32). Wealso emphasize that for the case a/µ = 0 .
5, the numerical data deviate from the standardDrude model, implying that there is a coherent/incoherent transition [51]. As a/µ increases,significant deviations from the Drude model can be observed and the conductivity looks in-coherent without a Drude peak. We expect the ac thermal and thermoelectric conductivitieswith momentum relaxation also to show a Drude peak at small a/µ as given in Ref. [51],and we postpone such computation to a future study.13 . DC Thermoelectric conductivities
In this section, we provide an alternative calculation of the dc electric conductivity,thermoelectric conductivities (Seebeck coefficients), and the thermal conductivity by usingthe method developed by Donos and Gauntlett [50]. In this setup, we consider a slightlydifferent form of the black bole fluctuations, δA z = − Et + a z ( r ) , h tz = h tz ( r ) , h rz = h rz ( r ) , χ = a z + δχ ( r ) , (44)where the temporal component of the four-potential a µ corresponds to a constant electricfield E along the z direction. The equations of motion for these linearized fluctuations aregiven by h (cid:48)(cid:48) tz − (cid:18) g (cid:48) tt g tt + g (cid:48) rr g rr + g (cid:48) zz g zz − g (cid:48) xx g xx (cid:19) h (cid:48) tz + (cid:18) g (cid:48) rr g (cid:48) zz g rr g zz + g (cid:48) tt g (cid:48) zz g tt g zz − g (cid:48)(cid:48) zz g zz + g (cid:48) zz g zz − g (cid:48) xx g (cid:48) zz g xx g zz − g rr e φ a g zz (cid:19) h tz + A (cid:48) t a (cid:48) z = 0 , (45) a (cid:48)(cid:48) z + (cid:18) g (cid:48) tt g tt − g (cid:48) rr g rr − g (cid:48) zz g zz + g (cid:48) xx g xx (cid:19) a (cid:48) z + A (cid:48) t g tt h (cid:48) tz − g (cid:48) zz A (cid:48) t g zz g tt h tz = 0 , (46)2 A (cid:48) t Eg rr g tt − g rr e φ aδχ (cid:48) + (cid:18) g (cid:48) rr g (cid:48) zz g rr g zz − g (cid:48) tt g (cid:48) zz g tt g zz − g (cid:48)(cid:48) zz g zz + g (cid:48) zz g zz + 2 g (cid:48)(cid:48) xx g xx − g (cid:48) rr g (cid:48) xx g rr g xx − g (cid:48) xx g (cid:48) zz g xx g zz + g (cid:48) tt g (cid:48) xx g tt g xx (cid:19) h rz = 0 . (47)Note that the derivative of the scalar potential is given by A (cid:48) t = − Q ( g rr g tt ) / g xx √ g zz . The equation(47) can be solved easily: h rz = EQ √ g rr a e φ g xx √ g tt g zz + δχ (cid:48) a . (48)In order to solve the equations of motion for a z and h tz , we need to impose proper boundaryconditions for the fluctuation fields at the event horizon r = r H and at the conformalboundary r → ∞ . We first assume that δχ (cid:48) is analytic at the event horizon and falls offfast at infinity. Regularity at the event horizon can be obtained by switching to Eddington-Finklestein coordinates: v = t − π ln( r − r H ) . (49)In this coordinate system, the gauge field is determined by the regularity as a z = − E πT ln( r − r H ) + O ( r − r H ) . (50)14rom Eq. (45), we know that the regularity at the event horizon requires h tz = EQ √ g zz a g xx (cid:12)(cid:12)(cid:12)(cid:12) r = r H + O ( r − r H ) . (51)Near the boundary ( r → ∞ ), we have the falloff of a z ∼ J z r − , where J z denotes the chargedensity current. As to h tz , from equation (45), we can see that there are two independentsolutions, one of which behaves as ∼ c r and the other as ∼ r − . We require that there beno sources associated with thermal gradients, and thus the coefficient c should be vanishing.We also demand that δχ (cid:48) fall off fast enough so that it has no contribution to the boundaryvalue of h rz .We now turn to the computation of the dc conductivity. We can see that the conservedcharge density current is indicated by the nonzero Maxwell equation (46), so we can definethe current as J z ≡ √− gf rz , f rz = a (cid:48) z h tz g rr g zz g tt + A (cid:48) t g rr g zz . (52)We emphasize that the charge density current derived from the Maxwell equation is notidentical to the radially conserved quantity given in (28). Since the current is radiallyconserved, J z can be evaluated both at the horizon and at the boundary. The dc electricconductivity along the z direction is expressed as σ DC = J z /E . By using (50) and (51), wefinally obtain the dc electric conductivity: σ DC = g xx √ g zz (cid:12)(cid:12)(cid:12)(cid:12) r = r H + Q a e φ g xx √ g zz (cid:12)(cid:12)(cid:12)(cid:12) r = r H = r H e φ ( r H ) / + Q a r H e − φ ( r H ) / . (53)This result exactly agrees with (33) obtained in the previous section.The conserved heat current Q is defined through introducing a two-form associated withthe Killing vector field K = ∂ t , and it is assumed as the following form [50]: Q = 2 √− g ∇ r K z + A t J z = (cid:114) g tt g rr g zz g xx (cid:18) − g tt h tz ∂ r g tt + ∂ r h tz (cid:19) + A t J z . (54)Notice that the quantity Q is also radially conserved. We can evaluate Q at the eventhorizon: Q = (cid:115) g xx g rr g tt g zz h tz ∂ r g tt (cid:12)(cid:12)(cid:12)(cid:12) r = r H . (55)The Seebeck coefficient can be obtained at the event horizon r = r H by using the expression α = s DC = 1 T Q E = 4 πQa e φ (cid:12)(cid:12)(cid:12)(cid:12) r = r H . (56)15n order to calculate the thermal conductivity, we need to consider perturbations with asource for the heat current. A consistent choice of the linearized fluctuations takes the form δA z = − Et + ζA t t + a z ( r ) , (57) h tz = − ζt √ g tt / √ g rr + h tz ( r ) , (58) h rz = h rz ( r ) , (59) χ = az + δχ ( r ) , (60)where ζ is a constant. According to the holographic dictionary, the coefficient ζ correspondsto the thermal gradient −∇ z T /T [54–56]. The choice of the fluctuations ensures that allthe time-dependent terms drop out of the conserved current J z and Q . The equations ofmotion for h tz and a z remain the same as given in (45) and (46). We can solve the equationfor h rz and obtain h rz = − Q ( ζA t − E ) √ g rr a e φ (cid:112) g tt g xx g zz − g zz ( g − zz ζ (cid:112) g tt /g rr ) (cid:48) a e φ (cid:112) g tt /g rr + δχ (cid:48) a , (61)where δχ can be a constant at the event horizon. The regularity at the event horizonrequires the gauge field to take the form a z = − E πT ln( r − r H ) + O ( r − r H ). It was suggestedin Ref. [50] that the horizon regularity condition for h tz can be obtained by switching tothe Kruskal coordinates h tz = g zz (cid:114) g tt g rr h rz (cid:12)(cid:12)(cid:12)(cid:12) r = r H − ζ √ g tt πT √ g rr ln( r − r H ) + O ( r − r H ) . (62)Again, we impose the boundary conditions at infinity: a z ∼ J z r − and h tz ∼ r − . Weemphasize that under the choice of fluctuations given in (57), the form of the conservedcurrent does not change. The conserved currents evaluated at the event horizon are givenby J z = (cid:20) E (cid:18) g xx √ g zz + Q a e φ g xx √ g zz (cid:19) + ζQg (cid:48) tt a e φ √ g rr g tt (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r = r H , (63) Q = (cid:20) E Qg (cid:48) tt a e φ √ g rr g tt + ζ g (cid:48) tt g xx √ g zz a e φ g rr g tt (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r = r H . (64)16e finally obtain the dc thermoelectric conductivities in the z direction: σ DC = ∂∂E J z = (cid:18) g xx √ g zz + Q a e φ g xx √ g zz (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r = r H , (65)¯ α = 1 T ∂∂E Q = 4 πQa e φ (cid:12)(cid:12)(cid:12)(cid:12) r = r H , (66) α = 1 T ∂∂ζ J z = 4 πQa e φ (cid:12)(cid:12)(cid:12)(cid:12) r = r H , (67)¯ κ = 1 T ∂∂ζ Q = 4 πsTa e φ (cid:12)(cid:12)(cid:12)(cid:12) r = r H . (68)Note that we have used the notation 2 κ = 1. One interesting point we need to clarify isthat a plays the role of both anisotropy and momentum relaxation source. Hence, a finite a is necessary for our whole computation. We would like to introduce the thermal conductivity at zero electric current, which is theusual thermal conductivity that is more readily measurable κ = ¯ κ − α ¯ αT /σ DC , and hence κ = 4 πsT e − φ r H Q + a e φ r H (cid:12)(cid:12)(cid:12)(cid:12) r = r H . (69)In conventional metals, the Wiedemann-Franz law holds, and thus the Lorentz ratio is givenby L ≡ κ/ ( σT ) = π / × k B /e . This reflects that for Fermi liquids the ability of thequasiparticles to transport heat is determined by their ability to transport charge so theLorenz ratio is a constant. It has been observed that the Wiedemann-Franz law does nothold in the high-temperature regime with linear temperature-dependent resistivity in heavyfermions [60], signaling the appearance of strong interactions. For our holographic setup,we expect similar non-Fermi behaviors. We find that the Lorenz ratios are given by¯ L ≡ ¯ κσT = s Q + a r H e φ (cid:12)(cid:12)(cid:12)(cid:12) r = r H , (70) L ≡ κσT = s a e φ r H ( Q + a r H e φ ) (cid:12)(cid:12)(cid:12)(cid:12) r = r H . (71)Deviations from the Wiedemann-Franz law can be observed from the above equations andalso as a →
0, ¯ L and κ approach finite, while L goes to zero and ¯ κ diverges. It is clear that if the anisotropic parameter a <
0, the ¯ α , α and ¯ κ will take negative values and thusbecome unphysical because negative thermal conductivity means that thermal current can flow from lowertemperature objects to higher temperature objects spontaneously. From this point of view, we can seethat unstable quasinormal modes in the bulk may result in unphysical transports of the dual field theory.It seems that we should abandon the case τ rel < V. SHEAR VISCOSITIES AND VISCOSITY BOUND
For the anisotropic fluid considered here, the viscosity tensor η ijkl yields two shear viscosi-ties out of five independent components [57]. In the u coordinate, we work with the h uν = 0and A u = 0 gauges and consider linearized fluctuations of the form h xy = e − iωt + ik z z h xy ( u ) forthe transverse shear viscosity, and h xz = e − iωt + ik y y h xz ( u ) for the longitudinal shear viscosity.For the transverse tensor mode h xy , the equation of motion is given by0 = h xy (cid:48)(cid:48) − u h xy (cid:48) + 12 H (cid:48) H h xy (cid:48) + F (cid:48) F h xy (cid:48) − φ (cid:48) h xy (cid:48) + B (cid:48) B h xy (cid:48) − k z h xy F H + ω h xy F B . (72)We introduce the following notation [58]: N µν = 12 κ g xx √− gg µµ g νν , (73)so that the equation of motion for h xy can be written as ∂ u ( N uy ∂ u h xy ) − k z N zy h xy − ω N ty h xy = 0 . (74)The Green function is G xy, xy = N uy ∂ u h xy h xy . (75)The shear viscosity is defined as η xy, xy = − G xy,xy iω . (76)The flow equation for the transverse viscosity is given by ∂ u η xy, xy = iω ( η N uy + N ty ) + iω N zy k z . (77)The transverse shear viscosity is easily obtained by demanding horizon regularity: η xy, xy = ( −N ty N uy ) (cid:12)(cid:12)(cid:12)(cid:12) u = u H = e − φH κ u H = s π . (78)For both prolate and oblate anisotropy, the transverse shear viscosities are exactly s π , andthus the viscosity bound is satisfied. However, for the longitudinal tensor mode, we havethe equation of motion for h zx :0 = h xz (cid:48)(cid:48) − u h xz (cid:48) − H (cid:48) H h xz (cid:48) + F (cid:48) F h xz (cid:48) − φ (cid:48) h xz (cid:48) + B (cid:48) B h xz (cid:48) − k y h xz F + ω h xz F B . (79)18e can recast it as ∂ u η xz, xz = iω ( η N uz + N tz ) + iω N yz k y . (80)The horizon regularity requires η xz, xz = s π H ( u H ) . (81)For prolate black brane solutions with H ( u H ) > η xz, xz s = π H ( u H ) < π violates the KSS bound [5, 58, 59]. We notice that the form of(81) exactly agrees with the result obtained by Rebhan and Steineder in Ref. [4], which isthe first example of a shear viscosity falling below the KSS bound in Einstein gravity withfully known gauge-gravity correspondence. One point that should be pointed out is that in(81) the factor H ( u H ) receives contributions from the gauge fields. Moreover, the transverseshear viscosity η xy, xy = s π reproduces the universal value for Einstein gravity with isotropichorizon geometry, also agreeing with Ref. [4].Comparing the shear viscosity obtained here with that of Einstein-Gauss-Bonnet gravity[52, 53, 61–68], we can find that the anisotropic parameter a plays the same role as theGauss-Bonnet (GB) coupling constant. When the (GB) coupling constant takes a positivevalue, the viscosity bound is violated, but there is no KSS bound violation for the negative-valued GB coupling. A. Causality analysis
In this subsection, we will show how an oblate anisotropy leads to pathological boundaryfield theory by exploring the causality analysis. We know that the anisotropy parameter a acts as an isotropy-breaking external source that forces the system into an anisotropicequilibrium state. The θ parameter is dual to the type-IIB axion χ with the form χ = az inwhich a only plays the role of anisotropy and does not add new degrees of freedom to theSYM theory. On the dual quantum field theory side, oblate solutions with imaginary a looklike a nonunitary deformation and could result in a negative field coupling. In the following,we give a clear explanation how this occurs. Intriguingly, for the oblate black brane solution with a < H ( u H ) <
1, the shear viscosity toentropy density ratio η xz, xz s = π H ( u H ) > π satisfies the KSS bound. However, in this case, the oblateanisotropy results in causality violation and then such a solution is unphysical. h xy ( t, u, z ), we obtain the equation of motion for h xy ∂ u ( N uy ∂ u h xy ) − k z N zy h xy − ω N ty h xy = 0 , (82)with the following notation: N µν = 12 κ g xx √− gg µµ g νν . (83)To see the causality on the boundary, we simply assume h xy = e − iωt + ik z z + ik u u . In the large-momentum limit, the effective geodesic equation can be recast as k µ k ν g eff µν = 0. The effectivemetric can be given by ds = F B ( − dt + HF B dz ) + 1 F du . (84)The local speed of light is given by c g = F BH . (85)In the standard Fefferman-Graham (FG) coordinate, the expansion of the functions F , B ,and H can be written as [3, 7] F = 1 + 11 a v + (cid:18) F + 11 a (cid:19) v + 712 a v log v + O ( v ) , (86) B = 1 − a v + (cid:18) B − a (cid:19) v − a v log v + O ( v ) , (87) H = 1 + a v − (cid:18) B − a (cid:19) v + a v log v + O ( v ) . (88)We can expand the local speed of light c g near the boundary v → c g − − a v + O ( v ) . (89)As the local speed of gravitons should be smaller than 1 (the local speed of the boundaryCFT), we require c g − − a v ≤ . (90)The above ansatz leads to a ≥
0. Following the procedure of Ref. [61], one can find thatthe group velocity of the graviton is given by v g = (cid:52) z (cid:52) t ∼ c g . (91)In a word, as near the boundary c g becomes greater than 1, the propagation of signals in theboundary theory with speed (cid:77) z (cid:77) t could become superluminal. Therefore, causality structureof this theory requires a ≥ . We also note that for the longitudinal modes h xz ( t, u, y ), the local speed of light is exactly 1, i.e. c g = 1 . GUBSER-MITRA CONJECTURE AND “WALL OF STABILITY” The Gubser-Mitra (GM) conjecture claims that gravitational backgrounds with a trans-lationally invariant horizon develop a dynamical instability precisely whenever the specificheat of the black brane geometry becomes negative [34]. The GM conjecture was later re-fined as working provided that there is a unique background with a spatially uniform horizonand specified conserved charges [35]. A holographic realization of the GM conjecture wasgiven in Ref. [36] by demonstrating that a tachyonic mode of the GM instability is dual toan imaginary sound wave in the gauge theory.On the other hand, the wall of stability refers to the regime τ rel ≥
0. This means that thetotal momentum density T zt of the field theory becomes unstable and grows in time when τ rel <
0, because it absorbs momentum, rather than dissipating it. Thus, the fluctuationswill grow exponentially in time. The wall of stability in fact imposes some constraints onthe anisotropic parameter from the dynamical side.It is our purpose in this section to consider the GM conjecture by comparing the dynam-ical to the thermodynamic instabilities in our anisotropic system. For completeness of ourstudy, we also extend our discussions to the massive gravity theory [8] and the Einstein-Maxwell linear scalar theory [10].
A. Dynamical and thermodynamic instabilities in the anisotropic background
In Sec. III, we proved that the relaxation time τ rel is proportional to a . That is to say, τ rel is positive for the prolate anisotropy, as we already knew that the prolate solution hasa thermodynamic instability at smaller horizon radii. This means that the dynamical insta-bility uncovered in this anisotropic background is not correlated with the thermodynamicinstability.It would be interesting to examine the GM conjecture by considering the sound modesin our anisotropic media: whenever the specific heat of the prolate black brane is negative,the speed of sound in such a system should be imaginary. The speed of sound determinedfrom the equation of state is given by v s = ∂P∂ E . (92)The thermodynamic potential in the grand canonical ensemble is found to be G = − P z =21 − T s − ρµ − a Φ [6, 7]. The entropy density can be written as s = − (cid:18) ∂G∂T (cid:19) µ, Φ = (cid:18) ∂P z ∂T (cid:19) µ, Φ . (93)The specific heat can be defined as c µ, Φ = (cid:18) ∂ E ∂T (cid:19) µ, Φ . (94)The speed of sound then can be expressed as v s = sc µ, Φ . (95)This implies that for the case c µ, Φ <
0, the speed of sound is purely imaginary, since theentropy density is always positive. It is clear that imaginary speed of sound in the gaugetheory is unphysical, and this unphysical quantity is related to the tachyonic mode of the GMinstability[36]. This is the meaning of the holographic interpretation of the GM conjecturepresented in Ref. [36]. Similarly, in our paper, a negative relaxation time scale leads totachyonic modes in the bulk gravitational theory and unphysical thermal conductivity onthe dual field side.We can in turn consider the case in which the relaxation time τ rel > B. Instabilities of the black brane in the massive gravity model
The application of massive gravity in holography with broken diffeomorphism invariancein the bulk introduces a mass term for the graviton in such a way that one has momentumrelaxation in the boundary dual field theory. The action of the four-dimensional massive22ravity model is given by [8, 12, 22] S = (cid:90) d x √− g (cid:20) κ (cid:18) R + 6 L + β (cid:0) [ K ] − [ K ] (cid:1)(cid:19) − F µν F µν (cid:21) + 12 κ (cid:90) z = z UV d x √− g b K , (96)where β is an arbitrary parameter having the dimension of mass squared and ( K ) µν ≡ g µρ f ρν , f µν = diag(0 , , , ds = L u (cid:20) − f ( u ) dt + dx + dy + 1 f ( u ) du (cid:21) , (97) A t = µ (1 − uu H ) , (98) f ( u ) = γ µ u L u H − γ µ u L u H − u u H − βu u H + βu + 1 . (99)The black hole temperature is written as T = 34 πu H − γ µ u H πL + βu H π . (100)It is easy to check that for the case β <
0, the local stability condition ∂T /∂u H < r H = 1 /u H increases, the black holetemperature goes up. However, for the case β >
0, there is a branch of black brane solutionshaving ∂T /∂u H >
0. That is what we mean, the instability of the black brane because theheat capacity could become negative. The heat capacity is computed in the usual way: c ρ = ∂ E ∂T = (cid:18) ∂ E /∂u H ∂T /∂u H (cid:19) ρ = (cid:18) − L u H κ − βL u H κ − µ u H (cid:19) β π − κ µ L π − πu . (101)Note that if ∂T /∂u H >
0, the heat capacity becomes negative, since ∂ E /∂u H < β > L + κ µ u L u := β c ,the black brane is thermodynamically unstable.It is interesting to note that the thermodynamic instability uncovered here is related tothe dynamical instability of the dual fluid. The momentum dissipation rate determined interms of the graviton mass and the equilibrium thermodynamical quantities is given by [9] τ − rel = − sβ π ( E + P ) . (102) We would like to thank Richard Davison for figuring out this point. β > τ rel < β than the black brane thermodynamics, and the regime of the dynamicalinstability does not coincide with the regime of the thermodynamic instability completely.This implies that our result provides a counterexample to the GM conjecture: the dynamicalinstability occurs even when the black brane is thermodynamically stable, as we saw abovein the window 0 < β < β c . C. Instabilities of the black brane in the Einstein-Maxwell linear scalars theory
In the holographic model consisting of Einstein-Maxwell theory with linear scalar fields,momentum relaxation can be realized through spatially dependent sources for operators dualto neutral scalars. The five-dimensional action can simply be written as S = (cid:90) M √− g (cid:34) R − − − (cid:88) I ( ∂χ I ) − F (cid:35) d x − (cid:90) ∂M √− γKd x, (103)where Λ = − / (2 l ) and χ I denotes an axion field. The resulting black branes are homo-geneous and isotropic: ds = − f ( r ) dt + dr f ( r ) + r dx i dx i , A t = µ (1 − r H r ) , χ I = β Ii x i , (104) f ( r ) = r − β − m r + µ r H r , m = r H (1 + µ r H − β r H ) . (105)The temperature of the black brane is given by T = 14 π (cid:18) r H − β r H − µ r H (cid:19) . (106)One may notice that through an analytical continuation β → iβ , the black brane tempera-ture here becomes exactly that of the massive gravity. We stress that the sign of the β isarbitrary. The specific heat can be read as c ρ = 8 πr H (cid:18) − β r H + 2 µ r H (cid:19)
14 + β r + µ r . (107)It is easy to find that for any given temperature T > β >
0, the heat capacityis positive. If the constant β is analytically continued to an imaginary value, the black brane24olution will generate an unstable branch with c ρ <
0. Also, the momentum dissipation rateis given by τ − rel = sβ π ( E + P ) , with a sign difference with that of massive gravity. Therefore,considering the parameter β in the range β ∈ ( −∞ , ∞ ), we can conclude that the regimesof thermodynamic and dynamical instabilities do not equal each other. VI. CONCLUSIONS
In summary, we have investigated various aspects of the dynamics of the linear perturba-tions, in particular the effect of the relaxation of momentum upon various observables, in thespatially anisotropic N = 4 super-Yang-Mills theory dual to the action (2). We computedthe dc thermoelectric conductivities analytically. The optical conductivity was obtainedthrough numerical computation. Actually, we uncovered a very interesting mechanism: theunderlying model is anisotropic, and coherent/incoherent metal transition is realized in ourmodel. This is because we only calculated metric perturbations with t and z components ofthe gauge field; the optical conductivity was obtained along the z direction which is provento be momentum dissipated. But in the x and y directions, it is still a metallic phase (seealso Ref. [18]). We also computed the shear viscosities and checked the viscosity boundfor the prolate anisotropy. Finally, we examined the relations between the GM conjectureand the wall of stability by comparing conditions for dynamical instabilities with conditionsfor thermodynamic instabilities in massive gravity and Einstein-Maxwell linear scalar the-ory. It is still an open question to us whether the GM conjecture is strictly obeyed by thisanisotropic system. It was noticed in Ref. [1] that anisotropically deformed N = 4 Yang-Mills plasma at zero chemical potential has low-temperature instabilities, leading to a newground state with anisotropic scaling [69]. It would be interesting to find a new black holesolution at nonzero chemical potential by considering the influence of such an instability andto study the transport properties, because this instability may relate to the GM conjecturediscussed here.The optical conductivity matches the Drude model for small a/µ and exhibits incoherentbehavior at significantly higher values of a/µ . In Ref. [70], it was found that the probefermions in this anisotropic background are a non-Fermi liquid type without well-definedquasiparticle sates. Nevertheless, our results in this paper show that the dual system showsDrude-type behavior with small anisotropy in ac conductivity, although there are no coherent25uasiparticle states. In the small-black-hole-radius branch, which is unstable, the dc electricconductivity shows the strange metal behavior with a linear temperature resistivity. It is afuture project to deform the present model to stabilize this branch, which is very motivatingphenomenologically.The ratio of the shear viscosity to entropy density violated the viscosity bound for theprolate black brane solution. On the other hand, for the momentum relaxation case, theoblate black brane solution is useless in calculating the conductivities: It may give unphys-ical, negative thermoelectric conductivity. In the future, it might be interesting to considerthe holographic transports of anisotropic black branes with higher-derivative gravity termsby adding chemical potential to the model constructed in Ref. [71] and investigate thediffusive bound as in Ref. [72]. Acknowledgements
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