Conductivity in the anisotropic background
aa r X i v : . [ h e p - t h ] F e b Conductivity in the anisotropic background
Bum-Hoon Lee a,b ∗ , Siyoung Nam a † , Da-Wei Pang b ‡ and Chanyong Park b § a Department of Physics, Sogang University, Seoul 121-742, Korea b Center for Quantum Spacetime (CQUeST), Sogang University, Seoul 121-742, Korea
ABSTRACT
By using the gauge/gravity duality, we investigate the dual field theories of the anisotropicbackgrounds, which are exact solutions of Einstein-Maxwell-dilaton theory with a Liouville po-tential. When we turn on the bulk gauge field fluctuation A x with a non-trivial dilaton coupling,the AC conductivity of this dual field theory is proportional to the frequency with an exponentdepending on parameters of the anisotropic background. In some parameter regions, we find thatthis conductivity can have the negative exponent like the strange metal. In addition, we alsoinvestigate another U (1) gauge field fluctuation, which is not coupled with a dilaton field. Weclassify all possible conductivities of this system and find that the exponent of the conductivityis always positive. ∗ e-mail : [email protected] † e-mail : [email protected] ‡ e-mail : [email protected] § e-mail : [email protected] ontents A x a ≤ a > The AdS/CFT correspondence[1, 2], which relates the dynamics of strongly coupled field theoriesto the corresponding dual gravity theories, has provided us a powerful tool for studying physicalsystems in the real world. Recently, inspired by condensed matter physics, the applicationsof the AdS/CFT correspondence to condensed matter physics(sometimes called the AdS/CMTcorrespondence) have been accelerated enormously. As a strong-weak duality, the AdS/CFTcorrespondence makes it possible to investigate the strongly coupled condensed matter systemsin the dual gravity side. Therefore it is expected that we can acquire better understandings forcertain condensed matter systems via the AdS/CFT correspondence. Some excellent reviewsare given by [3].According to the AdS/CFT correspondence, once the dual boundary field theory lies atfinite temperature, there should exist a corresponding black hole solution in the bulk. Thensome properties, like various conductivities [4, 5], superconductor [6, 7] and non-fermi liquid[8, 9], of the dual field theory can be inferred from the black hole. One particular class ofsuch black hole solutions is a charged dilaton black hole [10, 11, 12, 13], where the gaugecoupling of the Maxwell term is governed by a dilaton field φ . Charged dilaton black holesin the presence of a Liouville potential were studied in Ref. [14]. Such charged dialton blackhole solutions possess two interesting properties. First, for certain specific values of the gaugecoupling, the charged dilaton black holes can be embedded into string theory. Second, theentropy vanishes in the extremal limit, which may signify that the thermodynamic descriptionbreaks down at extremality. Such peculiar features suggest that their AdS generalizations may1rovide interesting holographic descriptions of condensed matter systems.Recently, holography of charged dilaton black holes in AdS with planar symmetry was ex-tensively investigated in [15]. The near horizon geometry was Lifshitz-like with a dynamicalexponent z determined by the dilaton coupling. The global solution was constructed via nu-merical methods and the attractor behavior was also discussed. The authors also examinedthe thermodynamics of near extremal black holes and computed the AC conductivity in zero-temperature background. For related works on charged dilaton black holes see [16, 17, 18, 19].In this paper we focus on conductivities of charged dilaton black hole solutions [20] with aLiouville potential at both zero and finite temperature. At first we obtain exact solutions, bothextremal and non-extremal, of the Einstein-Maxwell-dilaton theory, where the scalar potentialtakes the Liouville form. The extremal solution possesses anisotropic scaling symmetry whichreduces to the Lifshitz-like metric [21] in a certain limit. In Ref. [15, 22], after inserting anirrelevant operator which deforms the asymptotic geometry to AdS , the electric conductivitywas calculated. In this paper, we concentrate on the undeformed geometry and find exactsolution describing undeformed geometry. Secondly we calculate the electric conductivity byconsidering the fluctuation of the gauge field A x . The corresponding equation of motion for A x can be transformed into a Schr¨odinger equation, which enables us to evaluate the conductivityeasily [6]. In the usual (deformed) AdS spaces, because there is no non-trivial dilaton couplingto the gauge field the conductivity of the dual theory is usually proportional to the frequencywith the non-negative exponent, which is opposite to the strange metallic one. In the anisotropicbackground the electric conductivity generally is proportional to the frequency with an exponentdetermined by the non-trivial dilaton coupling and other background geometric parameters.Furthermore, in appropriate parameter regions we can find the strange metal-like conductivityproportional to the frequency with the negative exponent.Next we introduce another U (1) gauge field, which does not have a coupling with the dilatonfield, and calculate the conductivity by considering the fluctuations of this new gauge field. Thereare some motivations for considering new gauge field with different dilaton coupling. First, if ourfour-dimensional gravity theory is originated from the ten dimensional string theory, there existother many gauge fields having different dilaton coupling depending on how the string theoryis compactified. Second, if we promote this bottom-up approach to the top-down approach, thegauge field living on the probe brane world volume typically has a dilaton coupling like e − φ . Sostudying the dynamics of the bulk gauge fluctuations having various dilaton couplings would shedlight on understanding dual gauge theories. In this paper, we concentrate on a special case inwhich the gauge field fluctuation has no dilaton coupling and calculate the electric conductivitiesof the dual system. We find that the exponent of the conductivity for all parameter regions isalways positive. This is the main difference between the dual theory with or without a non-trivialdilaton coupling, which implies that that the non-trivial dilaton coupling is very important toexplain the strange metallic behavior. 2he rest of the paper is organized as follows: in Section 2 we find the exact solutions ofthe charged dilaton black holes with a Liouville potential, both extremal and non-extremal. Wecalculate the conductivity by turning on the gauge field fluctuations coupled with the dilatonfield in Section 3. In Section 4 we calculate the conductivity induced by a new U (1) gauge field,which is not coupled with the dilaton field. Summary and discussion will be given in the finalsection. Note added : In the final stage of this work, we noticed that similar solutions were studiedextensively [22]. However, one key difference is that they required the solutions to be asymp-totically
AdS while we treat the solutions to be global.
In the real condensed matter theory systems containing anisotropic scaling commonly appear.To describe this system holographically the Lifshitz background has been studied by manyauthors [21] by using the generalized AdS/CFT correspondence, so called the gauge/gravityduality. Here, we consider a different background having more general anisotropic scaling andinvestigate physical quantities of the dual field theory.We start with the following action S = Z d x √− g [ R − ∇ φ ) − e αφ F µν F µν − V ( φ )] , (1)where φ and V ( φ ) represent a dilaton field and its potential. Notice that the non-trivial dilatoncoupling in front of the Maxwell term can provide a different physics from the ordinary relativisticfield theory dual to AdS space. So main goal of the present work is to investigate the dilatoncoupling effect on the physical quantity like the conductivity of the dual field theory. Equationsof motion for metric g µν , dilaton field and U(1) gauge field are R µν − Rg µν + 12 g µν V ( φ ) = 2 ∂ µ φ∂ ν φ − g µν ( ∇ φ ) + 2 e αφ F µλ F νλ − g µν e αφ F , (2) ∂ µ ( √− g∂ µ φ ) = 14 √− g ∂V ( φ ) ∂φ + α √− ge αφ F , (3)0 = ∂ µ ( √− ge αφ F µν ) . (4)Now, we choose a Liouville-type potential as a dilaton potential V ( φ ) = 2Λ e − ηφ . (5)For η = 0, the dilaton potential reduces to a cosmological constant, which was studied inRef. [15]. To solve equations of motion, we use the following ansatz corresponding to a zerotemperature solution ds = − a ( r ) dt + dr a ( r ) + b ( r ) ( dx + dy ) , (6)3ith a ( r ) = t r a , b ( r ) = b r b , φ ( r ) = − k log r. (7)If we turn on a time-component of the gauge field A t only, from the above metric ansatz theelectric flux satisfying Eq. (4) becomes F tr = qb ( r ) e − αφ . (8)The rest of equations of motion are satisfied when various parameters appearing in the aboveare given by a = 1 + k η, b = (2 α − η ) (2 α − η ) + 16 , k = 4(2 α − η )(2 α − η ) + 16 , b = 1 ,t = − a + b )(2 a + 2 b − , q = − (cid:18) k a + b + η (cid:19) Λ α , (9)where Λ is negative. This solution described by three free parameters α , η and Λ is an exactsolution of equations of motion. Notice that b in Eq. (9) is always smaller than 1. For η = 0and Λ = − α = η ,the above solution becomes AdS × R . If we take a limit α → ∞ and at the same time set q = η = 0, we can obtain AdS geometry. When η is proportional to α like η = cα , the metricin the limit, α → ∞ , reduces to a Lifshitz-like one ds = − t r z dt + dr t r z + r (cid:0) dx + dy (cid:1) , (10)with z = 2 + c − c . (11)The exponent z is given by 2 for c = 2 / c = 1, etc.The above zero temperature solution can be easily extended to a finite temperature onedescribing a black hole. With the same parameters in Eq. (9), the black hole solution becomes ds = − a ( r ) f ( r ) dt + dr a ( r ) f ( r ) + b ( r ) ( dx + dy ) , (12)where f ( r ) = 1 − r a +2 b − h r a +2 b − . (13)Notice that since the above black hole factor does not include U(1) charge this solution corre-sponds to not a Reissner-Nordstr¨om but Schwarzschild black hole. The Hawking temperatureof this black hole is given by T ≡ π ∂ ( a ( r ) f ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) z = z h = (2 a + 2 b − t r a − h π , (14)where r h means the position of the black hole horizon.4 Fluctuation of background gauge field A x In this section, we consider the gauge field fluctuation A x and calculate the electric conductivityof the dual theory. For convenience, we introduce new coordinate variable u = r b . Then, themetric in Eq. (12) becomes ds = − g ( u ) f ( u ) e − χ ( u ) dt + du g ( u ) f ( u ) + u (cid:0) dx + dy (cid:1) , (15)where g ( u ) = t b u a + b − /b ,e χ ( u ) = b u b − /b , (16)with the black hole factor f ( u ) = 1 − u (2 a +2 b − /b h u (2 a +2 b − /b . (17)On this background, we turn on the gauge field fluctuation A x together with the metricfluctuation g tx . Although there is another relevant metric fluctuation g ux , in the g ux = 0 and A u = 0 gauge only A x and g tx describe the vector fluctuations of the bulk theory. From theaction for the gauge fluctuation δS = − Z d x √− g h F µν F µν , (18)where h = 4 e αφ is a coupling function, the equation of motion for A x becomes0 = 1 √− g ∂ µ (cid:0) √− g h g µρ g xν F ρν (cid:1) . (19)Under the following ansatz, A x ( t, u ) = Z dw π e − iwt A x ( w, u ) ,g tx ( t, u ) = Z dw π e − iwt g tx ( w, u ) , (20)the above equation for A x can be rewritten as0 = ∂ u (cid:16) e − χ gf h ∂ u A x (cid:17) + w e χ h gf A x + e χ h ( ∂ u A t ) (cid:18) g ′ tx − u g tx (cid:19) (21)and the ( u, x )-component of Einstein equation is given by g ′ tx − u g tx = − h ( ∂ u A t ) A x . (22)Combining above two equations gives0 = ∂ u (cid:16) e − χ gf h ∂ u A x (cid:17) + w e χ h gf A x − e χ/ h ( ∂ u A t ) A x . (23)5ntroducing a new variable and new wave function − ∂∂v = e − χ g ∂∂u ,A x = Ψ √ f h , (24)Eq. (23) simply reduces to a Schr¨odinger-type equation0 = Ψ ′′ + V ( v )Ψ (25)with the effective potential V ( v ) = (cid:18) w + ( f ′ ) (cid:19) f − (cid:18) f ′ h ′ h + f ′′ h g e χ ( A ′ t ) (cid:19) f − h ′′ h , (26)where the prime implies a derivative with respect to v . Using the first equation in Eq. (24) wecan easily find u as function of v . Here, we will concentrate on the case 2 a >
1, in which v isgiven by v = 1(2 a − t u a − b . (27)In v -coordinate the boundary ( u = ∞ ) is located at v = 0. In the black hole geometry, the blackhole horizon v h is given by v h = 1(2 a − t u a − b h . (28)So the zero temperature corresponds to put the black hole horizon to v = ∞ ( u = 0). We first consider the zero temperature case, in which f = 1 and f ′ = f ′′ = 0. The effectivepotential Eq. (26) reduces to V ( v ) = w − h g e χ ( A ′ t ) − h ′′ h . (29)Using Eq. (24), the above effective potential can be written as V ( v ) = w − cv , (30)with a constant c c = 4(16 + 4 α − η ) [8 + (2 α − η )( α + η )](16 + 4 α + 4 αη − η ) , (31)where we use Eq. (9). The exact solution Ψ of the Schr¨odinger equation becomesΨ = c √ vH (1) δ ( wv ) + c √ vH (2) δ ( wv ) , (32)where H ( i ) implies the i -th kind of Hankel function and δ = √ c . (33)6t the horizon ( v = ∞ ), the first or second term in Eq. (32) satisfies the incoming or outgoingboundary condition respectively. So to pick up the solution satisfying the incoming boundarycondition we should set c = 0. Then, the solution has the following expansion near the boundaryΨ ≈ Ψ (cid:18) v − δ − (cid:16) w (cid:17) δ Γ(1 − δ )Γ(1 + δ ) e − iπδ v + δ (cid:19) (34)with c = iπ Γ( δ ) (cid:16) w (cid:17) δ Ψ . (35)From this result together with Eq. (24), A x at the boundary ( u = ∞ ) becomes A x = A (cid:16) u [ αk − (2 a − − δ ) ] /b − ˜ c u [ αk − (2 a − + δ ) ] /b (cid:17) , (36)where A = Ψ (cid:2) (2 a − t (cid:3) − δ , ˜ c = (cid:16) w (cid:17) δ Γ(1 − δ )Γ(1 + δ ) e − iπδ (cid:2) (2 a − t (cid:3) δ . (37)The boundary action for the gauge fluctuation A x becomes S B = − Z d x √− g h g uu g xx A x ∂ u A x = 12 Z d k t ˜ c (cid:20) (2 a − δ ) − αk (cid:21) A , (38)so that the Green function is given by G xx ≡ ∂ S B ∂A ∂A = 4 t (cid:20) (2 a − δ ) − αk (cid:21) ˜ c ∼ w δ . (39)Finally, the AC conductivity of the dual system reads off σ = G xx iw ∼ w δ − . (40)For the DC conductivity we should set w = 0. Then, the DC conductivity of this system becomesinfinity for 2 δ < δ >
1. In the real world, there exist some condensed mattersystems having the AC conductivity with a negative exponent like the strange metal [23, 24]. Ifwe choose 2 δ − − .
65. the AC conductivity of this system can describes the strange metallicbehavior σ ∼ w − . . Notice that since there are two free parameters, α and η , for Λ = − α, η ) ≈ (1 , . , (2 , . , (2 , . , · · · , satisfy σ ∼ w − . and givethe regular value, q >
0. As will be shown in Sec. 4, if we consider the gauge field fluctuationwithout the dilaton coupling the conductivity becomes a constant when the spatial momentumis zero. If turning on the spatial momentum, the conductivity always grows up as the frequencyincreases. These behaviors are different with those of the strange metal, which implies that thedilaton coupling of the gauge field plays an important role for describing the strange metallicbehavior holographically. In the next section, we will show the strange metallic behavior atfinite temperature. 7 .2 At finite temperature
We consider the gauge and metric fluctuations at the finite temperature background. At thehorizon, the dominant term in the effective potential in Eq. (26) is U ( v ) ≈ (cid:18) w + ( f ′ h ) (cid:19) f h = (cid:18) w + (2 a + 2 b − a − v h (cid:19) f h , (41)where f h means the value of f at the horizon. Then, the approximate solution is given byΨ = c f ν − + c f ν + , (42)where ν ± = 12 ± i s w + (2 a + 2 b − a − v h − . (43)Notice that at the horizon the first and second term in Eq. (42) satisfy the incoming andoutgoing boundary condition respectively. Imposing the incoming boundary condition we canset c = 0.Now, we investigate the asymptotic behavior of Ψ. Since the leading behavior of the effectivepotential near the boundary is given by Eq. (30), the perturbative solution can be described byΨ = d v − δ + d v + δ , (44)where δ has been defined in Eq. (33) together with Eq. (31). By solving Eq. (25) numerically,we can determine the numerical value of d and d . In u -coordinate, the gauge field fluctuationhas the following perturbative form A x = A u [ ak − (2 a − − δ ) ] /b + d d (cid:2) (2 a − t (cid:3) δ u [ ak − (2 a − + δ ) ] /b ! , (45)where to determine the boundary value of A x as A we set d = 2 (cid:2) (2 a − t (cid:3) − δ A . (46)The boundary action for A x becomes S B = A Z d x t d (cid:2) (2 a − + δ ) + b − αk (cid:3) d (cid:2) (2 a − t (cid:3) δ . (47)Then, we can easily find the AC conductivity at finite temperature σ = 4 t iw (cid:20) (2 a − δ ) + b − αk (cid:21) d d (cid:2) (2 a − t (cid:3) δ , (48)where the last part d d [ (2 a − t ] δ can be numerically calculated by solving the Schr¨odingerequation together with the initial data at the horizon. In Figure 1, we plot the real and imaginaryAC conductivity. 8 Ω Σ Ω- ´ - ´ - ´ - ´ - ´ Σ Figure 1:
The conductivity at the finite temperature where we choose α = 2, η = 1 and Λ = − We can fit the dual AC conductivity of Figure 1 with the following expected form σ = a w − b . (49)If we choose α = 2, η = 1 and Λ = − a ≈ .
045 and b ≈ . As mentioned in Introduction, it is interesting to investigate the conductivity of the gauge fieldhaving different dilaton coupling. Here, we will concentrate on new gauge field fluctuation havingno dilaton coupling [25]. Even in this case, due to the parameters in the original action, we canfind several different conductivities depending on the parameter region.Before starting the calculation for the Green functions and conductivity in various parameterregions, we introduce a different coordinate z = 1 /r for later convenience. In the z -coordinate,the black hole metric is rewritten as ds = − t z a f ( z ) dt + z a dz t z f ( z ) + dx + dy z b (50)with f ( z ) = 1 − z a +2 b − z a +2 b − h , (51)where all parameters are same as ones in Eq. (9) and z h implies the event horizon of the blackhole. In this coordinate, the Hawking temperature becomes T = (2 a + 2 b − t πz a − h . (52)Now, we introduce another U(1) gauge field fluctuation, which is not coupled with a dilatonfield δS = − Z d x √− gf µν f µν , (53)9here f µν = ∂ µ a ν − ∂ ν a µ and we absorb a gauge coupling constant to the gauge field. Toobtain equations of motion for new gauge field, we first choose a z = 0 gauge and turn on the x -component of the gauge fluctuation only a x ( x ) = Z d k (2 π ) e − iwt + i~k · ~x a x ( k, z ) , (54)where a vector ~x corresponds to two-dimensional spatial coordinates. In the comoving frame, ~k = ( w, , k ), the equation of this gauge fluctuation becomes0 = a ′′ x + (cid:18) − a ) z + f ′ f (cid:19) a ′ x + (cid:18) w t z − a ) f − k t z − a − b f (cid:19) a x , (55)where the prime implies a derivative with respect to z and f is a black hole factor. In this section, we investigate various Green functions at zero temperature, which can be ob-tained by setting f = 1 and f ′ = 0. a ≤ b , as mentioned previously, is always smaller than 1 and the case, b = 1, can beconsidered as the limit α → ∞ . i) < a = b ≤ a = b ≤
1. In this case, the equation governing thetransverse gauge field fluctuation becomes0 = a ′′ x + 2 δz a ′ x + γz δ a x , (56)with δ = 1 − a and γ = w t − k t , (57)where 0 ≤ δ <
1. The exact solution of the above equation is given by a x = c exp( i √ γz − δ − δ ) + c exp( − i √ γz − δ − δ ) . (58)At the horizon ( z → ∞ ), the first or second term satisfies the incoming or outgoing boundarycondition respectively. Imposing the incoming boundary condition, the solution reduces to a x = c exp (cid:18) i √ γ − δ z − δ (cid:19) . (59)10 =0 k=2k=1 k=2 k=1 k=2 Figure 2:
The real and imaginary conductivity at t = 1 with a = b ≤ For δ > /
2, we can not purturbatively expand this solution near the boundary ( z = 0). Insuch case it is unclear how to define the dual operator, so we consider only the case δ < / a > /
2) from now on. In this case, a x has the following expansion near the boundary a x = a (cid:18) i √ γ − δ z − δ + · · · (cid:19) , (60)where a = c corresponds to the boundary value of a x , which can be identified with the sourceterm of the dual gauge operator. According to the gauge/gravity duality, the on-shell gravityaction can be interpreted as a generating functional for the dual gauge operator. The on-shellgravity action corresponding to the boundary action is given by S B = 12 Z d x √− g g zz g xx a x ∂ z a x = ia Z d kt √ γ. (61)Using this on-shell action we can easily calculate the Green function by varying the on-shellaction with respect to the source.For a x , the Green function becomes G xx ≡ ∂ S B ∂a ∂a = iw r − k t w , (62)and the conductivity of the dual system is given by σ = G xx iw = r − t k w . (63)For the time-like case ( w > k t ), the conductivity is real. In the space-like case, the imaginaryconductivity appears. In addition, the AC conductivity for k = 0 becomes a constant σ AC = 1,in which there is no imaginary part of the conductivity. In Figure 2, we plot the real andimaginary conductivity, in which we can see that as the momentum k increases the real orimaginary conductivity decreases or increases respectively. Furthermore, as shown in Eq. (63)and figure 2, the real and imaginary conductivities become zero at w = t k . Below or abovethis critical point, there exists only the imaginary or real conductivity respectively.11 i) < b < a ≤ a and b . So, insteadof solving Eq. (55) analytically, we will try to find a Green function and electric conductivitynumerically. To do so, we should first know the perturbative behavior of a x near the horizon aswell as the asymptotic boundary.At the horizon, since 1 /z − a ) term in Eq. (55) is dominant, the approximate solutionsatisfying the incoming boundary condition is given by a x = c exp (cid:18) i wt (2 a − z a − (cid:19) . (64)Near the boundary, 1 /z − a − b term in Eq. (55) is dominant, so Eq. (55) is reduced to0 = a ′′ x + 2(1 − a ) z a ′ x − k t z − a − b a x . (65)The leading two terms of asymptotic solution are a x = c + c z a − , (66)where c and c are integration constants. To find a relation between two integration constantsand c , we should solve Eq. (55) numerically with the initial conditions determined from Eq.(64). To control the limit z → ∞ we introduce an IR cut-off z , which is a very large number.At this IR cut-off, a x and a ′ x become a x ( z ) = c exp (cid:18) i wt (2 a − z a − (cid:19) ,a ′ x ( z ) = icw z a − t exp (cid:18) i wt (2 a − z a − (cid:19) . (67)Using these initial values, we can solve Eq. (55) numerically and find numerical values for a x ( ǫ )and a ′ x ( ǫ ), where ǫ ( ǫ →
0) means an UV cut-off. Then, two integration constants c and c canbe determined by a x ( ǫ ) and a ′ x ( ǫ ) c = a x ( ǫ ) ,c = a ′ x ( ǫ )(2 a − ǫ a − . (68)If we identify the boundary value of a x with a source term a a = lim ǫ → a x ( ǫ ) . (69) c and c correspond to a source and the expectation value of the boundary dual operatorrespectively. Then, from the boundary action the electric conductivity becomes σ = t a ′ x ( ǫ ) iwa x ( ǫ ) ǫ a − . (70)12 =0 k=1 k=2 k=0 k=1 k=2 Figure 3:
The real and imaginary conductivity at t = 1 with < b < a ≤ In Figure 3, we plot the real and imaginary conductivity. Notice that in this case there is nocritical point like the < a = b ≤ k the real conductivitybecomes zero as the frequency goes to zero. For k = 0, the real conductivity is a constant likethe previous case. For k = 0 the conductivity grows as the frequency increases, which is oppositeto the strange metallic conductivity. iii) < a < b ≤ k term in Eq. (55) is dominant at the horizon. Due to the sign of it, the nearhorizon behavior of this solution is space-like. So we should impose the regularity condition in-stead of the incoming boundary condition. More precisely, in this parameter region the equationgoverning a x at the horizon reduces to Eq. (65). The exact solution of it is given by a x = z a − (cid:0) d ′ I − ν ( x ) + d ′ I ν ( x ) (cid:1) , (71)with two integration constants d ′ and d ′ , where I ν ( x ) is a modified Bessel function and ν = 2 a − a + b −
1) and x = kz a + b − t ( a + b − . (72)At the horizon, the leading terms of it become a x = 1 z ( b − a ) / (cid:20) d exp (cid:18) − kz a + b − t ( a + b − (cid:19) + d exp (cid:18) kz a + b − t ( a + b − (cid:19)(cid:21) , (73)where d and d are different constants with d ′ and d ′ . In the above, the second term divergesat the horizon z → ∞ , so we can pick up the regular solution by imposing the regularity at thehorizon, which is the same as imposing the incoming boundary condition when k → − ik . Fromthe horizon solution a x = d exp (cid:16) − kz a b − t ( a + b − (cid:17) z ( b − a ) / , (74)13 =1k=0.5 Figure 4:
The imaginary conductivity at t = 1 with < a < b ≤ we can determine initial values for a x ( z ) and a ′ x ( z ) at the IR cut-off z . The near boundarysolution in < a < b ≤ a x in Eq. (74)is real due to the regularity condition at the horizon. So the resulting Green function is alsoreal, which implies that the conductivity is a pure imaginary number. Following the techniqueexplained in the previous section, the unknown integration constants c and c can be determinedby the boundary values a x ( ǫ ) and a ′ x ( ǫ ) after solving Eq. (55) numerically. Figure 4 shows theimaginary conductivity. a > a x becomes0 = a ′′ x − a − z a ′ x + (cid:18) w t z a − − k t z a + b − (cid:19) a x . (75)At the horizon( z → ∞ ), the first term proportional to w is dominant, so the approximatesolution satisfying the incoming boundary condition is given by a x ≈ exp (cid:18) iwz a − t (2 a − (cid:19) . (76)For k = 0, the above is an exact solution satisfying the incoming boundary condition, which isthe same as one in Eq. (59) with k = 0. In this case, from the result in Eq. (63) we can easilyfind that the conductivity becomes 1. Notice that in Figure 5 the numerical calculation for theconductivity at k = 0 gives the same result obtained by analytic calculation.Near the boundary ( z → k term is dominant, so the approximate solution becomes a x ≈ a (cid:0) c z a − (cid:1) , (77)where a is the boundary value of a x . To determine a constant c , we numerically solve thedifferential equation in Eq. (75) with the initial values given at the horizon. Then, we canfind the values for a x and a ′ x at the boundary. Comparing these results with Eq. (77), we can14 =0k=1 k=2 k=0k=1 k=2 Figure 5:
The real and imaginary conductivity when a = 5 / b = 3 / t = 1 . determine the unknown constant c numerically. Using these results, the Green functions andconductivity are given by G xx = t (2 a − c,σ = G xx iw . (78)In Figure 5, we present several conductivity plots depending on the momentum, which is verysimilar to one obtained in the < b < a ≤ < b < a ≤ a > k = 1 or 2. First, we consider the simplest case k = 0 in which we can obtain an exact solution. To solveEq. (55), we introduce a new coordinate du = dzz − a f . (79)Then, in the u -coordinate Eq. (55) for k = 0 reduces to0 = ∂ u a x + w t a x , (80)where the relation between u and z is given by u = z a − a − F (cid:18) a − a + 2 b − , a + b − a + 2 b − , z a +2 b − (cid:19) . (81)Near the horizon ( z ∼ z h ), the above relation becomes u ≈ − z a − h a + 2 b − z h − z ) , (82)so the horizon lies at u = ∞ for a > /
2. Near the boundary ( z → u is related to zu ≈ z a − a − . (83)15t the horizon, the solution of Eq. (80) satisfying the incoming boundary condition is given by a x = a exp (cid:18) i wut (cid:19) , (84)where a is the boundary value of a x . Near the boundary, the expansion of the above solutionbecomes a x = a (cid:18) i w (2 a − t z a − (cid:19) , (85)which is the same as one in the zero temperature case. Therefore, the Green function andconductivity for k = 0 is the same as the zero temperature result.Now, we consider the general case with non-zero k . In this case, it is very difficult to find ananalytic solution, so we adopt a numerical method. Since the black hole factor f becomes zeroat the horizon ( z → z h ), the differential equation in Eq. (55) near the horizon reduces to0 = a ′′ x + f ′ f a ′ x + w t z − a ) f a x , (86)where f = 1 − z a +2 b − z a +2 b − h . (87)The leading term of the solution satisfying the incoming boundary condition is a x = cf − iν (1 + · · · ) , (88)with ν = w z a − h (2 a + 2 b − t , (89)where the ellipsis implies higher order terms and c is an integration constant. Since at z = z h a x and a ′ x become zero, we pick up the initial values at z i = z h − ǫ , where ǫ is a very smallnumber, for solving the differential equation numerically. After solving Eq. (55), we can findnumerical values for a x ( ǫ ) and a ′ x ( ǫ ), where ǫ is a small number, up to multiplication constant c at the boundary.To understand the boundary behavior of a x , we should know the perturbative form of a x near the boundary. From Eq. (55), the perturbative solution near the boundary is given by a x = a (cid:0) A z a − (cid:1) , (90)where a is the boundary value of a x and A is an integration constant, which will be determinedlater. Using the above numerical results a x ( ǫ ) and a ′ x ( ǫ ), a x can be written as a x ( z ) = c (cid:18) a x ( ǫ ) + a ′ x ( ǫ )2 a − z a − ǫ a − (cid:19) , (91)Matching this value with Eq. (90) gives c = a a x ( ǫ ) and A becomes A = a ′ x ( ǫ )(2 a − ǫ a − a x ( ǫ ) . (92)16 =0 k=1k=1 k=2 k=0 k=1 k=2 Figure 6:
The real and imaginary conductivity at finite temperature T = 0 .
239 ( z h = 1) when a = 5 / b = 3 / t = 1 . Therefore, the Green function and conductivity can be written as G xx = (2 a − t A,σ ≡ G xx iw = t iwǫ a − a ′ x ( ǫ ) a x ( ǫ ) . (93)Figure 6 shows the conductivity at finite temperature. Similarly to the zero temperature cases,for k = 0 the real conductivity is still a constant. But for k = 1 or 2, the finite temperatureconductivity goes to a constant as the frequency goes to zero, while the zero temperature oneapproaches to zero. This implies that the finite temperature DC conductivity is a constant whilethe zero temperature DC conductivity is zero. We can also see that like the zero temperature casethe conductivity grows as the frequency increases, which is different with the strange metallicbehavior. Moreover, since our background geometry is not a maximally symmetric space, thedual boundary theory is not conformal. So we can expect that there exists the non-trivialtemperature dependence of the conductivity. In Figure 7 we plot the electric conductivitydepending on temperature. There exist two complementary approaches for studying applications of the gauge/gravity dualityto condensed matter systems. One is the bottom-up approach in which on a given gravitysolution we investigate the dual field theory. The other is the top-down approach in which, afterconsidering the probe brane configuration on the geometry obtained from the string theory, thedual field theory is investigated by studying various fluctuations on the probe brane. In thispaper we mainly adopt the bottom-up approach. For the top-down approach, see Ref. [26].Due to the peculiar properties, charged dilaton black holes may provide new backgroundsfor describing the gravity duals of certain condensed matter systems. In this paper we studiedconductivities of charged dilaton black holes with a Liouville potential both at zero and finitetemperature. In general, the anisotropic (charged dilaton) black hole has three free parameters.17epending on which parameters we choose, the anisotropic background can reduce to
AdS , AdS × S and the Lifshitz-like space. To investigate the dual theory which may describe someaspects of the condensed matter system, we have calculated its electric conductivity in variousparameter regions.First, we have considered the gauge fluctuation of the background gauge field, which iscoupled with the dilaton field. Due to this non-trivial dilaton coupling, the conductivity of thissystem depends on the frequency non-trivially. After choosing appropriate parameters, at bothzero and finite temperature we obtained the strange metal-like AC conductivity proportional tothe frequency with a negative exponent. Because our model have three free parameters, it mayshed light on investigating other dual field theory having more general exponent.Second, we have also investigated another dual field theory whose dual gravity theory hasthe another U(1) gauge field fluctuation without the dilaton coupling. Due to the parametersin the original action, there are several parameter regions giving different conductivities. Wehave classified all possible conductivities either analytically or numerically. Here, we foundthat the conductivities for k = 0 at zero and finite temperature become a constant becausethere is no dilaton coupling with the new gauge field fluctuation. This implies that to describethe non-trivial conductivity depending on the frequency like the strange metal, it would beimportant to consider the dilaton coupling effect at least in the bottom-up approach. We havealso investigated the conductivity depending on the spatial momentum and the temperature.As the spatial momentum increases we found that the real conductivity goes down. In addition,we found that the finite temperature DC conductivity becomes a non-zero constant while thezero temperature one is zero in this set-up.One further generalization is to incorporate the magnetic field and to find dyonic black holesolutions. Once we find such exact solutions carrying both electric and magnetic charges, itwould be interesting to study the thermodynamics and transport coefficients, such as the Hallconductivity [5], in the presence of the magnetic field. Another interesting direction is to studythe non-Fermi liquid behavior in the solutions we obtained, following [8, 9, 27, 28]. We hope toreport some results about such fascinating topics in the future. Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant fundedby the Korea government (MEST) through the Center for Quantum Spacetime (CQUeST) ofSogang University with grant number 2005-0049409. C. Park was also supported by BasicScience Research Program through the National Research Foundation of Korea(NRF) fundedby the Ministry of Education, Science and Technology(2010-0022369).18igure 7:
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