Holographic conductivity in the massive gravity with power-law Maxwell field
aa r X i v : . [ h e p - t h ] J un Holographic conductivity in the massive gravity with power-law Maxwell field
A. Dehyadegari, ∗ M. Kord Zangeneh,
2, 3, † and A. Sheykhi
1, 4, ‡ Physics Department and Biruni Observatory, Shiraz University, Shiraz 71454, Iran Physics Department, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz 61357-43135, Iran Center of Astronomy and Astrophysics, Department of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran
We obtain a new class of topological black hole solutions in ( n + 1)-dimensional massive gravity inthe presence of the power-Maxwell electrodynamics. We calculate the conserved and thermodynamicquantities of the system and show that the first law of thermodynamics is satisfied on the horizon.Then, we investigate the holographic conductivity for the four and five dimensional black branesolutions. For completeness, we study the holographic conductivity for both massless ( m = 0)and massive ( m = 0) gravities with power-Maxwell field. The massless gravity enjoys translationalsymmetry whereas the massive gravity violates it. For massless gravity, we observe that the realpart of conductivity, Re[ σ ], decreases as charge q increases when frequency ω tends to zero, while theimaginary part of conductivity, Im[ σ ], diverges as ω →
0. For the massive gravity, we find that Im[ σ ]is zero at ω = 0 and becomes larger as q increases (temperature decreases), which is in contrast tothe massless gravity. It also has a maximum value for ω = 0 which increases with increasing q (withfixed p ) or increasing p (with fixed q ) for (2 + 1)-dimensional dual system, where p is the powerparameter of the power-law Maxwell field. Interestingly, we observe that in contrast to the masslesscase, Re[ σ ] has a maximum value at ω = 0 (known as the Drude peak) for p = ( n + 1) / q . In this case ( m = 0) andfor different values of p , the real and imaginary parts of the conductivity has a relative extremum for ω = 0. Finally, we show that for high frequencies, the real part of the holographic conductivity havethe power law behavior in terms of frequency, ω a where a ∝ ( n + 1 − p ). Some similar behaviorsfor high frequencies in possible dual CFT systems have been reported in experimental observations. PACS numbers: 97.60.Lf, 04.70.-s, 71.10.-w, 04.70.Bw, 04.30.-w.
I. INTRODUCTION
A century after Einstein’s discovery namely general relativity, the domain of its applications has become as vast asit covers even condensed matter physics which seemed at the opposite end of physics building compared to gravity[1]. This strange topic which connects gravity to almost all fields of physics (see [2]) is called gauge/gravity duality(GGD); the extended version of AdS/CFT correspondence [3]. GGD has attracted increasing interests during recentyears and become one of the most promising fields of physics which is hoped to be able to solve many of unsolvedproblems in different fields of physics including condensed matter physics.Real materials in condensed matter physics do not respect the translational symmetry i.e. there is a dissipationin momentum. The momentum dissipation may come from the existence of a lattice or impurities. Although thisdissipation has no important influence on the values of some observable, it affects the behavior of some others forinstance conductivity. The DC conductivity in the presence of translational symmetry diverges, whereas in the absenceof this symmetry (when momentum is dissipating) it has a finite value. In the context of GGD, it is important tostudy a gravity model which includes holographic momentum dissipation. There are some attempts to construct suchgravity model [4]. One of these models proposed by D. Vegh [5], provides an effective bulk description of a theory inwhich momentum is no longer conserved. The conservation of momentum is due to the diffeomorphism invariance ofstress-energy tensor in dual theory. In [5], the proposal is to break this symmetry holographically by giving a massto graviton state. The resulting gravity is therefore massive gravity . One of the advantages of this theory is that theblack hole solutions of it are solvable analytically and therefore it is an excellent toy model to study holographicallythe properties of materials without momentum conservation. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
Thermal behaviors of black hole solutions in the context of massive gravity was explored extensively in recentyears [5–8]. Thermodynamics of linearly charged massive black branes has been investigated in [5]. In [6], a class ofhigher-dimensional linearly charged solutions with positive, negative and zero constant curvature of horizon in thecontext of massive gravity accompanied by a negative cosmological constant has been presented and thermodynamicsand phase structure of these black solutions have been studied in both canonical and grand canonical ensembles. In[7], van der Waals phase transitions of linearly charged black holes in massive gravity have been investigated and ithas been shown that the massive gravity can present substantially different thermodynamic behavior in comparisonwith Einstein gravity. Also it has been shown that the graviton mass can cause a range of new phase transitions fortopological black holes which are forbidden for other cases. The properties of massive solutions have been studiedin different scenarios [9]. From holographic point of view, the behaviors of different holographic quantities havebeen studied [5, 10–22]. The behavior of holographic conductivity for systems dual to linearly charged massiveblack branes has been explored in [5]. In [11], a holographic superconductor has been constructed in the massivegravity background. [13] studies holographic superconductor-normal metal-superconductor Josephon junction in themassive gravity. Also the holographic thermalization process has been investigated in this context [14]. Analytic DCthermo-electric conductivities in the context of massive gravity have been calculated in [12]. In massive Einstein-Maxwell-dilaton gravity, DC and Hall conductivities have been computed in [15]. [16] presents a holographic modelfor insulator/metal phase transition and colossal magnetoresistance within massive gravity. Inspired by the recentaction/complexity duality conjecture, it has been shown in [22] that the holographic complexity grows linearly withtime in the context of massive gravity.As we mentioned above, one of the quantities which is affected by momentum dissipation is conductivity. On theother hand, the choice of electrodynamics model has a direct influence on the behavior of conductivity. So, it is worthyto consider the effects of nonlinearity as well as massive gravity on the conductivity of the black hole solutions. It iswell-known that the nonlinear electrodynamics brings reach physics compared to the linear Maxwell electrodynamics.For example, Maxwell theory is conformally invariant only in four dimensions and thus the corresponding energy-momentum tensor is only traceless in four dimensions. A natural question then arises: Is there an extension of Maxwellaction in arbitrary dimensions that is traceless and hence possesses the conformal invariance? The answer is positiveand the invariant Maxwell action under conformal transformation g µν → Ω g µν , A µ → A µ in ( n + 1)-dimensions isgiven by [23], S m = Z d n +1 x √− g ( −F ) p , where F = F µν F µν is the Maxwell invariant, provided p = ( n + 1) /
4. The associated energy-momentum tensor of theabove Maxwell action is given by T µν = 2 (cid:18) pF µη F ην F p − − g µν F p (cid:19) . (1)One can easily check that the above energy-momentum tensor is traceless for p = ( n + 1) /
4. Also, quantum electro-dynamics predicts that the electrodynamic field behaves nonlinearly through the presence of virtual charged particlesthat is reported by Heisenberg and Euler [24]. Hence, nonlinear electrodynamics has been subject of much researches[25–27]. This motivates us to extend the linearly charged black hole solutions of massive gravity [5, 6] to nonlinearlycharged ones in the presence of power-law Maxwell electrodynamics and investigate the thermodynamics of them aswell as the behavior of conductivity corresponding to the dual system. In addition to power-law Maxwell electrody-namics, other types of nonlinear electrodynamics have been introduced in [28–30]. In spite of the special property for p = ( n + 1) /
4, different aspects of various solutions have been investigated for different p ’s [31–33]. In the context ofAdS/CFT correspondence, the power-law Maxwell field has been considered as electrodynamics source in [34–39].The layout of this letter is as follows. In section II, we present the action of the massive gravity in the presence ofpower-Maxwell electrodynamics and then by varying the action we obtain the field equations. We also derive a class oftopological black hole solutions of the field equations in higher dimensions. In section III, we study thermodynamicsof the solutions and examine the first law of thermodynamics for massive black holes with power-law Maxwell field.In section IV, we investigate the holographic conductivity of black brane solutions in the presence of a power-lawMaxwell gauge field. In particular, we shall disclose the effects of the power-law Maxwell electrodynamics as well asmassive gravity on the holographic conductivity of dual systems. We finish with closing remarks in section V. II. ACTION AND MASSIVE GRAVITY SOLUTIONS
The ( n + 1)-dimensional ( n ≥
3) action describing Einstein-massive gravity accompanied by a negative cosmologicalconstant Λ in the presence of power-law Maxwell electrodynamics is S = Z d n +1 x L , (2) L = √− g π " R −
2Λ + ( −F ) p + m X i c i U i ( g, Γ) , (3)where g and R are respectively the determinant of the metric and the Ricci scalar and Λ = − n ( n − / l is thenegative cosmological constant where l is the AdS radius. F = F µν F µν and F µν = ∂ [ µ A ν ] is electrodynamic tensorwhere A ν is vector potential. p determines the nonlinearity of the electrodynamic field. For p = 1, the linear Maxwellgauge field will be recovered. In action (2), Γ is the reference metric, c i ’s are constants and U i ’s are symmetricpolynomials of eigenvalues of the ( n + 1) × ( n + 1) matrix K µν ≡ √ g µα Γ αν so that U = [ K ] , (4) U = [ K ] − (cid:2) K (cid:3) , (5) U = [ K ] − K ] (cid:2) K (cid:3) + 2 (cid:2) K (cid:3) , (6) U = [ K ] − (cid:2) K (cid:3) [ K ] + 8 (cid:2) K (cid:3) [ K ] + 3 (cid:2) K (cid:3) − (cid:2) K (cid:3) , (7)where the square root in K is related to mean matrix square root i.e. (cid:16) √K (cid:17) µν (cid:16) √K (cid:17) νλ = K µλ and rectangular bracketsmean trace [ K ] ≡ K µµ . Here m is the massive gravity parameter so that in limit m →
0, one recovers the diffeomorphisminvariant Einstein-Hilbert action with a gauge field and a negative cosmological constant. The equations of motionfor gravitation and gauge field are R µν − R g µν + Λ g µν − pF µλ F λν ( −F ) p − −
12 ( −F ) p g µν + m χ µν = 0 , (8) ∇ µ (cid:0) F p − F µν (cid:1) = 0 , (9)which are obtained by varying the action (2) with respect to the metric tensor g µν and gauge field A µ respectively.In Eq. (8), we have χ µν = − c U g µν − K µν ) − c (cid:0) U g µν − U K µν + 2 K µν (cid:1) − c U g µν − U K µν +6 U K µν − K µν ) − c U g µν − U K µν + 12 U K µν − U K µν + 24 K µν ) . (10)The static spacetime line element takes the usual form ds = − f ( r ) dt + f − ( r ) dr + r h ij dx i dx j , (11)where f ( r ) is the metric function and h ij is a function of coordinates x i which spanned an ( n − n − n − k and volume ω n − . Without loss of generality, one cantake k = 0 , , −
1, such that the black hole horizon or cosmological horizon in (11) can be a zero (flat), positive (elliptic)or negative (hyperbolic) constant curvature hypersurface. The reference metric (fixed symmetric tensor) Γ µν can beconsidered as [5, 6] Γ µν = diag(0 , , c h ij ) , (12)where c is a positive constant. Using (11) and (12), one can easily calculates U i ’s as U = ( n − c r , U = ( n − n − c r , U = ( n − n − n − c r , U = ( n − n − n − n − c r . (13)Notice that U and U vanish for (3 + 1)-dimensional spacetime while U = 0 for (4 + 1)-dimensional spacetime. Usingthe metric (11), the electrodynamic field can be immediately found as F tr = − F rt = qr ( n − / (2 p − , (14)where q is a constant parameter related to the total charge of black hole. Inserting Eqs. (12), (13) and (14) into fieldequations (8), one receives f ′ r + ( n − fr − ( n − kr + 2Λ n − p − n − (cid:16) q r − n − p − (cid:17) p − c m r (cid:18) c + ( n − c c r + ( n − n − c c r + ( n − n − n − c c r (cid:19) = 0 , (15) f ′′ + 2( n − f ′ r + ( n − n − fr − ( n − n − kr + 2Λ − (cid:16) q r − n − p − (cid:17) p − ( n − c m r (cid:18) c + ( n − c c r + ( n − n − c c r + ( n − n − n − c c r (cid:19) = 0 , (16)where prime denotes the derivative with respect to r . Solving above equations, f ( r ) can be obtained as f ( r ) = k − m r n − − r n ( n −
1) + 2 p q p (2 p − ( n − n − p ) r np − p +1) / (2 p − + c m rn − (cid:18) c + ( n − c c r + ( n − n − c c r + ( n − n − n − c c r (cid:19) , (17)where m is an integration constant which is related to total mass of black hole as we see later. One may notethat the metric function (17) reduces to those of Refs. [5, 6] in the case p = 1. Also the solution (17), in the absentof massive parameter ( m = 0), leads to f ( r ) = k − m r n − − r n ( n −
1) + 2 p q p (2 p − ( n − n − p ) r np − p +1) / (2 p − , (18)which was presented in [32]. The mass parameter ( m ) in Eq. (17) can be found as m = kr n − − r n + n ( n −
1) + 2 p q p (2 p − ( n − n − p ) r ( n − p ) / (2 p − + c m r n − n − (cid:18) c + ( n − c c r + + ( n − n − c c r + ( n − n − n − c c r (cid:19) , (19)where r + is the radius of the event horizon given by the largest root of f ( r + ) = 0. According to Eq. (14) andregarding A t ( r ) = R F rt dr , the gauge potential A t can be calculated as A t ( r ) = µ + q (2 p − n − p ) r ( n − p ) / (2 p − . (20)In (20), µ is the chemical potential of the quantum field theory locates on boundary which can be found by demandingthe regularity condition on the horizon i.e. A t ( r + ) = 0 as µ = q (2 p − p − n ) r ( n − p ) / (2 p − . (21)One should note that the electric potential A t ( r ) has a finite value at infinity ( r → ∞ ) provided the parameter p isrestricted as 12 < p < n , (22) q ext + (cid:144) ext q ext - (cid:144) - - r f H r L (a) k = 0, q ext = 2 . q ext + (cid:144) ext q ext - (cid:144) - r f H r L (b) k = 1, q ext = 2 . q ext + (cid:144) ext q ext - (cid:144) - r f H r L (c) k = − q ext = 1 . FIG. 1: The behavior of f ( r ) versus r for n = 4, l = 1, p = 5 / m = 1, r + = 1, c = 1, c = 1, c = 3 / c = − / c = 1. obtained from ( n − p ) / (2 p − >
0. One can also obtain the electric potential as U = A ν χ ν | r → ref − A ν χ ν | r = r + , (23)where χ = C∂ t is the null generator of the horizon and C is a constant. When one applies the power-law Maxwellelectrodynamics, it is common to use a general Killing vector with a constant C [40, 41]. This is due to the factthat every linear combination of Killing vectors is also a Killing vector. Then, C is fixed so that the first law ofthermodynamics is satisfied [40, 41]. For linear Maxwell case ( p = 1), the constant C reduces to 1. Choosing infinityas the reference point, one can calculate the electric potential energy U = Cµ. (24)One can obtain the Hawking temperature of the black hole on the event horizon as T = f ′ ( r + )4 π = ( n − k πr + − r + π ( n −
1) + 2 p q p (1 − p )4 π ( n − r (2 p [ n − / (2 p − + c m π (cid:18) c + ( n − c c r + + ( n − n − c c r + ( n − n − n − c c r (cid:19) . (25)The extremal black hole, whose temperature vanishes, can be also determined by an extremal charge, q p ext = ( n − n − r p ( n − / (2 p − (2 p − p − Λ r p ( n − / (2 p − (2 p − p − + c m ( n − r [2 p ( n − / (2 p − (2 p − p (cid:18) c + ( n − c c r ext + ( n − n − c c r + ( n − n − n − c c r (cid:19) , (26)For q > q ext , there is a naked singularity in spacetime while q < q ext describes solutions with two inner and outerhorizons ( r + and r − ). These two horizons degenerate for q = q ext . The behaviors of the metric function f ( r ) versus r for different topologies of horizon are depicted in Fig. 1.Up to now, we have obtained the higher-dimensional black hole solutions in the context of massive gravity and inthe presence of power-law Maxwell gauge field. In the next section, we will study the thermodynamics of the obtainedsolutions. To do that, we shall obtain the Smarr-type formula and check the satisfaction of the first law of black holesthermodynamics. III. THERMODYNAMICS OF MASSIVE GRAVITY
The main purpose of this section is to examine the first law of thermodynamics for massive black holes with power-law Maxwell field. It was shown that the entropy of black holes in massive gravity still obeys the area law [6]. Itis easy to show that the entropy of black hole per unit volume ω n − as an extensive quantity of thermodynamics isgiven by [6] S = r n − , (27)which is a quarter of the event horizon area [6, 42]. The electric charge of black hole per unit volume ω n − can becalculated through the use of Gauss law Q = 14 π Z r n − ( −F ) p − F µν n µ u ν dr, (28)where n µ and u ν are respectively the unit spacelike and timelike normals to the hypersurface of radius r defined by n µ = 1 √− g tt dt = 1 p f ( r ) dt, u ν = 1 √ g rr dr = p f ( r ) dr. (29)Thus, one can obtain Q = 2 p − q p − π . (30)In order to obtain the mass of black holes in massive gravity one can apply the Hamiltonian approach presented inRef. [6]. The total mass ( M ) of massive black hole per unit volume ω n − can be calculated as [6] M = ( n − m π , (31)where m as a function of the horizon radius r + was given in Eq. (19). In order to check the first law of thermodynamic,we need to compute Smarr-type formula for mass M as a function of extensive quantities entropy and electric charge.Using relations (27), (30) and (31), one can obtain the Smarr-type formula for mass as M ( S, Q ) = k ( n − S ) ( n − / ( n − π − Λ (4 S ) n/ ( n − πn + Q p/ (2 p − (2 p − n − p ) (4 S ) n − p ( n − p − (cid:16) π p − (cid:17) / (2 p − + c m S π c + ( n − c c (4 S ) / ( n − + ( n − n − c c (4 S ) / ( n − + ( n − n − n − c c (4 S ) / ( n − ! . (32)Now, one can show that the thermodynamic quantities satisfy the first law of thermodynamic as dM = T dS + U dQ, (33)in which T = (cid:18) ∂M∂S (cid:19) Q and U = (cid:18) ∂M∂Q (cid:19) S , (34)provided C = p in (24). As it is clear, for linear Maxwell case ( p = 1), the constant C is reduced to 1. In theremainder of this work, we study the effect of power-law Maxwell electrodynamics on the holographic conductivity ofdual systems with and without translational symmetry. IV. HOLOGRAPHIC CONDUCTIVITY
In this section, we will obtain the electrical transport behavior of the dual field theory in the presence of a power-lawMaxwell gauge field. In order to do this, one should use the solution of the black brane ( k = 0) found in the pervioussection. First, we investigate the effects of the power-law Maxwell electrodynamics on the holographic conductivityof dual systems in which momentum is conserved ( m = 0). Next, we consider the solutions dual to the systems whichno longer possess momentum conservation ( m = 0). q Ω (cid:144) T Π R e @ Σ D p =
1, m = (a) n = 3 q Ω (cid:144) T Π R e @ Σ D p =
1, m = (b) n = 4 p Ω (cid:144) T Π R e @ Σ D q = = (c) n = 3 p (cid:144)
81 9 (cid:144) (cid:144) (cid:144) Ω (cid:144) T Π R e @ Σ D q = = (d) n = 4 q n Ω (cid:144) T Π R e @ Σ D m = (e) p = ( n + 1) / FIG. 2: The behaviors of real parts of conductivity σ versus ω/T for m = 0 with l = r + = 1. A. Vanishing m The planer ( n + 1)-dimensional metric can be rewritten as ds = −F ( u ) dt + l F ( u ) − u − du + l u − n − X i =1 dx i , (35)which is given by defining u = lr − in the metric (11). Accordingly, the event horizon of black brane is at u + = lr − and the n -dimensional thermal field theory lives at u = 0. The metric function of spacetime in absence of massiveparameter is F ( u ) = − m l − n u n − + u − + 2 p q p (2 p − ( n − − ( n − p ) − (cid:2) l − u (cid:3) np − p +1) / (2 p − , (36)obtained by substituting r = lu − and k = 0 in Eq. (18). Perturbing the vector potential component A x and themetric component g tx by turning on a x ( u ) e − iωt and g tx ( u ) e − iωt respectively, we can easily derive two linear equationsof motion for electrodynamics a ′′ x + (cid:16) (8 p − n −
3) (2 p − − u − + F ′ F − (cid:17) a ′ x + l ω u − F − a x + h ′ F − (cid:0) g ′ tx + 2 u − g tx (cid:1) = 0 , (37)and for gravity g ′ tx + 2 u − g tx + 2 p +1 ph ′ (cid:0) u l − h ′ (cid:1) p − a x = 0 , (38)where now the prime means derivative with respect to u and h ( u ) is electric potential in the form h ( u ) = µ + q (2 p − u ( n − p ) / (2 p − ( n − p ) l ( n − p ) / (2 p − , (39)which is obtained by transforming r → lu − in Eq. (20). By eliminating g tx between Eqs. (37) and (38), thedifferential equation for a x is a ′′ x + (cid:16) (8 p − n −
3) (2 p − − u − + F ′ F − (cid:17) a ′ x + a x F − (cid:16) l ω u − F − − p +1 ph ′ (cid:0) u l − h ′ (cid:1) p − (cid:17) = 0 . (40) q - - Ω (cid:144) T Π I m @ Σ D p =
1, m = (a) n = 3 q - - - - - Ω (cid:144) T Π I m @ Σ D p =
1, m = (b) n = 4 p - - - - - Ω (cid:144) T Π I m @ Σ D q = = (c) n = 3 p (cid:144)
81 9 (cid:144) (cid:144) (cid:144) - - - - - - Ω (cid:144) T Π I m @ Σ D q = = (d) n = 4 q n - - Ω (cid:144) T Π I m @ Σ D m = (e) p = ( n + 1) / FIG. 3: The behaviors of imaginary parts of conductivity σ versus ω/T for m = 0 with l = r + = 1. The behavior of above relation near the boundary ( u →
0) is a ′′ x + (4 p − n −
1) (2 p − − u − a ′ x + · · · = 0 , (41)which has the following solution a x ( u ) = a + a u ( n − p ) / (2 p − + · · · , (42)where a and a are two constant parameters. To calculate the expectation value of current for boundary theory, wecan use the following formula [43, 44] h J x i = ∂ L ∂ ( ∂ u δa x ) (cid:12)(cid:12)(cid:12)(cid:12) u =0 , (43)where δa x = a x ( u ) e − iωt and L was given in Eq. (3). So, it is obvious that the holographic conductivity can beobtained as σ = h J x i E x = − h J x i ∂ t δa x = − i h J x i ωδa x = 2 p − p ( n − p ) q p − a (2 p − πiωa . (44)It is easy to show that the holographic conductivity (44) reduces to σ = a / (4 πiωa ) for n = 3 and p = 1 [5, 43].In Figs. 2(a) and 3(a), the behaviors of real and imaginary parts of holographic conductivity for linear Maxwellcase ( p = 1) are illustrated as a function of ω/T and for various values of the charges of black brane q for n = 3.This figure shows that the real part of conductivity Re[ σ ] decreases as q increases (temperature decreases) for ω → σ ] diverges at ω = 0 independent of the value of the chargeparameter q . Also, the maximum value of Re[ σ ] is greater for greater q ’s. We observe that Re[ σ ] tends to a constantfor high frequencies independent of the value of the charge parameter. Next, we turn to study imaginary part of theconductivity Im[ σ ] plotted in Fig. 3(a). Imaginary part of conductivity includes a minimum for different charges.This minimum is deeper for larger charges (lower temperatures). At ω = 0, imaginary part of conductivity Im[ σ ]diverges (Fig. 3(a)). This fact supports our numerical computation which shows that real part of conductivity blowsup at zero frequency, according to Kramers-Kronig relation. For high frequencies, the imaginary part of conductivityvanishes independent of the value of charge. In Figs. 2(b) and 3(b), the behaviors of real and imaginary parts ofholographic conductivity for linear Maxwell in terms of frequency for different values of black brane’s charge q for q Ω (cid:144) T Π R e @ Σ D p =
1, m = (a) n = 3 q Ω (cid:144) T Π R e @ Σ D p =
1, m = (b) n = 4 p Ω (cid:144) T Π R e @ Σ D q = = (c) n = 3 p (cid:144)
81 9 (cid:144) (cid:144) (cid:144) Ω (cid:144) T Π R e @ Σ D q = = (d) n = 4 q n Ω (cid:144) T Π R e @ Σ D m = (e) p = ( n + 1) / FIG. 4: The behaviors of real parts of conductivity σ versus ω/T for m = 1 with l = r + = 1, c = 1, c = − c = 0. n = 4 are depicted. For low frequencies the behavior of holographic conductivity is the same as the case n = 3.However, for high frequencies the behaviors are different. In n = 3 case, the real (imaginary) part of conductivitytends to a constant for high frequencies whereas for n = 4 case it increases (decreases) as ω increases.Now, we intend to study the effect of nonlinearity of the electrodynamics (power parameter p of the power-lawMaxwell field) on holographic conductivity. Figs. 2(c), 2(d), 3(c) and 3(d) show the behavior of Re[ σ ] and Im[ σ ]as a function of ω/T for different values of p (restricted by 1 / < p < n/
2) for n = 3 and 4. In the ω → p leads to the smaller Re[ σ ]. For high frequencies, Re[ σ ] increases (decreases) as linear function of ω/T andits slope increases (decreases) as p decreases (increases) for p < ( n + 1) / p > ( n + 1) / p = ( n + 1) /
4, Re[ σ ]and Im[ σ ] tend to a constant for high frequencies as one can see in Figs. 2(e) and 3(e). Above behaviors show thatfor high frequencies Re[ σ ] ∝ ω a where a ∝ n + 1 − p . This result is important from holographic point of view sincesimilar results can be found in experimental observations [45, 46]. In [45], for a (2 + 1)-dimensional graphene system,it was reported that the value of Re[ σ ] tends to a constant for large frequencies. We observed such a behavior in theconformally invariant case, p = ( n + 1) /
4. For conductivity of a (2 + 1)-dimensional single-layer graphene inducedby mild oxygen plasma exposure, a positive slope with respect to frequency for high frequencies has been reported in[46]. We observed similar behavior for conductivity in case of p < ( n + 1) /
4. For all values of p , we see that Im[ σ ]blows up at zero frequency (Figs. 3(c) and 3(d)). For high frequencies, imaginary part of conductivity decreases forlow values of p , whereas it flattens for bigger p ’s. B. Nonvanishing m Now, we intend to demonstrate the influence of power-law Maxwell parameter p on the holographic conductivity inmassive gravity theory. Employing again r → lu − and setting k = 0 in (17), we obtain F ( u ) = − m l − n u n − + u − + 2 p q p (2 p − ( n − − ( n − p ) − (cid:2) l − u (cid:3) np − p +1) / (2 p − +( n − − c m lu − (cid:0) c + ( n − l − c c u + ( n − n − l − c c u + ( n − n − n − l − c c u (cid:1) . (45)Hereon,we should perturb the gauge field and the metric by turning on a x ( u ) e − iωt , g tx ( u ) e − iωt and g ux ( u ) e − iωt . Atthe linear regime, we have three independent differential equations for gauge field( F a ′ x ) ′ + (8 p − n −
3) (2 p − − u − F a ′ x + l ω u − F − a x + h ′ (cid:0) g ′ tx + 2 u − g tx + iωg ux (cid:1) = 0 , (46)0 q Ω (cid:144) T Π I m @ Σ D p =
1, m = (a) n = 3 q - - - - Ω (cid:144) T Π I m @ Σ D p =
1, m = (b) n = 4 p - - - - Ω (cid:144) T Π I m @ Σ D q = = (c) n = 3 p (cid:144)
81 9 (cid:144) (cid:144) (cid:144) - - - Ω (cid:144) T Π I m @ Σ D q = = (d) n = 4 q n - Ω (cid:144) T Π I m @ Σ D m = (e) p = ( n + 1) / FIG. 5: The behaviors of imaginary parts of conductivity σ versus ω/T for m = 1 with l = r + = 1, c = 1, c = − c = 0. and for massive gravity g ′ tx + 2 u − g tx + iωg ux + 2 p +1 ph ′ (cid:0) u l − h ′ (cid:1) p − a x + ic l − ω − u Ξ F g ux = 0 , (47) g ′′ tx + (5 − n ) u − g ′ tx − n − u − g tx + iωg ′ ux + 2 p +1 ph ′ (cid:0) u l − h ′ (cid:1) p − a ′ x − i ( n − ωu − g ux + c Ξ u − F − g tx = 0 , (48)in which Ξ = m (cid:0) c lu − + 2( n − c c + 3( n − n − c c l − u + 4( n − n − n − c c l − u (cid:1) . (49)Eliminating g tx between Eqs. (46), (47) and (48), one arrives at the two following second-order differential equations( F a ′ x ) ′ + (8 p − n −
3) (2 p − − u − F a ′ x + h l ω u − F − − p +1 ph ′ (cid:0) u l − h ′ (cid:1) p − i a x − ic l − ω − F h ′ Ξ u g ux = 0 , (50) l − u − (cid:16) u Ξ − F (cid:0) u Ξ F g ux (cid:1) ′ (cid:17) ′ − i p +1 pωc − u − (cid:20) Ξ − u F a x (cid:16) h ′ (cid:0) u l − h ′ (cid:1) p − (cid:17) ′ (cid:21) ′ + i ( n − p +1 pωc − u − h Ξ − u F a x h ′ (cid:0) u l − h ′ (cid:1) p − i ′ − ( n − l − u − ( u F g ux ) ′ + ω g ux − i p +1 pωh ′ (cid:0) u l − h ′ (cid:1) p − a x + c u l − Ξ F g ux = 0 . (51)One can show that the solution of differential equation (50) near boundary ( u →
0) is a ′′ x + (4 p − n −
1) (2 p − − u − a ′ x + · · · = 0 , (52)which is the same as (41) and also the holographic conductivity has the same form as (44). To solve above differentialequations numerically, we impose incoming boundary conditions at the horizon a x ( u ) , g ux ( u ) ∝ ( u + − u ) − iω/ πT , (53)1where T is the Hawking temperature.In Figs. 4 and 5, we depict the holographic conductivity for (2 + 1)- and (3 + 1)-dimensional dual systems includingmomentum dissipation in the presence of linear Maxwell and nonlinear electrodynamics. Fig. 5 shows that theimaginary part of conductivity near zero frequency does not have diverging behavior in the presence of momentumdissipation. Consequently, according to Kramers-Kronig relation, the real part of conductivity does not diverge at ω = 0 and includes a Drude peak (in contrast with the case of previous subsection with no momentum dissipationwhere imaginary part of conductivity blows up at zero frequency and accordingly real part diverges there). Also, realpart of DC conductivity becomes larger as q ( p ) increases. For high frequencies, the behaviors of real and imaginaryparts of conductivity for n = 3 and 4 in terms of black brane charge q and nonlinear parameterar p are similar to thecase of previous subsection with no momentum dissipation. V. CLOSING REMARKS
A gravity theory called massive gravity [5] was proposed in order to describe a class of strongly interacting quantumfield theories with broken translational symmetry via a holographic principle. In this letter, we consider the massivegravity theory when the gauge field is in the form of the power-Maxwell electrodynamics. First, we derive a class ofhigher dimensional topological black hole solutions of this theory. Then, we calculate the conserved and thermody-namic quantities of the system and check that these quantities satisfy the first law of black holes thermodynamics onthe horizon.The main purpose of this letter is to investigate the electrical transport behavior of the dual field theory in thepresence of a power-law Maxwell gauge field for the obtained solutions. In order to clarify the effects of the massivegravity on the holographic conductivity, we have first considered the holographic conductivity of the dual systems inwhich momentum is conserved ( m = 0). Then, we have extended our study to the case where translational symmetryis broken and consequently the system no longer possess momentum conservation ( m = 0). For both cases, we haveplotted the behaviour of the real and imaginary parts of the holographic conductivity in terms of the frequency pertemperature ( ω/T ) for (2+1)- and (3+1)-dimensional dual systems. In the former case ( m = 0), we observed that thereal part of conductivity Re[ σ ] for n = 3 decreases as q increases (temperature decreases) for ω →
0. Besides, Re[ σ ]has a maximum which is greater for greater charges. Also, Re[ σ ] tends to a constant for high frequencies independentof the value of charge. In addition, the imaginary part of conductivity Im[ σ ] diverges as ω →
0. For high frequencies,the imaginary part of conductivity vanishes independent of the value of charge. The low frequencies behavior ofholographic conductivity for n = 4 is the same as the case of n = 3. For high frequencies, in contrast with n = 3, thereal (imaginary) part of conductivity increases (decreases) as Re[ σ ] increases for n = 4. Next, we explored the effectof the power-law Maxwell field on holographic electrical transport. We observed that increasing p leads to the smallerRe[ σ ] for ω → σ ] ∝ ω a where a ∝ ( n + 1 − p ). Similar results for high frequenciescan be found in experimental observations on (2 + 1)-dimensional graphene systems [45, 46]. This is important fromholographic point of view.In the latter case ( m = 0), we find out that the imaginary part of the DC conductivity, Im[ σ ], is zero at ω = 0 andbecomes larger as q increases (temperature decreases). This is in contrast to the case without momentum dissipation.It also has a maximum value for ω = 0 which increases with increasing q (with fixed p ) or increasing p (with fixed q )for n = 3. For the real part of the conductivity, Re[ σ ], we see that in case of p = 1 the maximum value (Drude peak)achieves at ω = 0. Again this is in contrast to the former case ( m = 0) in which the minimum value of Re[ σ ], occursfor ω →
0. For different values of the power parameter, p , the real and imaginary part of the conductivity has relativeminimum and maximum, respectively. Finally, we observed that both real and imaginary parts of the holographicconductivity are similar to the previous case for high frequencies.In this work, we obtained the conductivity by applying the linear response theory where the electric field is treatedas a probe. This may restrict the study from fully explaining the effects of nonlinearity of electrodynamics model.Therefore, it is an interesting issue for future researches to consider the case where the properties of the system arefunctions of electric field. In such case, nonlinear response happens. Some examples of such studies can be found inliterature in Refs. [47–52]. Acknowledgments
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