Holographic DC Conductivity for Backreacted NLED in Massive Gravity
CCTP-SCU/2021001
Holographic DC Conductivity for Backreacted NLED in Massive Gravity
Shihao Bi a ∗ and Jun Tao a † a Center for Theoretical Physics, College of Physics,Sichuan University, Chengdu, 610065, China
Abstract
In this work a holographic model with the charge current dual to a general nonlinear electrodynamics(NLED) is discussed in the framework of massive gravity. Massive graviton can breaks the diffeomorphisminvariance in the bulk and generates momentum dissipation in the dual boundary theory. The expressionof DC conductivities in a finite magnetic field are obtained, with the backreaction of NLED field on thebackground geometry. General transport properties in various limits are presented, and then we turn to thethree of specific NLED models: the conventional Maxwell electrodynamics, the Maxwell-Chern-Simonselectrodynamics, and the Born-Infeld electrodynamics, to study the parameter-dependence of in-planeresistivity. Two mechanisms leading to the Mott-insulating behaviors and negative magneto-resistivity arerevealed at zero temperature, and the role played by the massive gravity coupling parameters are discussed. ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - t h ] J a n . Introduction The discovery of gauge/gravity duality makes it possible to deal with the strongly-coupledgauge theories on the boundary from the classical gravitational theories in the higher dimensionalbulk [1–6]. And the well-known prediction on the ratio of the shear viscosity to the entropy densityfor N = T c superconductors[59], is another example beyond the framework of energy band theory due to the strong Coulombrepulsive interaction. The strong electron-electron interaction would prevent the available chargecarriers to efficiently transport charges as if suffers the electronic traffic jam. The holographicconstruction of Mott insulators has been a heated topic and much progress has been made [32,60–67]. In Ref. [68], the holographic model coupled with a particular NLED named iDBI wasproposed to study the strong electron-electron interaction by introducing the self-interaction ofNLED field, and the Mott-insulating behaviors appear for large enough self-interaction strength.In this paper, we construct a holographic model coupled with a general NLED field toinvestigate the magneto-transport of the strongly-interacting system on the boundary from theperspective of gauge/gravity duality. The backreaction of NLED field on the bulk geometry istaken into consideration as in Ref. [69]. The massive gravity, with massive potentials associatedwith the graviton mass, is adopted to break the diffeomorphism invariance in the bulk producingmomentum relaxation in the dual boundary theory [70].This paper is organized as follows. In Sec. II we establish our holographic model of massivegravity with NLED field. Then the DC conductivities with non-zero magnetic field are derivedas the function of ρ , h , T , and related massive gravity coupling parameters in Sec. III, withvarious limit situation being discussed. Then in Sec. IV we present detailed investigation on thein-plane resistivity in the framework of conventional Maxwell electrodynamics, the CP-violatingMaxwell-Chern-Simons electrodynamics, and Born-Infeld electrodynamics. Finally, we make ourconclusion in Sec. V. We will use the unit (cid:125) = G = k B = l = II. Holographic Setup
The 4-dimensional massive gravity [70] with a negative cosmological constant Λ coupled to anonlinear electromagnetic field A µ we are considering is given by S = π (cid:90) d x √− g (cid:34) R − Λ + m ∑ j = c j U j ( g , f ) + L ( s , p ) (cid:35) , (1)where Λ = − / l , m is the mass parameter, f is a fixed symmetric tensor called the referencemetric, c j are coupling constants [85], and U j ( g , f ) denotes the symmetric polynomials of the3igenvalue of the 4 × K µν = (cid:112) g µλ f λ ν given as U = [ K ] , U = [ K ] − (cid:2) K (cid:3) , U = [ K ] − [ K ] (cid:2) K (cid:3) + (cid:2) K (cid:3) , U = [ K ] − [ K ] (cid:2) K (cid:3) + [ K ] (cid:2) K (cid:3) + (cid:2) K (cid:3) − (cid:2) K (cid:3) . (2)The square root in K means (cid:16) √K (cid:17) µλ (cid:16) √K (cid:17) λν = K µν and [ K ] ≡ K µµ . The NLED Lagrangian L ( s , p ) in Eq. (1) is constructed as the function of two independent nontrivial scalar using the field strengthtensor F µν = ∂ µ A ν − ∂ ν A µ s = − F µν F µν , (3a) p = − ε µνρσ F µν F ρσ . (3b)Here ε abcd ≡ − [ a b c d ] / √− g is a totally antisymmetric Lorentz tensor, and [ a b c d ] is thepermutation symbol. In the weak field limit we assume the NLED reduces to the Maxwell-Chern-Simons Lagrangian L ( s , p ) ≈ s + θ p with θ defined as L ( , ) ( , ) for later convenience.Varying the action with respect to g µν and A µ we obtain the equations of motion R µν − Rg µν + Λ g µν = T µν , (4a) ∇ µ G µν = . (4b)where the energy-momentum tensor is T µν = g µν (cid:18) L ( s , p ) − p ∂ L ( s , p ) ∂ p (cid:19) + ∂ L ( s , p ) ∂ s F λµ F νλ + m χ µν , (5)with χ µν = c (cid:0) U g µν − K µν (cid:1) + c (cid:16) U g µν − U K µν + K µν (cid:17) + c (cid:16) U g µν − U K µν + U K µν − K µν (cid:17) + c (cid:16) U g µν − U K µν + U K µν − U K µν + K µν (cid:17) . (6)And we introduce G µν = − ∂ L ( s , p ) ∂ F µν = ∂ L ( s , p ) ∂ s F µν + ∂ L ( s , p ) ∂ p ε µνρσ F ρσ . (7)4e are looking for a black brane solution with asymptotic AdS spacetime by taking the followingansatz [70, 71] for the metric and the NLED field, and the reference metric:d s = − f ( r ) d t + d r f ( r ) + r (cid:0) d x + d y (cid:1) , (8a) A = A t ( r ) d t + h ( x d y − y d x ) , (8b) f µν = diag (cid:0) , , α , α (cid:1) . (8c)where h is the magnetic field strength. From Eqs. (8a) and (8b) we find the nontrivial scalarsEq. (3b) are s = (cid:18) A (cid:48) t ( r ) − h r (cid:19) , (9a) p = − hA (cid:48) t ( r ) r . (9b)The equations of motion then are obtained as r f (cid:48) ( r ) + f ( r ) − r = c α m r + c α m + r (cid:0) A (cid:48) t ( r ) G rt + L ( s , p ) (cid:1) , (10a) r f (cid:48)(cid:48) ( r ) + f (cid:48) ( r ) − r = c α m + r ( L ( s , p ) + hG xy ) , (10b) (cid:2) r G rt (cid:3) (cid:48) = , (10c)where the non-vanishing components G µν are G rt = ∂ L ∂ p hr − ∂ L ∂ s A (cid:48) t ( r ) , (11a) G xy = ∂ L ∂ s hr + ∂ L ∂ p A (cid:48) t ( r ) r . (11b)Eq. (10c) leads to G tr = ρ / r with ρ being a constant. The event horizon r h is the root of f ( r ) ,i.e., f ( r h ) =
0, and the Hawking temperature of the black brane is given by T = f (cid:48) ( r h ) π . (12)Then at r = r h Eq. (10a) reduces to4 π Tr h − r h = c α m r h + c α m + r h (cid:0) A (cid:48) t ( r h ) G rth + L ( s h , p h ) (cid:1) , (13)where s h = (cid:18) A (cid:48) t ( r h ) − h r h (cid:19) , (14a)5 h = − hA (cid:48) t ( r h ) r h , (14b) G rth = L ( , ) ( s h , p h ) hr h − L ( , ) ( s h , p h ) A (cid:48) t ( r h ) (14c) III. DC Conductivity
From the perspective of gauge/gravity duality, the black brane solution Eqs. (8a) and (8b) in thebulk can describe an equilibrium state at finite temperature T given by Eq. (13). And the conservedcurrent J µ in the boundary theory is connected with the conjugate momentum of the NLED fieldin the bulk, which allows us calculate the DC conductivity in the framework of linear responsetheory [72, 73]. A. Derivation of DC Conductivity
The following perturbations on the metric and the NLED field are applied to derive the DCconductivity: δ g ti = r h ti ( r ) , (15a) δ g ri = r h ri ( r ) , (15b) δ A i = − E i t + a i ( r ) , (15c)where i = x , y . We first consider the t component. The absence of A t ( r ) in the Eq. (9b) leads to theradially independent conjugate momentum ∂ r Π i =
0, with Π t = ∂ L ( s , p ) ∂ ( A (cid:48) t ( r )) . (16)Then the expectation value of J t in the dual boundary field theory is given by (cid:10) J t (cid:11) = Π t . (17)In the linear level we have (cid:104)J t (cid:105) = ρ , which indicate that ρ can be interpreted as the charge densityin the dual field theory. At the event horizon r = r h the charge density ρ is given by ρ = L ( , ) ( s h , p h ) r h A (cid:48) t ( r h ) − L ( , ) ( s h , p h ) h . (18)6hen we consider the planar components. The NLED is explicitly independent of a i ( r ) , makingthe conjugate momentum of the field a i ( r ) Π i = ∂ L ( s , p ) ∂ (cid:0) a (cid:48) i ( r ) (cid:1) = ∂ L ( s , p ) ∂ ( ∂ r A i ) = √− gG ir , (19)being radially independent as well. And the charge currents in the dual field theory are given by (cid:10) J i (cid:11) = Π i , and can be expressed with the perturbed metric and field components h ti , h ri , a (cid:48) i and E i : (cid:104)J x (cid:105) = −L ( , ) ( s , p ) (cid:2) f ( r ) a (cid:48) x ( r ) + h f ( r ) h ry ( r ) + r A (cid:48) t ( r ) h tx ( r ) (cid:3) − L ( , ) ( s , p ) E y , (20a) (cid:104)J y (cid:105) = −L ( , ) ( s , p ) (cid:2) f ( r ) a (cid:48) y ( r ) − h f ( r ) h rx ( r ) + r A (cid:48) t ( r ) h ty ( r ) (cid:3) + L ( , ) ( s , p ) E x . (20b)The perturbed metric components h ti and h ri are coupled to E i in gravitational field equation andcan be reduced. Thus we take the first order of the perturbed Einstein’s equations of tx and ty components and we get: (cid:18) h r − α m c r L ( , ) ( s , p ) (cid:19) h tx ( r ) − hA (cid:48) t ( r ) f ( r ) h ry ( r ) = A (cid:48) t ( r ) f ( r ) a (cid:48) x ( r ) − E y hr , (21a) (cid:18) h r − α m c r L ( , ) ( s , p ) (cid:19) h ty ( r ) + hA (cid:48) t ( r ) f ( r ) h rx ( r ) = A (cid:48) t ( r ) f ( r ) a (cid:48) y ( r ) + E x hr . (21b)Next by using the regularity constraints of the metric and field near the event horizon [73]: f ( r ) = π T ( r − r h ) + · · · , A t ( r ) = A (cid:48) t ( r h ) ( r − r h ) + · · · , a i ( r ) = − E i π T ln ( r − r h ) + · · · , h ri ( r ) = h ti ( r ) f ( r ) + · · · , (22)the Eqs. (21a) and (21b) at the event horizon r = r h are hA (cid:48) t ( r h ) h ty ( r h ) − (cid:18) h r h − α m c r h L ( , ) ( s h , p h ) (cid:19) h tx ( r h ) = A (cid:48) t ( r h ) E x + hr h E y , (23a) (cid:18) h r h − α m c r h L ( , ) ( s h , p h ) (cid:19) h ty ( r h ) + hA (cid:48) t ( r h ) h tx ( r h ) = hr h E x − A (cid:48) t ( r h ) E y . (23b)Solving Eqs. (23a) and (23b) for h ti ( r h ) in terms of E i and inserting into Eqs. (20a) and (20b), onecan evaluate the current (cid:10) J i (cid:11) to the electric fields E i at the event horizon r = r h via (cid:10) J i (cid:11) = σ i j E j ,7here the DC conductivities are given by σ xx = σ yy = α m c r h (cid:18) α m c r h L ( , ) ( s h , p h ) − h r h + r h A (cid:48) t ( r h ) (cid:19)(cid:18) α m c r h L ( , ) ( s h , p h ) − h r h (cid:19) + h A (cid:48) t ( r h ) , (24a) σ xy = − σ yx = L ( , ) ( s h , p h ) r h A (cid:48) t ( r h ) h − (cid:18) α m c r h L ( , ) ( s h , p h ) (cid:19) (cid:18) α m c r h L ( , ) ( s h , p h ) − h r h (cid:19) + h A (cid:48) t ( r h ) − L ( , ) ( s h , p h ) . (24b)The resistivity matrix is the inverse of the conductivity matrix: R xx = R yy = σ xx σ xx + σ xy and R xy = − R yx = σ xy σ xx + σ xy . (25)The event horizon radius r h is the solution of Eq. (13) and will be affected by the temperature T ,the charge density ρ , the magnetic field h and the parameters c , , m and α . For a given Lagrangian L ( s , p ) one first solve A (cid:48) t ( r h ) from Eq. (18), and then plug it into Eq. (13) to solve r h , which willbe brought into Eqs. (24a) and (24b) to obtain the DC conductivity as a complicated function ofthe parameters mentioned above. B. Various Limits
The concrete examples of NLED will be discussed in Sec. IV and the properties of DCresistivity are discussed in detail. Before focus on the specific NLED model we first considersome general properties of the DC conductivity in various limit.
1. Massless and Massive Limits
The massless limit corresponds to m → σ xx = σ xy = ρ / h [74]. For comparison, in the massless limit the DCconductivities in Eqs. (24a) and (24b) become σ xx = α m | c | r h h + O ( m ) and σ xy = ρ h + O ( m ) . (26)8nd in the massive limit m − → + ∞ , the DC conductivities have the asymptotic behaviors σ xx = L ( , ) ( s h , p h ) + O ( m − ) and σ xy = −L ( , ) ( s h , p h ) + O ( m − ) . (27)which agree with the calculation by treating the NLED field as a probe one [75], for the geometryis dominated by the massive terms.
2. Zero Field and Charge Density Limits
In the zero field limit h = σ xx = L ( , ) (cid:18) A (cid:48) t ( r h ) , (cid:19) − ρ α m c r h and σ xy = −L ( , ) (cid:18) A (cid:48) t ( r h ) , (cid:19) . (28)where A (cid:48) t ( r h ) is obtained by solving ρ = L ( , ) (cid:18) A (cid:48) t ( r h ) , (cid:19) r h A (cid:48) t ( r h ) . (29)At zero charge density ρ =
0, the DC conductivities become σ − xx = L ( , ) (cid:18) − h r h , (cid:19) − h α m c r h and σ xy = −L ( , ) (cid:18) − h r h , (cid:19) . (30)We find that the DC conductivities are in general non-zero and can be interpreted as incoherentcontributions [76], known as the charge conjugation symmetric contribution σ ccs . There is anothercontribution from explicit charge density relaxed by some momentum dissipation, σ diss , dependingon the charge density ρ . The results show that, for a general NLED model, the DC conductivitiesusually depend on σ diss and σ ccs in a nontrivial way.
3. High Temperature Limit
Finally, we consider the high temperature limit T (cid:29) max (cid:110) √ h , √ ρ , | c | α m , (cid:112) | c | α m (cid:111) . Inthis limit, Eq. (13) gives T ≈ π r h . The longitudinal resistivity then reduces to R xx = + θ (cid:26) + π α | c | T (cid:20) h (cid:0) + θ (cid:1) + h θ ρ − − θ + θ ρ (cid:21)(cid:27) + O (cid:16) T − (cid:17) . (31)The nonlinear effect will be suppressed by the temperature and we only keep the leading order.Usually the metal and insulator have different temperature dependence. For the metallic materials9he phonon scattering enlarges the resistivity, while the thermal excitation of carriers in insulatingmaterials can promotes the conductivity. And the metal-insulator transition can occur when thecoefficient of 1 / T changes the sign, which is shown in Fig. 1 in the h / ρ − θ parameter space. The θ term can break the ( ρ , h ) → ( ρ , − h ) or ( ρ , h ) → ( − ρ , h ) symmetries for σ i j or R i j , but σ i j or R i j are invariant under ( ρ , h ) → ( − ρ , − h ) . And the phases are central symmetric in the parameterplane. - - - - / ρ θ High Temperature Limit
Metal InsulatorInsulator - - - - / ρ θ High Temperature Limit ∂ h R xx < ∂ ρ R xx > ∂ ρ R xx > Fig. 1: Left panel: The metal-insulator phase diagram in the high temperature limit. The red regionhas positive temperature derivative of R xx and thus describes a metal, while the blue region has negativetemperature derivative, and hence an insulator. The black solid lines are the phase boundaries. Right panel:The Mott-insulating region (blue) and the negative magneto-resistivity (red) in the high temperature limit The Mott-insulating and magneto-resistance behaviors are also presented in Fig. 1. The Mott-insulating region is where ∂ | ρ | R xx > θ > MR xx = R xx ( h ) − R xx ( ) R xx ( ) , (32)and we see that in the Fig. 1 the negative magneto-resistivity can only occur with non-zero θ . For10 =
0, Eq. (31) reduces to R xx = + (cid:0) h − ρ (cid:1) π α | c | T + O (cid:16) T − (cid:17) , (33)which gives metallic behavior for h | / ρ | < | / ρ | >
1. And one alwayshas ∂ | ρ | R xx < ∂ | h | R xx >
0, indicating the absence of Mott-insulating behavior and negativemagneto-resistivity.
IV. Various NLED Models
In this section, we will use Eqs. (13), (18) and (24a) to study the dependence of the in-planeresistivity R xx on the temperature T , the charge density ρ and the magnetic field h in somespecific NLED models. For convenience we rescale the c i ’s as c ∼ α m c and c ∼ α m c .The conventional Maxwell electrodynamics is firstly presented, with a detail discussion on the R xx and R xy ’s dependence on involved massive gravity coupling parameters c and c , the chargedensity ρ , the magnetic field h and the temperature T . Then the Chern-Simons θ term as anextension is introduced to investigate the CP-violating effect. Finally, we discuss the Born-Infeldelectrodynamics and the influence of non-linear effect on the DC resistivity. The high temperaturebehaviors have been discussed in Sec. III B 3, so we will mainly focus on the behavior of R xx around T = A. Maxwell Electrodynamics
We first consider the Maxwell electrodynamics, in which L ( s , p ) = s . From Eq. (18) we cansolve out that A (cid:48) t ( r h ) = ρ r h , (34)and bring it into Eq. (13) we get the equation4 π Tr h − r h − c r h − c + ρ + h r h = . (35)It is notice that c offers an effective temperature correction, and make the solution complicatedeven at T =
0. Although the c is a constant playing a similar role as the momentum dissipationstrength in Ref. [77], the effect on the resistivity is quite different. The effect of c and c on R xx and R xy at zero temperature is presented in Fig. 2. For c = R xx = R xy is constant, and11s independent of c . For non-zero c , the R xx increases and saturates, while the R xy decreases tozero as c becomes more negative. And a larger | c | makes the surface steeper. Fig. 2: The influence of c and c on R xx and R xy . We take T = ρ = . h = . I. Various NLED Models
In this section, we will use ???????? to study the dependence of the in-plane resistance R xx onthe temperature T , the charge density ρ and the magnetic field h in some specific NLED models.The high temperature behaviors have been discussed in ?? , so we will focus on the behavior of R xx around T = A. Maxwell Electrodynamics - - - - ρ T = c =- c = R xx - - - ρ R xx T = c =- c = h = = = = = - - M R xx T = c =- c = ρ = ρ = ρ = ρ = ρ = ρ = Fig. 1: Fig. 3: Upper left: R xx as a function of ρ and h . Upper right: Contour plot of R xx . Lower left: R xx vs ρ with h = , . , , . ,
2. Lower right: MR xx vs h with ρ = , . , . , . , . , .
5. We set T = c = −
1, and c =
12n the following Figs. 3 to 5 we separately investigate the resistivities as functions of ρ and h under different T and c . Because c acts as an effective temperature modification, we set c = − T =
0, and c =
0. No Mott-insulating behaviors andpositive magneto-resistivity is observed in Fig. 3. In addition, non-zero charge density cansuppress the resistivity as more charge carriers are introduced. - - - - ρ T = c =- c = R xx - - ρ R xx T = c =- c = h = = = = = - - M R xx T = c =- c = ρ = ρ = ρ = ρ = ρ = ρ = Fig. 3: Fig. 4: Upper left: R xx as a function of ρ and h . Upper right: Contour plot of R xx . Lower left: R xx vs ρ with h = , . , , . ,
2. Lower right: MR xx vs h with ρ = , . , . , . , . , .
5. We set T = . c = −
1, and c = In Fig. 5 the effect of c at zero temperature is presented. The figures show similar feature asthose in Fig. 4. The larger c also suppress the resistivity, while the influence is not so obvious asthe temperature. All the pictures presented here do not exhibit the Mott-insulating behaviors andnegative magneto-resistivities. Then the finite temperature situation with T = . c = is smoothen out. - - - - ρ T = c =- c =- R xx - - ρ R xx T = c =- c =- h = = = = = - - M R xx T = c =- c =- ρ = ρ = ρ = ρ = ρ = ρ = Fig. 5: Fig. 5: Upper left: R xx as a function of ρ and h . Upper right: Contour plot of R xx . Lower left: R xx vs ρ with h = , . , , . ,
2. Lower right: MR xx vs h with ρ = , . , . , . , . , .
5. We set T = c = −
1, and c = − Then we present the dependence of R xx on h / ρ and T / √ ρ for c = −
1. The effect of different c and c can be deduced by change the temperature correspondingly, based on the analysis above.For h < ρ , Fig. 6 shows that the temperature dependence of R xx is monotonic, and correspondsto metallic behavior. For h > ρ , the R xx increases first and then decreases monotonically afterreaching a maximum. The insulating behavior appears at high temperatures in this case. Moreover,if we take larger c i , the metallic behaviors at h > ρ would disappear and the R xx decreasesmonotonically with increasing temperature. So we come to the conclusion that increasing themagnetic field would induce a finite-temperature transition or crossover from metallic to insulatingbehavior. In the last sub-figure of Fig. 6 the influence of temperature on the magneto-resistivity isshown. As temperature increases, the magneto-resistivity is remarkably suppressed.14 / ρ T / ρ ρ = c =- c = R xx / ρ R xx ρ = c =- c = h / ρ = / ρ = / ρ = / ρ = / ρ = - - / ρ M R xx ρ = c =- c = T / ρ = / ρ = / ρ = / ρ = / ρ = / ρ = Fig. 7: Fig. 6: Upper left: R xx as a function of h / ρ and T / √ ρ . Upper right: Contour plot of R xx . Lower left: R xx vs T / √ ρ with h / ρ = , . , , . ,
2. Lower right: MR xx vs h with T / √ ρ = , . , . , . , . , .
5. We set T = c = −
1, and c = At the end of the discussion of Maxwell electrodynamics, the Hall angle θ H , defined as θ H = arctan σ xy σ xx , (36)is shown in Fig. 7. At zero temperature for large h or ρ the Hall angle get saturated, with thesign depend on those of h or ρ . The surface is anti-symmetric under the transformation ( ρ , h ) → ( ρ , − h ) and ( ρ , h ) → ( − ρ , h ) . As we expect, the temperature would evidently suppress the Hallangle. 15 ig. 7: Hall angel at T = T = . c = − c = B. Maxwell-Chern-Simons Electrodynamics
The Lorentz and gauge invariance do not forbid the appearance of CP-violating Chern-Simons θ term in the electrodynamics Lagrangian L ( s , p ) = s + θ p , (37)and the Chern-Simons theory is of vital importance for the both integer and fractional quantumHall effects in condensed matter physics [78–81]. And the value of θ can be related to the Hallconductivity in the unit of e / (cid:125) . After taking the Chern-Simons term into consideration, Eq. (18)gives A (cid:48) t ( r h ) = ρ + θ hr h , (38)and the Eq. (13) becomes4 π Tr h − r h − c r h − c + (cid:0) + θ (cid:1) h + θ h ρ + ρ r h = . (39)We then reinvestigate the dependence of DC resistivity R xx on ρ and h . As one expects, the similarsaddle surface to that of Maxwell electrodynamics is found, and the reflection asymmetry andcentral symmetry due to the θ term is shown in the upper panel of Fig. 8. The more surprisingthings are the appearance of Mott-insulating behavior ∂ | ρ | R xx > MR xx <
0. At zero field R xx is the even function of ρ , and as | ρ | increases, the value of R xx increasesand reaches a maximum, showing the feature of Mott-insulating behavior, and then decreasesmonotonically. While for finite magnetic field, the reflection symmetry for positive and negative16 is broken. Along the positive ρ direction the behavior of R xx is similar, however, for negative ρ it is more complicated. as | ρ | increases, the value of R xx decreases and reaches a minimum, andthen share similar behavior as the positive ρ case. B. Maxwell-Chern-Simons Electrodynamics - - - - ρ T = c =- c = θ = R xx - - - ρ R xx T = c =- c = θ = h = = = = = - - M R xx T = c =- c = θ = ρ =- ρ =- ρ = ρ = ρ = Fig. 9: - - - - ρ T = c =- c = θ = ∂ | ρ | R xx - - - - ρ T = c =- c = θ = ∂ | ρ | R xx .0 - - - - ρ T = c =- c = θ = ∂ | ρ | R xx .0 Fig. 10: Fig. 8: Upper left: R xx as a function of h and ρ . Upper right: Contour plot of R xx . Lower left: R xx vs ρ with h = , . , . , . , .
3. Lower right: MR xx vs h with ρ = − . , − . , , . , .
6. We set T = c = − . c = θ = We then further study how the value of θ affect the Mott-insulating behavior in Fig. 9. Toclearly show the region ∂ | ρ | R xx > θ =
1. Alarger θ makes it possible even with h =
0. On the other hand, the Mott-insulating region becomeslarger but ∂ | ρ | R xx becomes smaller as θ increases.17 . Maxwell-Chern-Simons Electrodynamics - - - - ρ T = c =- c = θ = R xx - - - ρ R xx T = c =- c = θ = h = = = = = - - M R xx T = c =- c = θ = ρ =- ρ =- ρ = ρ = ρ = Fig. 9: - - - - ρ T = c =- c = θ = ∂ | ρ | R xx - - - - ρ T = c =- c = θ = ∂ | ρ | R xx .0 - - - - ρ T = c =- c = θ = ∂ | ρ | R xx .0 Fig. 10: Fig. 9: Mott-insulating region for θ = , , c = − . c = The negative magneto-resistivity revealed in Fig. 8 is also shown in the ρ − h parameter planein Fig. 10, from which we can see that the negative magneto-resistivity emerges in the finiteinterval of h / ρ of about [ − . , ] . The positive magneto-resistivity region’s value is set to be zeroas well. At larger magnetic field, the negative magneto-resistivity phenomenon disappears, andthe magneto-resistivity increases almost linearly with h . For zero density the negative magneto-resistivity does not occur. - - - - ρ T = c =- c = θ = MR xx - .25- .20- .15- .10- .05 .00 Fig. 10: Negative magneto-resistivity region for θ = c = − . c = Finally we show the behavior of Hall resistivity R xy and Hall conductivity σ xy in Fig. 11. Thenegative Hall resistivity comes from the negative transverse conductivity σ xy depending on thedirection of magnetic field h and the type of charge carriers. From the figure we see that at18ero field the transverse conductivity σ xy is exactly − θ in regardless of charge density ρ , whichagrees with the scenario in quantum Hall physics [42, 80]. At finite magnetic field, for positive ρ the Hall conductivity will decrease to zero, and then changes its sign, while for negative ρ theHall conductivity remains negative. And we can also study the impact of magnetic field on theHall conductivity with fixed ρ . All the transverse conductivity for various ρ as strong field tendto be zero as the result of localization. For positive ρ , in the positive magnetic field, the Hallconductivity decreases to zero, then changes its sign and increases to a maximum, and finallydecreases monotonically to zero from the positive side. And in the negative magnetic field, theHall conductivity increases to a negative maximum, and then decreases to zero from the negativeside. The case of negative ρ can be known by simply making transformation ( ρ , h ) → ( − ρ , − h ) in the discussion above. - - - - ρ T = c =- c = θ = R xy - - - - - - - - - - - - ρ σ xy T = c =- c = θ = h = = = = = - - - - -
202 h σ xy T = c =- c = θ = ρ =- ρ =- ρ = ρ = ρ = Fig. 12:[1] D. Vegh, Holography without translational symmetry, arXiv:1301.0537 [hep-th].[2] A. Donos and J. P. Gauntlett, Novel metals and insulators from holography, JHEP , 007 (2014)doi:10.1007/JHEP06(2014)007 [arXiv:1401.5077 [hep-th]].[3] M. Blake and A. Donos, Quantum Critical Transport and the Hall Angle, Phys. Rev. Lett. , no. 2,021601 (2015) doi:10.1103/PhysRevLett.114.021601 [arXiv:1406.1659 [hep-th]]. Fig. 11: Upper left: R xy as a function of h and ρ . Upper right: Contour plot of R xy . Lower left: σ xy vs ρ with h = , . , . , . , .
3. Lower right: σ xy vs h with ρ = − , − , , ,
6. We set T = c = − . c = θ = . Born-Infeld Electrodynamics The Born-Infeld electrodynamics is described by a square-root Lagrangian [82, 83] L ( s , p ) = a (cid:16) − (cid:112) − as − a p (cid:17) , (40)where the coupling parameter a = ( πα (cid:48) ) relates to the Regge slope α (cid:48) . It is believed that sucha NLED governs the dynamics of electromagnetic fields on D-branes. If we take the zero-slopelimit α (cid:48) →
0, the Maxwell Lagrangian is recovered L ( s , p ) = s + O ( a ) . (41)The Born-Infeld electrodynamics takes the advantage of eliminating the divergence of electrostaticself-energy and incorporating maximal electric fields [84]. This can be seen from the solution ofEq. (18) A (cid:48) t ( r ) = ρ (cid:112) a ( ρ + h ) + r , (42)which is finite when r →
0. The r h is solved form4 π r h T − r h − c r h − c + ρ (cid:113) a ( h + ρ ) + r h + a (cid:118)(cid:117)(cid:117)(cid:116) (cid:0) ah + r h (cid:1) a ( h + ρ ) + r h − r h = . (43) I. Various NLED Models
In this section, we will use ???????? to study the dependence of the in-plane resistance R xx onthe temperature T , the charge density ρ and the magnetic field h in some specific NLED models.The high temperature behaviors have been discussed in ?? , so we will focus on the behavior of R xx around T = A. Maxwell ElectrodynamicsB. BI - - ρ R xx T = c =- c = =- h = = = = = = - - R xx T = c =- c = =- ρ = ρ = ρ = ρ = ρ = ρ = - ρ R xx T = c =- c = =- h = = = = - R xx T = c =- c = =- ρ = ρ = ρ = ρ = ρ = ρ = Fig. 1:[1] D. Vegh, Holography without translational symmetry, arXiv:1301.0537 [hep-th]. Fig. 12: R xx as a function of h and ρ and some intersection curves for fixed h and ρ for a = − . a = − h and ρ touch the upper bound. a the behavior of R xx is similar to the Maxwell case[69, 77], and the negative a can bring in more interesting phenomenon. However, for negative a the Eq. (42) suffers a singularity r = r s ≡ / (cid:112) | a | ( h + ρ ) , and the physical solution requiresthat r h > r s , setting an upper bound for h + ρ [77]. We then present two cases of a = − . a = −
1, respectively. For a = − .
4, We the saddle surface is similar to the previous results, andthere is no Mott-insulating behaviors and negative magneto-resistivity. However, if one increasesthe absolute value of the negative a , the region admitting physical solution for R xx shrinks andis truncated at the upper bound of h + ρ . For zero and finite small magnetic field, the Mott-insulating behavior is absent, while for a larger field the Mott-insulating behavior emerges. Andfor a relatively small ρ , the negative magneto-resistivity is present, and increasing ρ can destroyit. To expose the role of a we expand the interaction between electrons to the first order [77] F ( a ) = ρ A (cid:48) t ( r ) ∼ ρ r (cid:104) − a r (cid:0) h + ρ (cid:1)(cid:105) + O ( a ) , (44)in which the leading order is the familiar Coulomb interaction, and the nonlinearity parameterserves as an effective modification. A positive a suppress the interaction and we do not expectdifferent phenomenon compared with the Maxwell electrodynamics. However, for a negative a the interaction is enhanced at r h F ( a ) F ( ) = + (cid:18) r s r h (cid:19) , (45)and we expect that the NLED model with negative a can grasp some features of strongly correlatedsystems. Besides, in the leading order expansion Eq. (41) the Chern-Simons term does not appear,and can be deduced from Eq. (33) that the system will not exhibit Mott-insulating behavior andnegative magneto-resistivity. Thus we conclude that the Born-Infeld electrodynamics providesa new mechanism different from the Chern-Simons theory to give rise to the Mott-insulatingbehavior and negative magneto-resistivity, and the temperature can induce a transition at finitetemperature.One can also consider another kind of construction of square-root Lagrangian [84] L ( s , p ) = a (cid:16) − √ − as (cid:17) , (46)which has the same leading order expansion as Eq. (41), and will re-derive the results of Born-21nfeld electrodynamics in the presence of zero field. The solution of Eq. (18) gives A (cid:48) t ( r ) = ρ r (cid:115) ah + r a ρ + r , (47)which can be deduced that the effective interaction is F = ρ A (cid:48) t ( r ) ∼ ρ r (cid:104) − a r (cid:0) ρ − h (cid:1)(cid:105) + O ( a ) . (48)The radius equation is4 π r h T − r h − c r h − c + h r h + r h a (cid:32)(cid:115) a ρ + r h ah + r h − (cid:33) = . (49)We re-check the R xx in the square-root electrodynamics for negative a . For a = − . ρ , it is surprising that for larger ρ , the transition from positivemagneto-resistivity to negative magneto-resistivity is observed for about ρ > h . We then studiedthe case of larger negative a . To our surprise, the saddle surface changes its direction and looksas if it rotates 90 ◦ . We find that the Mott-insulating behavior appears and becomes significant atlarge ρ , and negative magneto-resistivity is observed for various ρ . I. Various NLED Models
In this section, we will use ???????? to study the dependence of the in-plane resistance R xx onthe temperature T , the charge density ρ and the magnetic field h in some specific NLED models.The high temperature behaviors have been discussed in ?? , so we will focus on the behavior of R xx around T = A. Maxwell ElectrodynamicsB. BI - - ρ R xx T = c =- c = =- h = = = = - - R xx T = c =- c = =- ρ = ρ = ρ = ρ = - ρ R xx T = c =- c = =- h = = = = - R xx T = c =- c = =- ρ = ρ = ρ = ρ = Fig. 1:[1] D. Vegh, Holography without translational symmetry, arXiv:1301.0537 [hep-th]. Fig. 13: R xy as a function of h and ρ and some intersection curves for fixed h and ρ for a = − . a = − . Conclusion In this work the black brane solution of four-dimensional massive gravity with backreactedNLED is obtained, and with the dictionary of gauge/gravity duality, the transport propertiesof the strongly correlated systems in the presence of finite magnetic field in 2+1-dimensionalboundary is studied. In our holographic setup the bulk geometry and NLED field are perturbed,and the DC conductivities are obtained in the linear response regime. Then some generalproperties are obtained in various limit, which agrees well with the previous work. To make itconcrete, we present the study of the conventional Maxwell electrodynamics, the topological non-trivial Maxwell-Chern-Simons electrodynamics, and the Born-Infeld electrodynamics with string-theoretical correction taken into consideration. We concentrate on two interesting phenomena, i.e.,the Mott-insulating behavior and negative magneto-resistivity, and results at zero temperature aresummarized in Tab. 1.
Lagrangian Parameter Mott-insulating behavior Negative magneto-resistivityMaxwell s No NoMaxwell- s + θ p θ See Fig. 9 See Fig. 10Chern-SimonsBorn-Infeld 1 a (cid:16) − (cid:112) − as − a p (cid:17) a > a = − . a = − h and small ρ . Finite h and small ρ .See Fig. 12 See Fig. 12Square 1 a (cid:0) − √ − as (cid:1) a > a = − . ρ . See Fig. 13 a = − ρ . See Fig. 13 YesTab. 1: Conclusion of Mott-insulating behavior and negative magneto-resistivity for various NLED modelsat zero temperature T = The massive gravity coupling parameters’ influence on the in-plane resistivity is compared withthe temperature, where we find the c behaves as the effective temperature correction, and non-zero one can lead to the DC conductivity. While the c has less significant effect on R xx than c .23oreover, the dependence on ρ and h is shown and we find the field can induce the metal-insulatortransition or Mott-insulating behavior and negative magneto-resistivity. Two different mechanism,the Chern-Simons term, and the negative nonlinearity parameter are proved that can give rise toMott-insulating behavior and negative magneto-resistivity. We hope our work could explain someexperimental phenomenon in strongly correlated systems. Acknowledgment
We are grateful to thank Peng Wang for useful discussions. This work is supported by NSFC(Grant No.11947408). [1] T. Banks, W. Fischler, S.H. Shenker and L. Susskind,
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