Linear and quadratic in temperature resistivity from holography

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Received: July 13, 2016 Revised: November 14, 2016 Accepted: November 15, 2016 Published: November 22, 2016

Linear and quadratic in temperature resistivity from

holography

Xian-Hui Ge,a,d,e Yu Tian,b,d Shang-Yu Wuc and Shao-Feng Wua,d,e

a_{Department of Physics, Shanghai University,}

Shanghai 200444, China

b_{School of Physics, University of Chinese Academy of Sciences,}

Beijing, 100049, China

c_{Department of Electrophysics, National Chiao Tung University,}

Hsinchu 300, China

d_{Shanghai Key Laboratory of High Temperature Superconductors,}

Shanghai 200444, China

e_{Shanghai Key Lab for Astrophysics,}

100 Guilin Road, 200234 Shanghai, China

E-mail: [email protected],[email protected],[email protected],

[email protected]

Abstract: We present a new black hole solution in the asymptotic Lifshitz spacetime with a hyperscaling violating factor. A novel computational method is introduced to compute the DC thermoelectric conductivities analytically. We find that both the linear-T and quadratic-T contributions to the resistivity can be realized, indicating that a more detailed comparison with experimental phenomenology can be performed in this scenario.

Keywords: Black Holes, Holography and condensed matter physics (AdS/CMT), Gauge-gravity correspondence

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Contents

1 Introduction 1

2 A new black brane solution in Lifsthitz spacetime with linear axion fields

and hyperscaling violating factor 3

3 DC transport coefficients 5

3.1 DC electrical and thermoelectric conductivities 5

3.2 DC thermal and thermoelectric conductivities 11

4 Discussions and conclusions 15

1 Introduction

The normal states of high temperature superconductors and heavy fermion compounds have become one of the most challenging topics in condensed matter physics. A clear understanding of the normal-state transport properties of cuprates is considered as a key step towards understanding the pairing mechanism for high-temperature superconductiv-ity. There is still a lack of a satisfying explanation of the linear temperature dependence of resistivity at sufficiently high temperatures in materials such as organic conductors, heavy fermions, Fullerenes, Vanadium Dioxide, and Pnictides. In addition, the quadratic tem-perature dependence of the Hall angle, the violation of Kohler’s rule and the divergence of the resistivity anisotropy are those puzzled the theorists for more than two decades [1].

The transport properties of the normal states of high temperature superconductors are highly anisotropic with a much higher conductivity parallel to CuO2 plane than the perpendicular direction. The in-plane resistivity of hole-doped cuprates shows a systematic evolution with doping. In the underdoped cuprates, the in-plane resistivity varies approx-imately linearly with temperature at high temperature. But as the temperature cools down, the in-plane resistivity deviates downward from linearity, suggestive of a higher power T-dependence. The optimally doped cuprates are characterized by a linear-T resis-tivity for the range above the critical temperature T > Tc, whilst on the overdoped side, the linear-T relation is replaced by T2-dependence. On the other hand, the T2-dependence of the Hall angle can be observed in a wide range of doping from underdoped region to overdoped region.

The AdS/CFT correspondence provides a powerful prescription for calculating trans-port coefficients of strongly coupled systems by analyzing small perturbations about the black holes that describe the equilibrium state [2–4]. Recently, some of us studied conduc-tivity anisotropy holographically in [5]. In [6], Blake and Donos attempted to attack the

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mystery of the linear temperature resistivity and the quadratic temperature Hall angle phe-nomena by proposing two different relaxation time scales. One central point of their obser-vations is that the Hall angle is only proportional to the momentum dissipation-dominated conductivity i.e. θH ∼ Bσdiss/q, where σdiss is the momentum dissipation conductivity, B is the magnetic field strength and q is related to the charge density. Hence, the temper-ature dependence of the Hall angle is different from the DC conductivity because the DC conductivity is decomposed into the sum of a coherent contribution due to momentum relaxation and an incoherent contribution due to intrinsic current relaxation1 [7]. They further predicted that the resistivity would take the general form ρ ∼ T2/(∆ + T ), where ∆ is a model dependent energy scale. In the low temperature limit T  ∆, the resistivity is governed by the Fermi-liquid T2 behavior. The T2-dependence of the Hall angle also signifies the Fermi-liquid phenomena. Conversely, in the high temperature limit T  ∆, it shows linear resistivity of strange metals. In [9], the authors studied DC electrical and Hall conductivity in the massive Einstein-Maxwell-Dilaton gravity. They found that the linear-T and quadratic-T resistivity can be simultaneously achieved in Lifshitz spacetimes at a dynamical exponent z = 6/5 and a hyperscaling violating exponent θ = 8/5. Other works addressing on the linear-T resistivity and Hall angle can be found in [10–22] for an incomplete list.

In this paper, we report our construction of a new asymptotic Lifshitz black hole solu-tion in the Einstein-Maxwell-dilaton-axion model with a hyperscaling violating exponent. The solution is supported by two gauge fields and a dilationic scalar, the former playing very different roles. One gauge field is responsible for generating the Lifshitz-like vacuum of the background. The other plays a role analogous to that of a standard Maxwell field in asymptotically AdS space. The general expressions of transport coefficients are then calculated. When focusing on special cases with z = 1 in which the metric corresponds to asymptotically AdS space, one can easily achieve a resistivity with two time scales in the asymptotic AdS spacetime. It is well known that in real materials, the spatial transla-tion invariance is broken and the momentum of charge carriers is not conserved because of the presence of impurities and lattices [23–55]. In this paper, the translational symmetry breaking is realized through introducing linear-spatial coordinates dependent axions. An established means to test whether quasiparticles and thus Landau’s Fermi-liquid theory valid, is to compare the thermal conductivity and the electrical conductivity [1]. If quasi-particles can be well defined, the Wiedemann-Franz law characterizes the zero temperature value of the Lorenz number L0 = π2/3 × kB2/e2, where kB is Boltzmann’s constant and e is the charge of an electron. If in a system L/L0 equals one, we say that Fermi liquid description is exactly satisfied. On the other hand, L/L0 > 1 means that there are ad-ditional carriers which contribute to the heat current but not to the charge current. By contrast, L/L0 < 1 at zero temperature implies the breakdown of Landau’s Fermi-liquid picture [28, 48]. In this paper, all the thermoelectric conductivities and the Lorenz ratio will be computed in this model. We also would like to check the Wiedemann-Franz law at

1_{As it was clarified in [7,8], it is not proper to say that DC conductivity has one term stemmed from}

momentum relaxation and the other term from incoherent contribution since it is inconsistent with the known behavior of the incoherent hydrodynamic DC conductivities.

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zero temperature. Although in the holographic setup, the metal has no relationship what-ever with real Fermi liquids, the strange metal scaling geometries presented here maybe able to mimic Fermi liquid behavior in transport [56–58].

The structure of this paper is organized as follows. In section 2, we present a new black hole solution in general (d + 2)-dimensional Lifshitz spacetime. We then calculate the DC electrical conductivity, thermal conductivity and thermoelectric conductivity in terms of the horizon data in section 3. We develop a new method in calculating the DC transport coefficients. Discussions and conclusions are presented in section 4.

2 A new black brane solution in Lifsthitz spacetime with linear axion

fields and hyperscaling violating factor

Let us begin with a general action

S = 1 16πGd+2 Z dd+2x√−g  R+V (φ)−1 2(∂φ) 2₋1 4 n X i=1 Zi(φ)F(i)2 − 1 2Y (φ) d X i (∂χi)2  , (2.1) where we have used the notation Zi = eλiφ and Y (φ) = e−λ2φ. Note that R is the Ricci scalar and χi is a collection of d−massless linear axions. The action consists of Einstein gravity, axion fields, and U(1) gauge fields and a dilaton field. For simplicity, we only consider two U(1) gauge F_rt(1) and F_rt(2) in which the first gauge field plays the role of an auxiliary field, making the geometry asymptotic Lifshitz, and the second gauge field makes the black hole charged, playing a role analogous to that of a standard Maxwell field in asymptotically AdS space.

Solving the equations of motion, we are able to obtain a spacetime which is asymptot-ically Lifshitz and hyperscaling violated. The action yields a Lifshitz black brane solution with a hyperscaling violating factor

ds2 = r−2θd  − r2zf (r)dt2+ dr 2 r2_{f (r)} + r 2_d~x2 d  , (2.2) f (r) = 1 − m rd+z−θ + Q2 r2(d+z−θ−1) − β2 r2z−2θ/d, (2.3) F(1)rt = Q1 p 2(z − 1)(z + d − θ)rd+z−θ−1, (2.4) F_(2)rt = Q2 p 2(d − θ)(z − θ + d − 2)r−(d+z−θ−1), (2.5) λ1 = − 2d − 2θ + 2θ_d p2(d − θ)(z − 1 − θ/d), (2.6) λ2 = r 2z − 1 − θ/d d − θ , (2.7) eφ = r √ 2(d−θ)(z−1−θ/d)_{, V (φ) = (z + d − θ − 1)(z + d − θ)r}2θ/d_{, (2.8)} χi = βiaxa, β₀2 = 1 d d X i=1 − → βa· − → βa, −→βa· − → βb = β20δab for i ∈ {1, d}. (2.9) where β2 = d2β20

2(d−θ)(d2_{+2θ−(z+θ)d)}. This solution is Lifshitz-like even in the UV. When the

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The black hole solution can return to the result given in [59] and [60] under the condition of β = 0 and θ = 0, respectively. The transport coefficients have been studied in [61]. We emphasize that the choice of couplings Y (φ) and Zi(φ) is our choice here and we believe that different choices of coupling would leads to different power scalings of the transport. Intriguingly, in a later paper, the exact solution presented here was found again by the authors of [62]. The event horizon locates at r = rH satisfying the relation f (rH) = 0. We

can express the mass m in terms of rH

m = rd+z−θH + Q

2

2rH2−d−z+θ− β

2_rd−z−θ+2θ/d

H . (2.10)

By further introducing a coordinate z = rH/r, we can recast f (r) as

f (z) = 1−zd+z−θ+ Q 2 2 r2(d+z−θ−1)H z2(d+z−θ−1)_−zd+z−θ_+ β2 r2z−2θ/dH zd+z−θ_−z2z−2θ/d_{. (2.11)} The corresponding Hawking temperature is given by

T = (d + z − θ)r z H 4π  1 −d + z − θ − 2 d + z − θ Q 2 2r −2(d+z−θ−1) H − d2+ 2θ − (z + θ)d d(d + z − θ) r 2θ/d−2z H β 2  . (2.12) The entropy density is given by s = rHd−θ/4G. The specific heat of this black hole can

be evaluated via c = T (∂s/∂T )Q,β. We find that the thermodynamical stability and the positiveness of the specific heat require θ < d. The near horizon geometry can be evaluated by introducing two new coordinates u and τ :

r − rH= rH2 l2_u, t = τ rHz−1 .

We can see that at zero temperature T = 0, the solution near the horizon develops an AdS2× Rd−1 _{geometry. The near horizon geometry is defined by the limit  → 0:}

ds2= r−2− 2θ d H  −dτ2_{+ du}2 l2_u2  + r2− 2θ d H d~xd. (2.13)

The effective AdS2 radius is given by: l_ads2 ₂ = r −2−2θ d H l2 , (2.14) l2 = (d − 1)(d − θ)(d + z − θ − 2)Q2₂rH2(θ−d−z)/d + (d + z − θ)(dz − θ)r −2 H /d . (2.15)

We observe that even in the absence of the U(1) gauge field, the black brane could still be extremal with near horizon of AdS2 as we just demonstrated. It means that at low temperature the theory flows to an IR fixed point in the presence of the linear axion fields. Black hole solution at (d+z−θ−2) = 0. One may notice that as (d+z−θ−2) → 0, Q2 and f (r) appear to diverge. At well-defined solution can be achieved in an alternative form:

f (r) = 1 − m rd+z−θ − q₂2ln r 2(d − θ)rd+z−θ − β2 r2z−2θ/d, (2.16) = 1 − m r2 − q₂2ln r 2(2 − z)r2 − β2 r2z−2θ/d, F_(2)rt = q2r−1, (2.17)

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where m and q2 = Q2p2(d − θ)(z − θ + d − 2) are finite physical parameters without divergence as (d + z − θ − 2) → 0. A careful examination of (2.16) and (2.17) reveals that they satisfy the corresponding Einstein equation and Maxwell equation. We can express f (r) in terms of the event horizon radius

f (r) = 1 − r 2 H r2 + q₂2 2r2_{(2 − z)}ln rH r − β2 r2z−2θ/dH  r2 H r2 − r2z−2θ/dH r2z−2θ/d  . (2.18)

The Hawking temperature is given by

T = r z H 2π  1 − q 2 2 4(2 − z)r2 H −β 2_{(d + θ − dz)} dr2z−2θ/dH  . (2.19) 3 DC transport coefficients

Firstly, we would like to introduce a new method by taking advantage of the matrix theory and the equations of motion, which maybe called the matrix method, to calculate the DC electrical and thermoelectric conductivities. The standard calculational method will be presented in section 3.2 as a consistent check and the thermal conductivity will be computed. In what follows, we work in the special case with d = 2. Later, we will extend our discussions to more general conditions.

3.1 DC electrical and thermoelectric conductivities

For simplicity, we rewrite the metric in d = 2 dimensional spacetime as

ds2 = −gttdt2+ grrdr2+ gxxdx2+ gxxdy2. (3.1)

For the purpose of computing the electrical conductivity, we consider the linear perturba-tions of the form

A_(1)x = a1(r)e−iωt, (3.2)

A_(2)x = a2(r)e−iωt, (3.3)

htx = htx(r)e−iωt, (3.4)

χ1 = βx + ¯χ1(r)e−iωt, (3.5)

and let the other metric and gauge perturbations vanishing. Since we choose the conduc-tivity along the x− direction, it is consistent to set all scalar fluctuations to be vanished except for the one with the linear piece along the direction x. We can arbitrarily denote this scalar by χ and write χ = βx + ¯χ(r)e−iωt. The equation of motion for the linear

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perturbation can be obtained as r gtt grr Z2a0₁ 0 +A 0 (1)tZ1gxx √ gttgrr  gxxhtx 0 + ω2r grr gtt Z2a1 = 0, (3.6) r gtt grrZ2a 0 2 0 +A 0 (2)tZ2gxx √ gttgrr  gxxhtx 0 + ω2r grr gttZ2a2 = 0, (3.7) r gtt grrgxxZ2χ¯ 0 0 + ω2r grr gttgxxZ2χ − iωβ¯ 2_Z2r grr gtthtx = 0, (3.8)  gxxhtx 0 + i ¯χ 0_g tt ωZ2 + Z1A 0 (1)ta1+ Z2A0(2)ta2 = 0, (3.9)  g2_xx √ grrgtth 0 tx 0 − q1a0₁− q2a0₂− β2gxxYr grr gtthtx− iωgxxY r grr gttχ = 0,¯ (3.10) where the prime denotes a derivative with respect to r. Note that the derivative of the scalar potential is given by A0_(1)t = − q1

Z1(φ) √ gttgrr gxx and A 0 (2)t = − q2 Z2(φ) √ gttgrr gxx , where q1 = Q1p2(z − 1)(z + d − θ) and q2 = Q2p2(d − θ)(z − θ + d − 2). Equation (3.9) is a constrained equation, which implies that the linear perturbations a1, a2, htxand ¯χ are not all linearly independent.

After introducing ˜χ = f rz−5χ¯0/(iω) and eliminate htx, we are able to rewrite the equations (3.6)–(3.9) in a more explicit form

(rz−3+θf a0₁)0 = A1a1+ B1a2+ C1χ,˜ (3.11) (r3z−1−θf a0₂)0 = A2a1+ B2a2+ C2χ,˜ (3.12) (r3(z−1)f ˜χ0)0 = A3a1+ B3a2+ C3χ,˜ (3.13) where A1 =  q2₁ r5−z−θ − ω2 r5+z−θ_f  , A2 = B1 = q1q2 r5−z−θ, B2 =  q2₂ r5−z−θ − ω2 r3−z+θ_f  , B3 = C2= − βq2 r5−z−θ, C3 =  β2 r5−z−θ − ω2 r5−z_f  , A3 = C1= − βq1 r5−z−θ.

We notice that the combination (3.11) + (3.13) × q1/β and (3.12) + (3.13) × q2/β leads to  rz−3+θf a0₁+q1 βr 3(z−1)_{f ˜χ}0 0 = 0, (3.14)  r3z−1−θf a0₂+q2 βr 3(z−1)_{f ˜χ}0 0 = 0. (3.15)

A massless mode can be extracted from (3.14) and (3.15). From the membrane paradigm approach [63] we know that the realization of the currents in the boundary theory can be identified with radially independent quantities in the bulk. From (3.6) to (3.9), one can

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easily find that the equivalent expressions of the conserved electric currents in the zero frequency limit read

J1 = −rz−3+θf a0₁+ q1rθ−2htx, (3.16) J2 = −r3z−1−θf a0₂+ q2rθ−2htx. (3.17) The DC conductivity is the zero frequency limit of the optical conductivity

σ_ijDC = lim ω→0σ DC ij (ω) = lim_ω→0 ∂Ji(ω) ∂Ej(ω) (3.18)

The DC conductivity can be evaluated at the horizon whenever we have massless mode since it does not evolve between the horizon and the boundary [44]. Then let us define a matrix ˜σ from u w v rz−3+θf a0₁ r3z−1−θf a0₂ r3(z−1)f ˜χ0 }  ~= ˜σ u w v iωa1 iωa2 iω ˜χ }  ~,

where the special notation_{J. . .K should be considered as a square matrix which is introduced} for convenience, for example

u w v a1 a2 ˜ χ }  ~≡    a1 a(2)1 a (3) 1 a1 a(2)2 a (3) 2 ˜ χ ˜χ(2) χ˜(3)   , (3.19)

in which a(i)₁ , a(i)₂ and ˜χ(i) _{are linearly independent sources, introduced to guarantee the} source term invertible. After inverting the components in J. . .K, a

(i) 1 , a

(i)

2 and ˜χ(i) will be not important in further calculations and it is better for us to hide them in _{J. . .K.} We emphasize that the matrix ˜σ is not the exact conductivity tensor of the system as we can see below. We take the derivative of ˜σ and obtain

˜ σ0 = u w v rz−3+θf a0₁ r3z−1−θf a0₂ r3(z−1)f ˜χ0 }  ~ 0u w v iωa1 iωa2 iω ˜χ }  ~ −1 − iω˜σ u w v a1 a2 ˜ χ }  ~ 0u w v iωa1 iωa2 iω ˜χ }  ~ −1 = u w v A1a1+ B1a2+ C1χ˜ A2a1+ B2a2+ C2χ˜ A3a1+ B3a2+ C3χ˜ }  ~ u w v iωa1 iωa2 iω ˜χ }  ~ −1 − iω˜σ u w v a0₁ a0₂ ˜ χ0 }  ~ u w v iωa1 iωa2 iω ˜χ }  ~ −1 = 1 iω    A1 B1 C1 A2 B2 C2 A3 B3 C3   − iω˜σ    (rz−3+θf )−1 0 0 0 (r3z−1−θf )−1 0 0 0 (r3(z−1)f )−1   σ.˜ (3.20)

The prime denotes the derivative with respect of r. The advantage of this method is that it reduce second order ordinary differential equations to non-linear first order ordinary differential equations. Multiplying both sides of equation (3.20) with f , we obtain

f ˜σ0 = f iω    A1 B1 C1 A2 B2 C2 A3 B3 C3   − iω˜σ    r3−θ−z 0 0 0 r1+θ−3z 0 0 0 r3−3z   σ.˜

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At the event horizon f (rH) = 0 and ˜σ0 is finite. So the above equation reduces to

0 =    rθ−z−5H 0 0 0 rHz−θ−3 0 0 0 rz−5H   − ˜σ0    r3−θ−zH 0 0 0 r1+θ−3zH 0 0 0 r3−3zH   σ˜0.

The regularity condition at the event horizon yields

˜ σ0=    rH−4−θ 0 0 0 rH2z−2+θ 0 0 0 r2z−4H   .

From the definition of the matrix ˜σ, we obtain the boundary condition at the event horizon f a0₁ → iωr−z−1_H a1    rH , (3.21) f a0₂ → iωr−z−1_H a2   _r H , (3.22) f ˜χ0 → iωr−z−1_H χ˜    rH . (3.23)

Considering the above relation (3.21)–(3.23), we then can impose the regularity condition at the horizon from equation (3.10) and obtain

htx r=rH =  − iω q1 β2_Y a1− iω q2 β2_Y a2− iω ¯ χ β2_Y     r=rH . (3.24)

The last term in the right hand of (3.24) will be dropped out in the following calcula-tion since it does not contribute to the transport. Further utilizing (3.21), (3.22), (3.16) and (3.17), we can determine the value of currents

J1 = −  rHθ−4+ q₁2 β2r 2z−4 H  iωa1−q1q2 β2 r 2z−4 H iωa2, (3.25) J2 = −  rH2z−2−θ+ q2₂ β2r 2z−4 H  iωa2− q1q2 β2 r 2z−4 H iωa1. (3.26)

The DC electric conductivity can be computed via σij = _∂E∂Ji_j, where Ej = −iωaj. Finally we obtain σ11= rθ−4H + q₁2 β2r 2z−4 H , σ12= q1q2 β2 r 2z−4 H , σ21= σ12, σ22= r2z−2−θH + q₂2 β2r 2z−4 H . (3.27)

This result is consistent with [62]. The physical interpretation of the DC conductivity tensor obtained here is somehow subtle: we consider electric perturbations only along the x−direction but obtain a 2 × 2 conductivity matrix with non-vanishing off-diagonal components. We also observe that taking z → 1, θ → 0 and then q1→ 0, but the quantity σ11 = rH−4 is not vanishing. However, if we set z = 1 and θ = 0 from the very beginning

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in the action (2.1), the auxiliary gauge field F_(1)rt naturally does not appear and the black hole solution is the Reissner-Nordstr¨om-AdS metric with vanishing σ11and σ12. So we have a discontinuity in the z → 1, θ → 0 and q1 → 0 limit. This means that once we change the asymptotic structure from an AdS to a Lifshitz one and turn on the perturbation δA(1)x, it could not have a continuous limit back to the perturbation considered in the Reissner-Nordstr¨om-AdS spacetime by simply taking z → 1, θ → 0 and q1→ 0 limit.

The original purpose of introducing the auxiliary U(1) gauge field F(1)rt is to construct the Lifshitz-like nature of the vacuum. One may notice that not only A(1)t , but also a1 diverges in the asymptotic r → ∞ regime:

a1= a10+ a20

rz−4+θ, (3.28)

where the second term diverges when z − 4 + θ < 0 at the infinite boundary. So that we must impose the regular condition a20= 0. That is to say, a1 does not introduce a charge current on the asymptotic boundary. In this sense, we should set the boundary condition J1 = 0. From (3.25), (3.26) and σij = _∂E∂Ji_j, we obtain

σDC = rH2z−2−θ+ q2 2 (β2_{+ q}2 1r 2z−θ H ) r2z−4H . (3.29)

This is a very intriguing result because (3.29) means that even without translational sym-metry breaking, finite DC electric conductivity can still be realized because of the presence of the auxiliary U(1) charge q1 [64]. By embedding the Lifshitz solution in AdS, the di-vergence encountered here is no longer a problem since an AdS embedding modifies the UV properties without affecting the horizon behavior. However, it is not of our purpose to realize such an AdS embedding in this paper.

Another interesting situation is the case without translational invariance breaking (i.e. β = 0). We also arrive at a finite conductivity

σDC = rH2z−2−θ+ q2₂ q2 1r 2z−θ H rHθ−4. (3.30)

The linear and quadratic in temperature resistivity can be reached via z = 6/5 and θ = 8/5. Note that these are the exact exponents given in [9]. This feature of the construction of a finite conductivity without the need to break translational invariance has been reported and explained by Sonner in [64]. Throughout this paper, we mainly consider the situation with J1 = 0, because it is mathematically inconsistent to turn off a1. However, it is also physically unclear of the boundary correspondence of the source a1 because the auxiliary gauge field is only introduced to realize Lifshitz-like vacuum. Therefore, it is consistent to set J1= 0.

Considering two gauge fields resulting a 2×2 electric conductivity matrix, one naturally expects that the thermoelectric conductivity has more than one component. One may notice the equation of motion for htx at zero frequency is given by

h00_tx−1 2  g0_rr grr +g 0 tt gtt  h0_tx+ g 0 rrgtt0 2grrgtt + g 02 tt 2g2_tt − g_tt00 gtt +Z2A 02 (2)t gtt +Z1A 02 (1)t gtt  htx +Z2A0(2)ta 0 2+ Z1A0(1)ta 0 1 = 0. (3.31)

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Clearly, the vector type of perturbations htx is coupled to a1 and a2. Together with equations of motion of the Maxwell fields to the linear order, we can write down a radially conserved heat current

Q =r gtt grr  − gtt_h tx∂rgtt+ h0tx  − A_(1)tJ1− A(2)tJ2. (3.32) After imposing the regularity condition at the event horizon, that is to say

htx(r = rH) =  − iω q1 β2_Y a1− iω q2 β2_Y a2+ . . .     r=rH , (3.33)

we can simply evaluate the conserved heat current at the event horizon Q = −4πT iωr 2z−2−θ H β2 (q1a1+ q2a2)     r=rH . (3.34)

We have used the boundary condition A(1)t(rH) = A(2)t(rH) = 0. The thermoelectric

conductivity can be obtained at the event horizon r = rHby using the expression ¯αi = _{T ∂E}∂Q_i. We finally obtain ¯ α1 = ∂Q T ∂E1 = 4πq1 β2 r 2z−2−θ H , (3.35) ¯ α2 = ∂Q T ∂E2 = 4πq2 β2 r 2z−2−θ H . (3.36)

There are no off-diagonal components of the thermoelectric coefficient as can be seen above. Both components obey the same temperature scaling. If one turns on magnetic field, the off-diagonal components of the thermoelectric conductivity can be observed.

Special case: z = 1, θ = 1 and J1 = 0. For the case θ = 1, z = 1 and thus q1 = 0, the temperature is given by

T = rH 2π  1 − q 2 2 4r2 H − β 2 2rH  . (3.37)

In this case, the entropy density s = rH/4G is proportional to the temperature in the small

q and β limit. We find that the DC electric conductivity (3.29) behaves as σDC = 1 rH + q 2 2 β2_r2 H ∼ 1 2πT + q2 2 4π2_β2_T2, (3.38)

where we have use the large horizon radius approximation rH ∼ 2πT . The resistivity in

the small β limit can be expressed as ρ ≈ 4β 2_π2_T2 q₂2+ 2β2_πT = ˜ T2 ˜ T + ∆, (3.39)

where ˜T = 2πT and ∆ = q2₂/β2. Equation (3.39) shows us that for ˜T  ∆, the resistivity is dominated by the linear-T behavior, while ˜T  ∆, the system obeys the Fermi-liquid

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Figure 1. The resistivity as a function of temperature. (Left) The resistivity shows linear-T behavior at higher temperature with q2

2/β4= 8. (Right) The resistivity shows quadratic-T behavior

at lower temperature with q2

2/β4 = 10. The dashed red lines correspond to fitting functions ρ ∼

10.78T /β and ρ ∼ 6.09T2/β2, respectively.

like law. As a demonstration, we plot the resistivity as a function of temperature in figure 1. In the higher temperature regime, the resistivity shows linear in temperature dependence, analogous to the experimental behavior of bad metals. In the low temperature regime, the resistivity varies as T2, retaining Landau’s Fermi-liquid description, although the quasiparticle picture is not well defined here. One can also understand equation (3.38) as follows: for small β but fixed temperature T and charge density q2, (3.39) shows Fermi-liquid-like property, while large β results in strange metal behavior. At zero temperature, the DC conductivity becomes

σDC = 4 β2 + β2 q2 2 −pβ 4_{+ 4q}2 2 q2 2 . (3.40)

This equation implies that as the disorder goes to zero, the system becomes an ideal metal with infinite DC conductivity, while β → ∞ the ground state is an insulator.

3.2 DC thermal and thermoelectric conductivities

In irreversible thermodynamics, the dissipative properties of a system are closely related to the entropy production in a unit time

ds dt =

X

i

TiXi, (3.41)

where Xi is the thermal force which is determined by the gradients of energy, temperature, chemical potential etc. Ti denotes the current driven by Xi which can be written in the linear approximation as

T_i=X j

L_ijX_i, (3.42)

where Lij represent the transport coefficients. We can see that both the thermal force Xi and the transport coefficients Lij contribute to the entropy production rate. The thermal force represents the external factor describing the environment and the transport coef-ficients are the intrinsic causes reflecting the responsibility of the system driven by the thermal force.

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In what follows, we would like to introduce a linear in time source for the background metric. So that even in the absence of hydrodynamics the transport coefficients investigated here retain their essential interpretation: they characterize the rate of entropy production when the equilibrium state is subjected to a slowly varying source. Therefore, it is reason-able to write the linear perturbation with both time- and radial-coordinates dependence: δgµν = tc0+ hµν(r) with c0 a source. For instance, we are able to write gauge perturbation A(i)x = aie−iωt= ai+ Eit + O(t2).

In order to compute the thermoelectric and thermal conductivities, we need to consider perturbations with sources for both the electric and the heat currents.

gtx= tδh(r) + htx, A(1)x = E1t + ta1(r) + δA1, A(2)x= E2t + ta2(r) + δA2, (3.43) The conserved currents can be written as

J1 = − r gtt grrZ1(φ)  ta0₁+ δA0₁  − q1gxx  tδh(r) + htx  , (3.44) J1 = − r gtt grrZ2(φ)  ta0₂+ δA0₂− q2gxx  tδh(r) + htx  . (3.45)

The conserved heat current becomes ˜ Q =r gtt grr h − gtthtx+ tδh(r)  ∂rgtt+  tδh0(r) + h0_tx i − A_(1)tJ1− A_(2)tJ2. (3.46) In order to evaluate the thermoelectric conductivities, we assume δh(r) = −ζgttand ai(r) = −Ei+ ζA(i)t, so that the time-dependent terms of the conserved currents are canceled and the form of the currents remain unchanged. According to the holographic dictionary, the coefficient ζ corresponds to the thermal gradient −∇xT /T . We can then express the conserved currents (3.44) and (3.46) as

J₁ = −r gtt grrZ1(φ)δA 0 1− q1gxxhtx, (3.47) J₂ = −r gtt grr Z2(φ)δA02− q2gxxhtx, (3.48) ˜ Q = r gtt grr  − gtthtx∂rgtt+ h0tx  − A_(1)tJ1− A(2)tJ2. (3.49) In the previous section, we choose the gauge hrx = 0. Here we would like to turn on hrx. The linearized rx-component of the Einstein equations now is given by

hrx = gxxδχ 0 1 β + Z2(φ)gxxA0_(2)tE2+ Z1(φ)gxxA0_(1)tE1 Y (φ)gttβ2 +gxxδh 0_{(r) − g}0 xxδh(r) gttβ2Y (φ) . (3.50) We assume that δχ0₁ is analytic at the event horizon and falls off fast at the infinity so that it has no contribution to the boundary value of hrx. After switching to the Eddington-Finklestein coordinates (v, r) with v = t +R pgrr/gttdr and imposing the regularity con-dition at the event horizon, from (3.43) we obtain

δA1= E1 Z p grr/gttdr, (3.51) δA2 = E2 Z p grr/gttdr. (3.52)

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In the Eddington-Finklestein coordinates, we need explore relationship between htx and hrx. The linear perturbative part of the metric can be expressed as

2htxdvdx + 2htxr grr gtt

drdx + 2hrxdrdx. (3.53)

In order to cancel out the divergence at the event horizon, we need to impose the condition htx(r = rH) = − r gtt grr hrx     r=rH =  −E1q1+ E2q2 Y (φH)β2 − 4πT ζgxx Y (φH)β2     r=rH . (3.54)

Therefore, the conserved currents can be expressed by their values at the event horizon J₁ =  E1Z1(φ) + E1q12+ E2q1q2 β2_{Y (φ)g} xx +4πT q1ζ β2_{Y (φ)}     r=rH , (3.55) J₂ =  E2Z2(φ) + E2q22+ E1q1q2 β2_{Y (φ)g} xx +4πT q2ζ β2_{Y (φ)}     r=rH , (3.56) ˜ Q =  4πT q1E1+ 4πT q2E2 β2_{Y (φ)} + 16π2T2ζgxx β2_{Y (φ)}     r=rH . (3.57)

The electrical conductivity matrix can be written down as σ11 = ∂J1 ∂E1 = r θ−4 H + q₁2 β2r 2z−4 H , σ12= ∂J1 ∂E2 = q1q2 β2 r 2z−4 H , (3.58) σ21 = ∂J2 ∂E1 = σ12, σ22= ∂J2 ∂E2 = r 2z−2−θ H + q₂2 β2r 2z−4 H . (3.59)

Therefore, we reproduce the result presented in (3.27). The thermoelectric conductivity α and thermal conductivity ¯κ are then evaluated as

α1 = 1 T ∂J1 ∂ζ = 4πq1 β2_{Y (φ)}     r=rH = 4πq1 β2 r 2z−2−θ H , (3.60) α2 = 1 T ∂J2 ∂ζ = 4πq2 β2_{Y (φ)}     r=rH = 4πq2 β2 r 2z−2−θ H , (3.61) ¯ κ = 1 T ∂ ˜Q ∂ζ = 16π2T gxx β2_{Y (φ)}     r=rH = 16π 2_T β2 r 2z−2θ H . (3.62)

The thermal conductivity is not influenced by the gauge fields and only one component appears at this moment.

One can continue of the analysis given before (3.29) and imposes the condition J1 = 0. The DC thermoelectric and thermal conductivities become

¯ αDC = 4πq2 (β2_rθ−2z H + q2₁)r2H . (3.63) ¯ κDC = 16π2T β2_r2θ−2z H + rθHq 2 1 . (3.64)

As z = 1, θ = 1 and q1 = 0, the Seebeck coefficient behaves as ¯α ∼ 1/T . On the other hand, setting β to be zero, one obtains ¯α ∼ 1/T2. At zero temperature, ¯α = 8πq2/β4.

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In brief, the thermoelectric conductivity is influenced by temperature and impurities. It would be interesting to check the Wiedemann-Franz law by introducing the thermal con-ductivity at zero electric current, which is the usual thermal concon-ductivity that is more readily measurable, κ = ¯κDC− αDCα¯DCT /σDC, and thus

κDC = 16π2T rH2z+2−2θ β2_r2 H+ q 2 2rθH+ q 2 1r2+2z−θH . (3.65)

In the case θ = 1, z = 1 and thus q1 = 0, both the ratios κ/T = 16π2/β2 and ¯κ/T = 16π2_/β2 _{are a constant. This reflects that the thermal conductivity is dominated by} im-purity scattering.

In conventional metals, the WF law is characterized by the constant Lorenz ratio L0. The WF law asserts that the ratio of the electronic contribution of the thermal conductivity to the electrical conductivity of a conventional metal, is proportional to the temperature. This implies that the ability of the quasiparticles to transport heat is determined by their ability to transport charge so the Lorenz ratio is a constant. In our set-up, the Lorenz ratios are given by

¯ L ≡ ¯κDC σDCT = 16π2rH4−θ β2_r2 H+ q 2 2rθH+ q 2 1r 2+2z−θ H , (3.66) L ≡ κDC σDCT = 16π 2_r6 H(β 2_rθ H+ q 2 1r2zH ) (β2_r2+θ H + q22r2θH + q12r 2+2z H )2 (3.67) At zero temperature with θ = 1 and z = 1, (3.66) and (3.67) yield ¯L = 4π2 + 4π2β2/p4q2

2 + β4 > L0 and L = 4π2β4/(4q22+ β4) + 4π2β2/p4q22+ β4. Usually, we regard L as the quantity comparable with the experiments. Eq. (3.67) implies that as β → 0 deviations from the Fermi-liquid behavior can be obtained, while β → ∞, so L = 8π2 the system shows Fermi-liquid-like behavior. This is quiet different from the behavior of the electric conductivity given in (3.38).

d + 2-dimensional DC transport coefficients. In what follows, we extend our results to the d + 2-dimensional case

σ11 =  g d−2 2 xx Z1(φ) + q₁2 β2_{Y (φ)g}d/2 xx     r=rH = rH2θ−2θ/d−2d+ q2₁ β2r 2z+θ−2−d−2θ/d H , (3.68) σ12 = σ21= q2q1 β2 r 2z+θ−2−d−2θ/d H , (3.69) σ22 =  g d−2 2 xx Z2(φ) + q₂2 β2_{Y (φ)g}d/2 xx     r=rH = rHd+2z−θ−4+ q₂2 β2r 2z+θ−2−d−2θ/d H , (3.70) ¯ α1 = 4πq1 β2_{Y (φ)}     r=rH = 4πq1 β2 r 2z−2−2θ/d H , (3.71) ¯ α2 = 4πq2 β2_{Y (φ)}     r=rH = 4πq2 β2 r 2z−2−2θ/d H , (3.72) ¯ κ = 16π 2_{T g}d/2 xx β2_{Y (φ)}     r=rH = 16π 2_T β2 r d+2z−2−θ−2θ/d H . (3.73)

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We would also like to generalize the transport coefficients under the the condition J1 = 0, so that the transport coefficients reduce to diagonal components

σDC = rHd+2z−θ−4+ q2r2θ+2z−d−2θ/dH β2_r2+θ H + q₁2r 2z+d H , αDC = ¯αDC = 4πr2z+θ−2θ/dH β2_r2+θ H + q2₁r 2z+d H , ¯ κDC = 16π2T rH2z+d−2θ/d β2_r2+θ H + q12rH2z+d .

Similarly, we have the thermal conductivity at zero electric current κDC = 16π2T rH3d+2z q2 2r 4+3θ H + (β2r 2+θ H + q₁2r d+2z H )r 2d+2θ/d H . (3.74)

The corresponding Lorenz ratio in (d + 2)-dimensional spacetime are obtained as ¯ L ≡ ¯κDC σDCT = 16π 2_r2d+4+θ H q₂2r4+3θH + (β2r 2+θ H + q12rHd+2z)r 2d+2θ/d H , (3.75) L ≡ κDC σDCT = 16π 2_r4+4d+θ+2θ/d H (β2r 2+θ H + q12rd+2zH ) (q₂2rH4+3θ+ (β2r 2+θ H + q₁2r d+2z H )r 2d+2θ/d H )2 . (3.76)

At zero temperature with vanishing charge density qi = 0, which is associated with the quantum critical regime. As z = 1, the above Lorenz ratios are a constant at zero tem-perature ¯L = L = 16π2 d2− d(θ + 1) + 2θ
/d(d − θ + 1). Note that in the absence of charge density, the electric conductivity is dominated by the particle-hole creation of the boundary field theory. While for non-vanishing charge density, the Lorenz ratios decrease as the chemical potential increases. By contrast, β = 0 and z = 1 at zero temperature leads to ¯L = 8π2/(d − θ)(d + z − θ) and L = 0. In general, the Lorenz ratios become tem-perature independent when θ = d regardless of the value of z, which in turn corresponds to a vanishing specific heat.

4 Discussions and conclusions

In the previous sections, we did not study the Hall angle, which we would like to defer to a future publication. After turning on a magnetic field on the background , one can easily find that the blacken factor is given by

f (r) = 1 − m rd+z−θ + Q2₂ r2(d+z−θ−1) − β2 r2z−2θ/d + B2r2z−6 4(1 − θ/d)(4 + 2θ/d − 3z). (4.1) The Hall angle2 can be evaluated by following [6]

θH ∼ Bq2 β2 r

2z+θ−2−d−2θ/d

H . (4.2)

2_{Consistency of the resulting perturbation equations requires both gauge fields to fluctuate. This in turn}

leads to some subtleties in the analysis. In general, two gauge fields with magnetic fields lead to a 4 × 4 DC electrical conductivity matrix. Here we mainly consider Hall angle generated by the second gauge field in the action (1).

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For the case d = 2, z = 1 and θ = 1 and in the higher temperature limit rH ∼ T , we

have θH ∼ T−2 which is what observed in cuprates. The transport coefficients in the presence of a magnetic field are studied in [65] and comparisons with the experimental phenomenologies are discussed.

In summary, we obtained a new black hole solution in Lifshitz spacetime with a hy-perscaling violating factor. At zero temperature, the black hole approaches AdS2 × Rd geometry near the horizon with non-vanishing entropy density. One can reproduce the black hole solution given in [60] and [59] as θ = 0 and β = 0, respectively. The black hole solution is different from the one obtained in [66], where the authors constructed a class of Lifshitz spacetimes in five dimensions that carry electric fluxes of a Maxwell field.

We then studied holographic DC thermoelectric conductivities in this model with mo-mentum dissipation. The novel matrix method was introduced to compute the transport coefficients. Since two gauge fields are presented, these result in a 2 × 2 DC electric conduc-tivity matrix. The results cannot recover electric conducconduc-tivity in Reissner-Nordstr¨om-AdS background by simply taking z → 1, β → 0 and θ → 0 limits, although the metric can re-cover that of Reissner-Nordstr¨om-AdS type in these limits. This reflects that once we turn on the gauge perturbation in Lifshitz spacetime, it is not possible to have a continuous limit to the perturbation that is normally considered in Reissner-Nordstr¨om-AdS background. When we physically setting the electric current J1 of the auxiliary gauge field to be vanish-ing, the components of the conductivity matrix with respect to the auxiliary gauge fields disappear, but mixture between q1 and the transport coefficients can be observed. It is only when we take z = 1, q1 vanishes. We expect that when we turn on the magnetic field in Lifshitz spacetime, the resulting electric conductivity should be a 4 × 4 matrix. More complicated situations would then be observed. It deserves further investigation on such complication and mixture.

The most intriguing result is that linear and quadratic in temperature resistivity can be realized simultaneously under the condition z = 1, d = 2 and θ = 1. The exponents taken here agree with the scaling approach provided in [67], but different from [68,69]. We notice that the exponents taken here violates the null energy condition in the bulk. But a careful examination of the local thermodynamic stability and the causal structure of the boundary field theory reveals that it is true that the system is locally thermodynamically stable at all temperatures and charges without superluminal signal propagation on the boundary.

This work can be considered as a concrete example realizing what were proposed by Blake and Donos in their paper [6]. For the resistivity, at the low temperature, it behaves as the Fermi-liquids, while in the high temperature, it reduces to linear in temperature resistivity same as strange metals. We also studied the thermoelectric conductivities and the Lorenz ratios in this paper. Although in the holography, there are no quasiparticles and thus the system has no relationship with real Fermi liquids, the scaling geometries presented here are able to mimic Fermi liquid behavior for certain regime of q₂2/β4 as shown in (3.67).

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Acknowledgments

We would like to thank Elias Kiritsis, Hong L¨u, Sang-Jin Sin and John McGreevy for helpful discussions. Y. Tian would like to thank Prof. Glenn Barnich for his hospitality at Universit´e Libre de Bruxelles and International Solvay Institutes. XHG was partially sup-ported by NSFC, China (No. 11375110). Y.T. is partially supsup-ported by NSFC with Grant No. 11475179 and the Grant (No. 14DZ2260700) from Shanghai Key Laboratory of High Temperature Superconductors. SYW was partially supported by the Ministry of Science and Technology (grant no. MOST 104-2811-M-009-068) and National Center for Theo-retical Sciences in Taiwan. SFW was supported partially by NSFC China (No. 11275120, No. 11675097).

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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