DC conductivity with external magnetic field in hyperscaling violating geometry

Neha Bhatnagar1, ∗ and Sanjay Siwach1, †

1Department of Physics, Banaras Hindu University,Varanasi-221005, India

We investigate the holographic DC conductivity of (2+1) dimensional systems while consideringhyperscaling violating geometry in bulk. We consider Einstein-Maxwell-Dilaton system with twogauge fields and Liouville type potential for dilaton. We also consider axionic fields in bulk tointroduce momentum relaxation in the system. We apply an external magnetic field to studythe response of the system and obtain analytic expressions for DC conductivity, Hall angle and(thermo)electric conductivity.

I. INTRODUCTION

String theory provides us a valuable tool to investigate strongly coupled gauge theories using AdS/CFTduality(holography)[1–3]. The duality establishes a connection between strongly coupled gauge theory in d-dimensionson the boundary and its weakly coupled gravity dual in (d+1)-dimensional bulk spacetime. Many important phe-nomenon of nuclear physics, condensed matter physics and high Tc superconductors are being explored using thisduality [4–9].

Recently, considerable interest is seen to study realistic condensed matter systems using holography. The inves-tigation of different phases of strongly coupled systems requires new holographic models. Significant results havebeen obtained in this area after including momentum dissipation term in holographic models. Realistic examples ofstrongly coupled systems have finite DC conductivity either due to the presence of impurity or as a result of brokentranslational invariance. One can introduce momentum relaxation in holographic systems through various ways; byintroducing impurity in holographic set-up [10, 11] or by introducing spatial source field which breaks translationalsymmetry [12, 13]. Also the breaking of diffeomorphism invariance using massive gravity term in the bulk theory,results in finite DC conductivity [14–18].

Further, efforts are being made to study strongly coupled condensed matter systems near quantum critical pointsusing holography. These critical points are realized in the boundary system by opting for anisotropic scaling betweentemporal and spatial directions in the gravity set-up. Although the introduction of anisotropy results in breakingof Lorentz invariance, the metric remains scale invariant. The system with this anisotropy while possessing scalingsymmetry is known as Lifshitz-like geometry,

x→ λx t = λzt. (1)

where x is spatial coordinate, t is temporal coordinate and z > 1 is known as the dynamical critical exponent.Several efforts are being made to realize this type of geometry (Lifshitz-like) in gravity set-up. The most common way

is working with Einstein-Maxwell-Dilaton (EMD) theories in the gravity system. This geometry was first introducedby [19] and found wide range of application in analyzing thermodynamical and hydrodynamical aspects of stronglycoupled systems [20–28]. Generalized system with warped metric is also used for the detailed studies of realisticcondensed matter system [29–36].

ds2d+2 = r2θd

(−r2zdt2 +

dr2

r2+ r2

d∑i=1

dx2i

), (2)

where θ is known as hyperscaling violating parameter in d-dimensions.In this work, we investigate the DC conductivity and (thermo)electric conductivity of (2+1) dimensional systems

with the hyperscaling violating term for Lifshitz-like geometry. Hence, we consider two different gauge fields in gravityset-up, one field will introduce Lifshitz-like geometry while other introduces finite charge density. Linear axionic fieldshave been introduced in the system to break translational invariance and obtain finite DC conductivity. Introducing

∗Electronic address: bhtngr.neha@gmail.com†Electronic address: sksiwach@hotmail.com

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an external magnetic field [37–40] has given us an edge for detailed study of the holographic system and is themain motivation of our present work. We discuss the dependence of transport on the dynamical exponents in thepresence of external magnetic field and multiple gauge fields. To read transport coefficients for boundary theory usingcorrelation function, one can use various approaches [41–56]. In this work, we have simplified the calculation whileusing Wilsonian RG(renormalization group) flow approach[57]. The advantage of this approach is that second ordercoupled differential equations reduces to first order ordinary differential equations [43, 46, 49]. We have studied theseRG flow equations and extract the transport coefficients of the boundary theory.

The paper is organized as follows. We introduce basic holographic set-up of Lifshitz-like geometry with hyperscalingviolation term in the next section. We use Wilsonian RG flow approach to study DC conductivity and (thermo)electricconductivity for the system in the presence of the external magnetic field in the following section. We discuss thedependence of transport coefficients on magnetic field and strength of momentum relaxation by various plots. Thetemperature dependence of various counductivities is also investigated. The concluding section contains the detaileddiscussion and summary of the work done.

II. EMD SYSTEM

Let us consider the EMD system with two gauge fields and axionic fields. The gravity action is given by,

S =

∫d4x√−g

(R− 1

2(∂φ)2 + V (φ)− 1

4

2∑i=1

eλiφF 2i − eλ3φ

2∑i=1

∂χ2i

), (3)

where F1 and F2 are two U(1) gauge fields, the role of first gauge field is to break Lorentz-invariance and introduceLifshitz-like geometry while second gauge field introduced the charge in the gravitational set-up. V (φ) is dilatonpotential and χi are the axionic fields. In this work, we take the potential of form, V (φ) = −2Λeλ0φ for furthercalculation.

The Einstein equation from the above action is given by,

Rµν =1

2∂µφ∂νφ+ Λeλ0φgµν +

1

2eλ3φ(∂µχi∂νχi) +

2∑i=1

1

2eλiφ(FiµρF

ρi ν −

1

4F 2i gµν). (4)

The matter fields equations of motion are obtained as,

�φ =1

2λ3

2∑i=1

(∂χi) +1

4

2∑i=1

λieλiφF 2

i + 2λ0Λeλ0φ, (5)

0 = ∇µ(eλ1φFµν1

), 0 = ∇µ

(eλ2φFµν2

), (6)

0 = ∇µ(eλ3φ∇µχi

). (7)

The metric ansatz for the above action (3) is given by Lifshitz-like, hyperscaling violating black-brane solution as in[27–29, 36],

ds2 = rθ(−r2zf(r)dt2 +

dr2

r2f(r)+ r2(dx2 + dy2)

). (8)

The external magnetic field is introduced in the set-up in the given form,

A2 = A2(r)dt+Bxdy. (9)

The axion fields are linear in spatial direction given by, χ1 = αx and χ2 = αy where α is considered as the strengthof the momentum dissipation.Using the gravity solution, the parameters of given model are related by, [29]

γ =√

(θ + 2)(θ + 2z − 2), λ0 =−θγ, λ1 =

−(4 + θ)

γ, (10)

λ2 =(θ + 2z − 2)

γ, λ3 =

−γθ + 2

, q1 =√

2(z − 1)(θ + z + 2), (11)

3

with Λ = −12 (θ + z + 1)(θ + z + 2) and φ = γ log r.

From the temporal component of the gauge field equation (6) we obtain,

J ti = qi = r−z+3eλiφ(Ai)′t, (12)

considering qi as the charges of two gauge fields. Here the role of q1 is the used to introduce Lifshitz-scaling whereasq2 is interpreted as the black hole charge.The black hole factor with an external magnetic field (B), mass (m) and charge (q2) along with axionic strength(α)is given by,

f(r) = 1− m

rθ+z+2+

q22 +B2

2(θ + 2)(θ + z)r2(θ+z+1)+

α2

(θ + 2)(z − 2)rθ+2z. (13)

The Hawking temperature of black hole is obtained from the expression given below,

T =rz+1h f ′(rh)

4π, . (14)

Thus,

T =z + 2 + θ

4πrzh −

q228π(2 + θ)

1

rz+2+2θh

− α2

4π(2 + θ)

1

rz+θh

. (15)

The constraint from the gravity solution is that every point in space-time follows the null energy condition (NEC) i.e.,TµνV

µV ν ≥ 0, where V µ is the light like vector. Thus, the allowed values of ‘z’ and ‘θ’ consistent with the gravitysolution are [29],

(2 + θ)(2z − 2 + θ) ≥ 0, (16)

(z − 1)(2 + z + θ) ≥ 0. (17)

Later, we shall see that the consistency of coupled equations demand that θ = z− 1 and both the conditions reqirez ≥ 1. Further at z = 2, the solution of black hole factor (13) is not valid as the last term changes sign. So we shallconsider the range 1 ≤ z < 2, which corresponds to 0 ≤ θ < 1.

To study the response of system, we introduce the following perturbations,

δA1i(t, r) =

∫ ∞−∞

dω

2πa1i(r)e

−iωt, (18)

δA2i(t, r) =

∫ ∞−∞

dω

2πa2i(r)e

−iωt, (19)

δχi(t, r) =

∫ ∞−∞

dω

2πbi(r)e

−iωt, (20)

δgti(t, r) =

∫ ∞−∞

dω

2πrθ+2hti(r)e

−iωt. (21)

The coupled linearized equations of motion for the fields are obtained as,

0 = (rz−3−θfa′1i)′ +

ω2a1ir5+z+θf

+ q1h′ti,

0 = (r3z−1+θfa′2i)′ +

ω2a2ir3−z−θf

+ q2h′ti + εij

iωBhtjfr3−z−θ

,

0 = (r5−zfb′i)′ +

ω2bifr3z−3

− iωαhtifr3z−3

.

(22)

where εij is the Levi-Civita tensor. The constraint equation of the set-up is given by,

0 = ωr5−z+θh′ti + ωq1a1i + ωq2a2i + iαr5−zfb′i − q2Bhti −Bfa′zir3z−1+θ. (23)

whereas metric perturbation equation is obtained in the given form,

0 = (r5−z+θh′ti)′ − q1rz−1−θa′1i − q2rz−1−θa′2i +

(α2r−θ−2z+2 +B2r2z−4)htif

+iαωr−θ−2z+2bi

f+ εij

iωBr2z−4a2if

. (24)

4

III. HOLOGRAPHIC APPROACH

To study the transport properties of (2+1) dimensional boundary system, one has to solve coupled equations ofmotion (22), (23) and (24) using the standard procedure as mentioned in [39, 52]. However in this work we follow theapproach introduced by [57] and developed through various studies[43, 46, 49]. Here, DC conductivities are obtainedfrom first order RG flow equations in the near horizon limit. To study the conductivity of a system we simply applyOhm’s Law as given below.

J = σE. (25)

Applying holographic techniques to obtain transport coefficients we use the Onsager relation (J = τX) in the matrixform as shown, (

J1i J1jJ2i J2j

)= τ

(X1i X1j

X2i X2j

), (26)

where ‘Xi’ are the linear independent sources and ‘Ji’ are the responses of system. τ matrix are the coefficientsevaluated in the near horizon limit. The detailed discussion and application of the formalism is presented in [49, 53, 57].Also, we could use the following notation ∥∥∥∥J1J2

∥∥∥∥ = τ

∥∥∥∥X1

X2

∥∥∥∥ (27)

And τ = JX−1 can be expressed as,

τ =

∥∥∥∥J1J1∥∥∥∥∥∥∥∥X1

X2

∥∥∥∥−1 (28)

The linearized equations of motion (22) and (24) can be put in the matrix form as,

τ =

∥∥∥∥∥∥∥∥−rz−3−θfa1i′−r3z−1+θfa′2i−r5−zfb′i−r5−z+θh′ti

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥iωa1iiωa2iiωbiiωhti

∥∥∥∥∥∥∥−1

(29)

Now, taking the radial derivative and substituting the equations of motion we get,

τ ′ =

∥∥∥∥∥∥∥∥∥∥

ω2a1ir5+z+θf

+ q1h′ti

ω2a2ir3−z−θf

+ q2h′ti + εij

ωBhtjfr3−z−θ

ω2bifr3z−3 − iωαhti

fr3z−3

q1rz−1−θa′1i + q2r

z−1−θa′2i +ωBr2z−4εija2j

f − iαωr−θ−2z+2bif − (α2r−θ−2z+2+B2r2z−4)hti

f

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥iωa1iiωa2iiωbiiωhti

∥∥∥∥∥∥∥−1

−τ

−iω

rz−3−θf0 0 0

0 −iωr3z−1+θf

0 0

0 0 −iωr5−zf 0

0 0 0 −iωr5−z+θ

τ

Further simplifying we get,

τ ′ =

−iω

rz+5+θf0 0 0

0 −iωr3−z−θf

0 iBr3−z−θf

0 0 −iωr3z−3f − α

r3z−3f

0 −iBr2z−4

f−αr−θ−2z+2

f − (B2r2z−4+α2r−θ−2z+2)iωf

+

−q1

r5−z+θ0 0 0

0 0 0 −q2r5−z+θ

0 0 0 0−q1r−2f

−q2r2z+2θf

0 0

τ − τ

−iω

rz−3−θf0 0 0

0 −iωr3z−1+θf

0 0

0 0 −iωr5−zf 0

0 0 0 −iωr5−z+θ

τ (30)

5

Multiplying the above metric by black hole factor ‘f(r)’ and taking its near horizon limit, where f(rh) = 0 we obtainτh.Considering the constraint equation (23) in the matrix form as,0 0 0 0

0 iB α −iω0 0 0 00 0 0 0

τ =

0 0 0 0

−q1 −q2 0 −q2Bω

0 0 0 00 0 0 0

(31)

we obtain,

iBτ11 + ατ21 − iωτ31 = −q1, (32)

iBτ12 + ατ22 − iωτ32 = −q2, (33)

iBτ13 + ατ23 − iωτ33 = 0, (34)

iBτ14 + ατ24 − iωτ34 = −q2Bω. (35)

To maintain the consistency of the equations, we fixed θ = z − 1 and the τh matrix is given as,

τh =

r−3−zh 0 0 0

0 r−3+3zh 0

−Br−3+3zh

ω

0 0 r−2z+4h

αr−2z+4h

iωq1iω

iBr−3+2zh +q2iω

r4−2zh

iω −B2r3z−3h +α2r4−2z

h −iq2Bω2

(36)

By substituting τh in equation (29), we find the flow equation in the near horizon limit,

(−r−2h fa1i)′ = r−3−zh iωa1i,

(−r4z−2h fa2i)′ = r−3+3z

h iωa2i + εijiBr−3+3zh htj ,

(−r5−zh fbi)′ = r−2z+4

h iωbi + αr−2z+4h hti,

(−r4hhti)′ = (iBr−3+3zh + q2)a2i + r4−2zh αbi + q1a1i −

(iB2r3z−3h + α2r4−2zh + q2B)htjω

.

(37)

Near horizon limit of equation (24) is given by,

(B2r2z−4h + α2r−3z+3h )hti = q1a

′1i + q2a

′2i + εijωBa2jr

2z−4h − iαωr−3z+3

h bi. (38)

Simplifying the above expression while using the flow equations (37) we obtain the expression for the metric pertur-bation as,

hti|r=rh = −iω(q1r

−z−1h )a1i + εij(q2r

−z−1h + iBr2z−4h )a2j

(B2r2z−4h + α2r−3z+3h )− iq2Br−z−1h

−iαωr−3z+3

h bi

(B2r2z−4h + α2r−3z+3h )− iq2Br−z−1h

. (39)

A. DC conductivity

Using gauge field equation (22), we get the expression for conserved currents for first gauge field as,

J1i = −rz−3−θfa′1i − q1hti. (40)

On substituting the equation (37) the expression takes the form,

J1i = r−3−zh iωa1i − q1hti. (41)

Now from equation (39) neglecting the second term (i.e. axion perturbation part) we obtain the DC conductivityusing,

σij =∂Ji∂Ej

, where Ej = iωaj . (42)

6

Thus we obtain,

σ11xx = σ11

yy = r−3−zh +q21(B2rz−5h + α2r−4z+2

h

)(B2r2z−4h + α2r−3z+3

h )2 +B2q22r−2z−2h

, (43)

Also, we have some mixed terms for DC conductivity, where charges of both the fields effect the conductivity asshown,

σ12xx = σ12

yy =q1q2α

2r−4z+2h

(B2r2z−4h + α2r−3z+3h )2 +B2q22r

−2z−2h

, (44)

σ11xy = −σ11

yx =q21q2Br

−2h

(B2r2z−4h + α2r−3z+3h )2 +B2q22r

−2z−2h

, (45)

σ12xy = −σ12

yx = q1BB2r4z−8h + α2r−z−1h + q22r

−2−2zh

(B2r2z−4h + α2r−3z+3h )2 +B2q22r

−2z−2h

. (46)

Since these expression are quite complex to analyse we numerically study the dependence of conductivities on magneticfield and momentum dissipation term for two different values of the dynamical exponent, z = 1 and z = 4/3 in Fig.1to Fig.4.

FIG. 1: Variation of σ11xx with B and α for z = 1(left) and z = 4/3(right)

FIG. 2: Variation of σ11xy with B and α for z = 1(left) and z = 4/3(right)

Keeping z = 1 reduces the geometry to RN-AdS and we observe trivial DC conductivity flow. However, for z 6= 1non-trivial dependence of DC conductivity on magnetic field and momentum relaxation strength is seen. We alsoobserve a discontinuity while considering z = 1 (θ = 0) case, as σ11

xx = r−4h . On the other hand for RN-AdS black holethe conductivity is given is a constant term given by, σ11

xx = 1.Similarly, for the second gauge field we can evaluate the conductivity accordingly. The conserved current for the

boundary theory is obtained using equation (22),

J2i = −r3z−1+θfa′2i − q2hti, (47)

7

FIG. 3: Variation of σ12xx with B and α for z = 1(left) and z = 4/3(right)

FIG. 4: Variation of σ12xy with B and α for z = 1(left) and z = 4/3(right)

On substituting equation (37) we obtain the modified expression for J2i as,

J2i = r−3+3zh iωa2i + εijiBr

−3+3zh htj − q2hti. (48)

Then the DC conductivity is evaluated concerning the second gauge field.

σ22xx = σ22

yy =α2r2z−4h [B2 + r−5z+7

h (q22r−z−1h + α2)]

B2q22r−2z−2h + (B2r2z−4h + α2r−3z+3

h )2, (49)

σ22xy = −σ22

yx =q2Br

4z−8h

[B2 + r−5z+7

h

(q22r−z−1h + 2α2

)]B2q22r

−2z−2h +

(B2r2z−4h + α2r3−3zh

)2 , (50)

The above expressions are complicated to interpret. So we study the dependence of these conductivities on magneticfield and α using plots as shown in Fig.5 to Fig.6.

Also from the plots of different DC conductivities (for both the fields) we observe at fixed momentum relaxationstrength conductivity σijxx shows a monotonic dependence on the external magnetic field whereas in σijxy this behavioris not seen.Let us consider the limit B → 0.

σ22xx = r3z−3h + r2z−4h

q22α2, σ22

xy = 0 (51)

From the above expression it is noted that electric conductivity obeys inverse Matthiessen’s rule given by,

σDC = σQ + σD (52)

where σQ is the charge conjugation symmetric part and σD is the momentum dissipation part.The temperature dependence of conductivity is governed by equation (15) and T ∼ rzh. Here we observe the

following scaling in the DC conductivity,

8

FIG. 5: Variation of σ22xx with B and α for z = 1(left) and z = 4/3(right)

FIG. 6: Variation of σ22xy with B and α for z = 1(left) and z = 4/3(right)

i For z = 1, σ22xx ∼ 1 +

q22α2T 2

ii For z = 4/3, σ22xx ∼ T 3/4 +

q22Tα2

ii For z → 2, σ22xx ∼ T 3/2 +

q22α2

In our holographic model, we are able to capture the low temperature behavior of the DC conductivity obeyingFermi-Liquid law ( σDC ∼ 1

T 2 ) for z = 1 along with a constant term. This behavior changes to unconventional

metallic behavior (σDC ∼ 1T ) as we increase the Lifshitz scaling to z = 4/3 and becomes constant in the limiting case

z → 2. Thus, there is non-trivial dependence of conductivity on temperature for hyperscaling range (1 < z < 2).

B. Halls Angle

We can obtain the expression for Hall angle using equations (49) and (50)[39, 53, 54]. Thus,

tan θH =σ22xy

σ22xx

(53)

θH =Bq2r

2z−4h

[B2 + r7−5zh

(2α2 + q22r

−z−1h

)]α2[B2 + r7−5zh

(α2 + q22r

−z−1h

)] (54)

From the above expression it is observed that, θH ∝Bq2r

2z−4h

α2 as the terms in the bracket is consider as a geometricquantity [52].

Comparing the result with that of DC conductivity Hall angle consists of only dissipation part (σD), unlike DCconductivity which is the combination of two different terms (shown in equation (51)). This is responsible for the

9

FIG. 7: Variation of θH with B and α for z = 1(left) and z = 4/3(right)

presence of different scaling in strange metals[58]. We plot the dependence of Hall angle on the magnetic field appliedand the strength of momentum relaxation in Fig.7 for fixed q2. Let us consider the temperature dependence of Hallangle,

i For z = 1, θH ∼ 1/T 2

ii For z = 4/3, θH ∼ 1/T

ii For z → 2, θH ∼ 1/T 0

For z = 1, we observe the temperature dependence is same as measured in cuprates [59]. However the behaviorchanges to θH ∼ 1/T with the non-trivial scaling z 6= 4/3. Further θH reduces to a constant for z → 2. We show thetemperature and magnetic field dependence on the Hall angle from plots given in Fig.8 and Fig.9.

FIG. 8: θ vs T at α= 0.1(blue),0.5(red),1(green) for z = 1(left) and z = 4/3(right)

-100 -50 0 50 100

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

B

ΘH

FIG. 9: θ vs B at T = 0.5(blue),1(red),1.5(green) for z = 1(left) and z = 4/3(right)

10

C. Thermoelectric conductivity

Next, we evaluate thermoelectric conductivity using heat current expression.

Qi = −4πTgxxhti, and αij =∂QiT∂Ej

(55)

We get the following expression for thermoelectric conductivity depending on external magnetic field and dynamicalexponent for our model as shown in Fig. 10 and Fig. 11.

α22xx = α22

yy =4πr−3z+3

h q2α2

(B2r2z−4h + α2r−3z+3h )2 +B2q22r

−2z−2h

, (56)

α22xy = −α22

yx =4πBrz+1

h (B2r4z−8h + α2r−z−1h + q22r−2−2zh )

(B2r2z−4h + α2r−3z+3h )2 +B2q22r

−2z−2h

. (57)

In the limiting case B → 0, we obtain,

α22xx =

4πq2r3z−3h

α2, α22

xy = 0 (58)

Thus, the thermoelectric conductivity also shows non-trivial dependence on hyperscaling and unconventional temper-ature dependence.

FIG. 10: Variation of α22xx with B and α for z = 1(left) and z = 4/3(right)

FIG. 11: Variation of α22xy with B and α for z = 1(left) and z = 4/3(right)

11

D. Seebeck Coefficient

The generation of transverse electric field in the system is given by the thermoelectric power (Seebeck coefficient).Using the results of the conductivity we obtain,

S =α22xx

σ22xx

=4πr−5z+7

h q2

[B2 + r−5z+7h (q22r

−z−1h + α2)]

· (59)

FIG. 12: Variation of S with B and α for z = 1(left) and z = 4/3(right)

FIG. 13: S vs T at α=0.5(blue),1(red),1.5(green) for z = 1(left) and z = 4/3(right)

FIG. 14: S vs B at T=0.5(blue),1(red),1.5(green) for z = 1(left) and z = 4/3(right)

The variation of Seebeck coefficient with applied magnetic field and momentum relaxation strength is shown inFig. 12. The result from experiments suggest that at high temperature Seebeck coefficient remains constant. Thedependence of Seebeck coefficient on model parameters is shown in Fig.13 and Fig.14. We observe at differenttemperature the behavior of Seebeck coefficients does not change appreciably for non-trivial scaling. The temperaturescaling of the coefficient is still unclear from experimental results[60].

12

E. Thermal conductivity

Using the results for thermoelectric and DC conductivity, we can also obtain the thermal conductivity [49]. Thethermal conductivity for non-zero magnetic field can be obtained using the relation given below,(

< Ji >< Qi >

)=

(σij αijTαiiT κijT

)(Ej

−(∇jT )/T

)(60)

Considering the thermal current Qx = 0 and Ey = 0 we obtain the expression for the thermal conductivity using,

κ22xx =T (α22

xx)2

σ22xx − σ22

xx(0), and κ22xy =

Tα22xxα

22xy

σ22xx

(61)

where σ22xx(0) is the electric conductivity for Qx = 0 (vanishing heat currents). Thus, we obtain,

κ22xx = κ22yy =16π2r5+4z

h T (B2r5zh + r7hα2)

B4r10zh + r14h α4 +B2r6+4z

h (q22 + 2r1+zh α2)(62)

κ22xy = −κ22yx =16π2TBq2r

8+6zh

B4r10zh + r14h α2 +B2r6+4z

h (q22 + 2r1+zh α2)(63)

Variation of thermal conductivity with different model parameters is shown in Fig.15 and Fig.16 which also shows

FIG. 15: Variation of κ22xx with B and α for z = 1(left) and z = 4/3(right)

non-trivial dependence on hyperscaling.Let us discuss the limiting case B → 0,

κ22xx =16π2Tr4z−2h

α2, κ22xy = 0, (64)

F. Lorenz ratio

To complete the discussion, we check the expression for Lorenz ratio. First we obtain the Hall Lorentz ratio using[53],

L =κxyTσxy

=16π2r2z+8

h

B2r6zh + q22r6h + 2α2rz+7

h

, (65)

The expression for the Lorenz ratio is,

L =κxxTσxx

=16π2r2z+1

h

(B2r5zh + α2r7h

)α2(B2r6zh + q22r

6h + α2rz+7

h

) . (66)

13

FIG. 16: Variation of κ22xy with B and α for z = 1(left) and z = 4/3(right)

FIG. 17: Variation of L with B and α for z = 1(left) and z = 4/3(right)

The explicit dependence of Lorenz ratio on the dynamical scaling and momentum relaxation strength is shown inFig.17.

For B → 0,

L =16π2r2z+2

h

q22 + α2rz+1h

. (67)

We obtain the Lorenz ratio ratio at zero temperature for vanishing magnetic field keeping z = 1 as,

L =κxxTσxx

|T,B→0 =4

3π2

(1 +

α2√α4 + 12q22

)(68)

The given expression indicates, at B = 0 the WF law is valid and we obtain Fermi-liquid type ground state for z = 1.The temperature dependence of Lorenz ratio is shown in Fig.18 for different momentum relaxation strength. Ac-

cording to Wiedemann-Franz (WF) law the Lorenz ratio is constant for normal metals. Since our results show explicitdependence on temperature and also on Lifshitz scaling, the WF Law is violated in this model. The temperaturedependence can not be extracted in a simple manner because of the interplay of different parameters in the system.We also plot the variation of the Lorenz ratio with the magnetic field in Fig.19.

IV. CONCLUSIONS AND SUMMARY

In this paper, we have investigated the DC transport of holographic systems with Lifshitz-like geometry andhyperscaling violation. The geometry is dual to non-relativistic (z 6= 1) condensed matter systems under the appliedexternal magnetic field. We considered the near horizon limit of linearized equations of motion and calculated DCconductivity, thermoelectric and thermal conductivity analytically.

We introduced the perturbations in both the gauge fields and obtained the expressions for the transport coefficients.The behavior of transport coefficients is depicted by numerical plots for different values, z = 1 and z = 4/3, of the

14

FIG. 18: L vs T at α=0.1(blue),0.5(red),1(green) for z = 1(left) and z = 4/3(right)

FIG. 19: L vs B at T =0.5(blue),1(red),1.5(green) for z = 1(left) and z = 4/3(right)

dynamical exponent. While we have non-trivial scaling for z = 4/3, the geometry reduces to RN-AdS black holefor z = 1. Dependence of DC transport on magnetic field and momentum relaxation strength is also studied whileconsidering different Lifshitz scaling. The pattern of different type of conductivities ( σijxx, α

ijxx and κijxx) are quite

similar showing monotonic dependence on the magnetic field but this behavior is absent in σijxy etc .We also obtained the Hall angle, Seebeck coefficient and Lorenz ratio for the system and plotted them as a function

of temperature and magnetic field. Hall angle shows 1/T 2 in temperature behavior as given in equation (54) forz = 1 but it changes to 1/T for z = 4/3. The Wiedemann-Franz law is violated for our holographic model depictingunconventional metallic behavior. However, at zero temperature and B → 0 the Fermi-liquid behavior is obtainedfor z = 1. Seebeck coefficient and Lorenz ratio also showed non-trivial dependence on the hyperscaling parameter.This unconventional dependence of transport coefficient on temperature can be useful to study the strange metalphenomenon [52, 56].

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