NUPHB14509S0550-3213(18)30338-910.1016/j.nuclphysb.2018.11.022The Author(s)High Energy Physics – TheoryHyperscaling violating geometry with magnetic field and DC conductivitySubirMukhopadhyay⁎subirkm@gmail.comChandrimaPaulplchandrima@gmail.comDepartment of Physics, Sikkim University, 6th Mile, Gangtok 737102, IndiaDepartment of PhysicsSikkim University6th MileGangtok737102India⁎Corresponding author.Editor: Stephan StiebergerAbstractWe consider black hole with magnetic field in hyperscaling violating Lifshitz theories arised in a four dimensional Einstein–Maxwell-dilaton system along with axion fields. Considering the linearised equation of relevant fluctuations in metric and gauge fields, we analytically compute thermoelectric conductivity of the dual theory using Dirichlet boundary condition and find agreement with conductivities obtained in near horizon analysis. We also study temperature dependence of the conductivities.1IntroductionHolographic techniques have been proved to be quite successful in analysing strongly coupled systems arised in condensed matter [1–8]. In the original proposal [9–11] it was for asymptotically anti de-Sitter spacetime and thus are amenable to theories characterised by relativistic invariance at the boundary. Soon it transpires it can be generalised to other asymptotic spacetimes as well [12–19]. In particular, this has been extended to systems having anisotropic scaling symmetry along temporal and spatial direction. For such systems, asymptotically Lifshitz spacetimes turns out to be the pertinent set up on the gravity side. An essential motivation for these is to understand the novel behaviour of strongly correlated phases of matter, which cannot be explained using conventional theories, as it does not show quasiparticle description. Application of holographic methods for such phases are expected to provide new insights and deeper understanding about dynamics of these systems.In this vein, a number of works have considered non-relativistic geometries which are asymptotically Lifshitz theories characterised by hyperscaling violation [20–24]. A four dimensional Einstein–Maxwell–Axion–Dilaton theory gives rise to such geometries characterised by two parameters z and θ, corresponding to Lifshitz scaling and the hyperscaling violation respectively. The Axion is chosen linear in space coordinates to introduce inhomogeneity in order to model the feature of underlying lattice structure [25–27]. It involves two U(1) gauge fields, one of which is required to introduce Lifshitz like behaviour, other playing the role of electromagnetic field.Electrically charged black hole background in this theory has been considered and electrical DC conductivity was computed [20] using near horizon analysis [28]. In [21], a magnetic field has been introduced in addition and thermoelectric conductivity was studied using near horizon analysis, once again. However, near horizon analysis [28], though very useful, does not provide the conserved current in the boundary theory. In addition, it is not flexible to incorporate different boundary conditions of the fields in the bulk. Instead, it chooses one boundary condition out of multiple possibilities.In view of these, a different approach has been proposed in [22]. It considered linearised fluctuations around the electrically charged black hole and from analysis of asymptotic behaviour of the solutions they determine counterterms, obtain the physical observables in the dual theory and compute the thermoelectric conductivities. Unlike near horizon analysis, this approach is amenable to incoroporate different boundary conditions on the fields, such as Dirichlet and Neumann or a combination of them.In the present work, we have extended the approach of [22] in presence of magnetic field. We consider a black hole background with a magnetic field and from the analysis of linear fluctuations of necessary fields we have computed the full thermal conductivity matrix. This provides the dependence on magnetic field B and in particular enable one to compute Hall angles. This analysis can accommodate different boundary conditions which may lead to different behaviour of thermal conductivities. In the present case, we have used Dirichlet boundary condition on spatial components of one of the gauge fields and find agreement of conductivities derived in approach of near horizon analysis [21]. We have discussed temperature dependence of thermoelectric conductivities and Hall angle in several scaling regimes.This paper is structured as follows. In the next section we introduce the asymptotically Lifshitz hyperscaling violating solution. In section 3 we introduce the fluctuations in metric and gauge fields, consider their linearised equations of motion and obtain solution in low frequency limit. In section 4 we compute the thermoelectric coefficients and discuss their temperature dependence. We conclude in section 5. Some of the materials related to the necessary canonical transformation of the fields has been discussed in the appendix.2Hyperscaling violating Lifshitz black holeIn the present section we will discuss the asymptotically Lifshitz hyperscaling violating solution, which we will use as the background. The electrically charged solution has been discussed in [20,22] and the electrically charged solution with magnetic field has been mentioned in [21,23]. They appear as a classical solution of an Einstein–Maxwell-dilaton-axion system. We will consider two gauge fields coupled through a symmetric invertible matrix ΣIJ, I,J=1,2 which is a function of the dilaton ϕ, having positive eigenvalues. In addition, there are two axion fields, χa, with a running over 1,2 required to violate the momentum conservation, which is necessary for computation of direct conductivity. The Axion term in the action has a dilaton dependent prefactor Z(ϕ).The four dimensional action is given by(2.1)S=∫d4x−g[R−α(∂ϕ)2−ΣIJFμνIFJμν−Z(ϕ)(∂χa)2−V(ϕ)]+12κ2∫∂Md3x−γ2K,where κ2 in the second term (Gibbons–Hawking boundary term) is given by 8πG. We consider two axion fields and two guage fields with I=1,2. V(ϕ) is the potential, which is functions of dilaton fields.From the action (2.1) we get the following equation of motion. The Einstein, Maxwell, axion and dilaton equations are(2.2)Rμν=α∂μϕ∂νϕ+12V(ϕ)gμν+Z(ϕ)∂μχa∂νχa+2ΣIJ(ϕ)(FμλIFνJλ−14gμνFρσIFJρσ),∇μ(ΣIJ(ϕ)FμνJ)=0,∇μ(Z(ϕ)∂μχa)=0and2α□ϕ−V′(ϕ)=ΣIJ′(ϕ)FρσIFJρσ,respectively.In order to obtain asymptotically Lifshitz hyperscaling violating solution we choose the following ansätz for the metric, axion and the gauge fields.(2.3)dsB2=γμνdxμdxν=dr2+e2A(−f(r)dt2+dxadxa),χBa=pxa,ϕB=ϕB(r),AI=aI=atI(r)dt+BI4ϵabxadxb, where γab denotes background metric tensor. We have chosen a linear axion to break the translation invariance to incorporate momentum relaxation. The first gauge field is required to generate a Lifshitz like behaviour of the metric, while the second one gives rise to the electrical charge and magnetic field of the solution. For the sake of generality, we have kept the constant magnetic field FabI=12BIϵab associated with both the gauge fields.Substituting the ansatz (2.3) in the second equation of (2.2) implies the electric charges qI=−f−1/2eAΣIJ∂ratJ is constant. The first and the last equation (2.2), on substitution of the ansatz (2.3) reduces to the following equations:(2.4)f″2f+3A′f′2f−f′24f2=p2Z(ϕ)e−2A+2e−4A(ΣIJ(ϕ)qIqJ+14ΣIJ(ϕ)BIBJ),A″+A′(3A′+f′2f)+p2Z(ϕ)e−2A+12V+e−4A(ΣIJ(ϕ)qIqJ+14ΣIJ(ϕ)BIBJ),(6A′2+4A′f′2f)=α(∂rϕ)2−2p2Z(ϕ)e−2A−V−2e−4A(ΣIJ(ϕ)qIqJ+14ΣIJ(ϕ)BIBJ),2α[∂r2ϕ+(3A′+f′2f)∂rϕ]−V′(ϕ)=2e−4A(ΣIJ′(ϕ)qIqJ+14ΣIJ′(ϕ)BIBJ).Given a form of Z(ϕ) and ΣIJ(ϕ) one can solve these equations to find out the metric, the Maxwell field, the dilaton and the potential.Like the electrically charged black hole, these equations do admit an exact black hole solution [21,23], which depends on two parameters z and θ. We will consider the range z≥1 and θ≤z+2, which follows from null energy condition, as explained in [22]. We present the solution in radial coordinate v, which is particularly suited for asymptotic behaviour. The metric in terms of this radial coordinate v is given by(2.5)ds2=v−θ[−v2zF(v)dt2+dv2v2F(v)2+v2(dx2+dy2)], where in our ansatz (2.3) we set e2A=v2−θ and the blackening factor F(v) is given by(2.6)F(v)=1+p2(2−θ)(z−2)v2z−θ−mv2+z−θ+8q22(2−θ)(z−θ)v2(z+1−θ)+B2v2z−616(4+θ−3z)(2−z). In terms of v coordinate, the role of the blackening factor is played by F(v). This v coordinate is related to r through(2.7)dr=−sgn(θ)v−θ/2F−1/2(v)dvv.Other fields and functions are given as follows: ΣIJ(ϕ) and Z(ϕ) are(2.8)Σ11(ϕ)=14e[(θ−4)/μ]ϕ,Σ22(ϕ)=14e[(2z−2−θ)/μ]ϕ,Σ12=0,Z(ϕ)=12e[μ/(θ−2)]ϕ,where α=1/2 and μ is given by 2μ2α=(2−θ)(2z−2−θ). The dilaton, the axion and the gauge fields are given by(2.9)ϕ=μlogv,χa=pxa,at1=4sgn(θ)q12+z−θ(v2+z−θ−vh2+z−θ),at2=4sgn(θ)q2θ−z(vθ−z−vhθ−z). The charge q1 and the potential V(ϕ) are(2.10)q12=(2+z−θ)(z−1)/8,V(ϕ)=−(2+z−θ)(1+z−θ)eθϕ/μ−2z−2−θ4(z−2)B2e(θ+2z−6)(ϕ/μ).Unless otherwise mentioned we will keep our analysis general without committing to specific solution. The reason is as follows. For electrically charged case, BI=0 it can be shown that general solution with asymptotic behaviour exists. We expect a similar general solution with specific asymptotic behaviour in the case of this black hole, as well. Therefore the present set up may be used to deal with general solutions. Though, while studying the coefficients of conductivities we will use the specific exact solution only.3FluctuationWe will be interested in the thermoelectric coefficients, which are related to the correlation function of operators. In order to compute those we consider linear fluctuations in the metric and the gauge fields around its background solution.(3.1)γij=γBij+hij,AiI=ABiI+aiI,ϕ=ϕB+φ,χa=χBa+τa, where i,j takes values on t, x and y. Defining Sij=γjkhik, one can set Stt=Sxx=Syy=Sxy=0 and φ=atI=0 consistently, leaving nonzero fluctuations to be Sta, Sat, aaI and τa. Sat is related to Sta and so we will not consider the former. In what follows, we will assume these fields depend on t and r only. With such dependence the linearised equations satisfied by these fluctuations for the background given in the ansatz (2.3) turn out to be as follows:(3.2)[∂r2+(3∂rA−∂rf2f)∂r−e−2A(2p2Z+e−2AΣIJBIBJ)]Sta=−2e−2A[pZ(∂tτa)+2ΣIJ(∂ratI)(∂raaJ)+e−2AΣIJ(∂tabI)ϵabBJ],∂r∂tSta+2e−2AΣIJ(∂ratI)BJϵabStb=−2pfZ∂rτa−4e−2AΣIJ∂ratI∂taaJ−2fe−2AΣIJBJϵab∂rabI∂r{ΣIJeAf−1/2[(∂ratJ)Sta+f∂raaJ]}=f−1/2e−AΣIJ(∂t2aaJ+12ϵab∂tStbBJ,∂r2τa+(3∂rA+∂rf2f+∂rZZ)∂rτa−e−2Af∂r2τa=−f−1e−2Ap∂tSta, where we have not included equations for Sat, which follows from the above set of equations. Considering the time dependence of the various functions is given by eiωt, the above set of equations reduce to the following(3.3)[∂r2+(3∂rA−∂rf2f)∂r−e−2A(2p2Z+e−2AΣIJBIBJ)]Sta=−2e−2A[−iωpZτa+2ΣIJ(∂ratI)(∂raaJ)+iωe−2AΣIJabIϵabBJ],iω∂rSta+2e−2AΣIJ(∂ratI)BJϵabStb=−2pfZ∂rτa−4iωe−2AΣIJ∂ratIaaJ−2fe−2AΣIJBJϵab∂rabI,∂r{ΣIJeAf−1/2[(∂ratJ)Sta+f∂raaJ]}=f−1/2e−AΣIJ(−ω2aaJ+iω2ϵabStbBJ),∂r[e3Af1/2Z∂rτa]=−iωpZeAf−1/2(Sta−iωpτa). Following [22] we introduce new field(3.4)Θa=Sta−iωpτa. The boundary operator associated with Θa plays the role of energy operator in the boundary theory. Introducing Ω=ω2−2p2fZ we write down the equations in terms of this new field Θa. Some of the terms, however, we have written in terms of Sta, which can be expressed in terms of Θa and τa.(3.5)∂r[2p2fZΩ−1(−f−1/2e3A∂rΘa+4qIaaI)−2iωΩ−1BIϵab(qIStb−f1/2eAΣIJ∂rabJ)]−eAf−1/2(2p2Z+e−2AΣIJBIBJ)Θa=iωpe−Af−1/2ΣIJBIBJτa−2iωe−Af−1/2ϵabΣIJBJabI,−f−1/2eA∂r(−f1/2eAΣIJ∂raaJ−2p2fZΩ−1qIΘa)−2p2ω2Ω2f−1/2eA∂r(fZ)qIΘa+ω2f−1(ΣIJ−4Ω−1e−2AfqIqJ)aaJ+2iωΩ−1e−2AϵabqIBJ(qJStb−eAf1/2ΣJK∂rabK)−iω2f−1ΣIJϵabStbBJ=0,∂rSta+4e−2AΣIJ(∂ratI)aaJ=2iωe−2AΣIJBJϵab(∂ratJStb+f∂rabJ)+2ipfZω∂rτa,∂r[e3Af1/2Z∂rτa]=−iωpZeAf−1/2Θa.In order to obtain near horizon limit, we will use another radial coordinate u, which is related to r through du=−f(r)1/2e−A(r)dr. In terms of u the metric becomes(3.6)ds2=e2A(u)(−f(u)dt2+du2f(u)+dxadxa). The derivative in u is related to that in r through(3.7)∂r=−fe−A∂u,∂u=−f−1/2eA∂r u is related to v through the relation du=sgn(θ)vz−3dv where z and θ are parameters determining behaviour of the metric. The horizon of the black hole solution is given by u=uh, where f(uh)=0 and at the near horizon limit f(r)≡4πTρ+O(ρ2), where ρ=uh−u. A, Z and ΣIJ approaches constant values at the near horizon limit. The near horizon limit of the four equations can be arranged in the following manner.(3.8)2p2Zω2[f∂u(f∂u(e2AΘa))]+2p2Ze2AΘa−2iωϵabΣIJBI[f∂u(f∂uabJ)+ω2abJ]+8p2Zω2qIf∂u(faaI)−2iωqIBIϵab∂uStb+ΣIJBIBJSta=0,ΣIJ[f∂u(f∂uaaJ)+ω2aaJ]−2p2ZqIω2f2∂uΘa−4e−2AqIqJfaaJ+2iωe−2AqIBJϵabΣJKf2∂uabK+2iωe−2AϵabqIqJBJfStb−iω2ΣIJϵabStbBJ=0,∂uSta−2iωe−2AϵabqIBIStb+4e−2AqIaaI−2iωe−2AϵabΣIJBJf∂uabJ−2ipωf∂uτa=0,f∂u(fZ∂ue2Aτa)=−iωpZe2AΘa. Considering the terms contributing in leading order of ρ we obtain(3.9)2p2Zω2[f∂u(f∂u(e2AΘa))]+2p2Ze2AΘa−2iωϵabΣIJBI[f∂u(f∂uabJ)+ω2abJ]+ΣIJBIBJSta=0,ΣIJ[f∂u(f∂uaaJ)+ω2aaJ]−iω2ΣIJϵabStbBJ=0. Introducing(3.10)ηaI=aaI+12pBIϵabτb, and choosing the in-falling behaviour, we obtain the following near horizon behaviour(3.11)e2AΘa∼ρ−iω4πT,ηaI∼ρ−iω4πT. We will use the above near horizon behaviour to determine the relations among the constants that appear in the solutions of the various fields.In order to study direct conductivity, we require the solution of the fields Θa, aaI and τa. However, the differential equations are quite involved and since we will be interested in the direct conductivity which depends on the behaviour of the fields at low frequency limit we will expand the fields in powers of frequency and from there we will determine the low frequency behaviour of the fields. So we consider the following expansions(3.12)Θa=Θa(0)+ωΘa(1)+ω2Θa(2)+...,aaI=aaI(0)+ωaaI(1)+ω2aa(2)+...,τa=τa(0)+ωτa(1)+ω2τa(2)+.... We will substitute these expansions in the equations and will determine the fields at different orders of frequency in an iterative manner.First we will consider the equations at the order of zero frequency. Substituting the expansions of (3.12) in (3.5) we obtain from the second equation in (3.5)(3.13)∂r(f1/2eAΣIJ∂raaJ(0)−qIΘa(0))=0, which suggests it is convenient to define a new function(3.14)CIa=f1/2eAΣIJ∂raaJ−qIΘa. Then (3.13) implies CIa(0) is a constant. From the first equation in (3.5)) we get(3.15)∂r[e3Af3/2∂r(f−1Θa(0))+4atICIa(0)]=0, where we have used the equation of background fields (2.4). From the third equation of (3.5) one obtains for axion(3.16)∂uτa(0)=ϵabCIb(0)BIe2AZf−1.From (3.15), (3.14), (3.16) we write the solutions in terms of integrals(3.17)Θa(0)=fΘ1a+fΘ2a∫due2Af2−4fCIa(0)∫atIdue2Af2,aaI(0)=aa0I(0)−CIa(0)∫ΣIJfdu−qJΘ1a∫ΣIJdu−qJΘ2a∫ΣIJ∫due2Af2−4qJCKa(0)∫duΣIJ∫atKdue2Af2,τa(0)=τ0a(0)+ϵabCIb(0)BI∫due2AfZ, where Θ1a, Θ2a, aa0I(0) and τ0a(0) are constants of integration.At the near horizon limit, A, Z and ΣIJ are approaching constant value A(h), Z(h) and ΣIJ(h). Behaviour of f(u) near u→uh is f∼4πTρ and atI∼O(ρ), which leads to(3.18)Θa(0)=(4πTρ)Θ1+Θ2a4πTe2A(h)−4CIa(0)∂uatI4πTe2A(h)ρlogρ,aaI(0)=aa0I(0)+(qJΘ2a4πTe2A(h)+CJa(0))ΣIJ(h)4πTlogρ+qJΘ1aΣIJ(h)ρ,τa(0)=τ0a(0)−ϵabCIb(0)BI4πTe2A(h)Z(h)logρ. The equations at the zeroth order of frequency are very much similar to that obtained in absence of magnetic field [22] as in the equations BI appears at the first order of ω.Next we will consider the equations at first order of frequency. As we have already mentioned, we will use a recursive procedure to determine the solutions at different orders of ω, by using solutions obtained in the lower orders. Substituting the (3.12)in the second equation in (3.5)) we get(3.19)f−1/2eA∂r(f1/2eAΣIJ∂raaJ(1)−qIΘa(1))−2i2p2fZe−2AϵabqIBJ(qJΘb(0)−eAf1/2ΣJK∂rabK(0))−i2f−1ΣIJϵabΘb(0)BJ=0, which leads to(3.20)∂uCIa(1)=−if−1ϵabBJ(−qICJb(0)p2Ze2A+ΣIJΘb(0)2). By integrating (3.20) we can write CIa(1) in terms of the zeroth order terms. Similarly, Θa(1) and τa(1) satisfy(3.21)∂u[e2Af2∂u(f−1Θa(1))]−4CIa(1)∂uatI+2iϵabΣIJBJabI(0)+ip2ϵabBIfCIa(0)∂u(1fZ)−ipΣIJBIBJτa(0)=0,∂u[e2AfZ∂uτa(1)]=ipZe2Af−1Θa(0), while aaI(1) can be obtained from(3.22)∂uaaI(1)=−f−1ΣIJCJa(1)−qIf−1Θa(1). Like CIa(1), all these equations can be integrated to obtain expressions at first order in terms of the zeroth order fields.The near horizon behaviour of the fields at first order can be obtained by integrating the above equations after substituting the near horizon behaviour of f, A, Z and Σ and using the expressions obtained for the zeroth order fields. For CIa(1) we obtain,(3.23)CIa(1)=CI0a(1)+iϵabBJe2A(h)4πT[(−qICJb(0)p2Z(h)+Θ2bΣIJ(h)8πT)logρ+12Θ1aΣIJ(h)e2A(h)4πTρ+2ΣIJCKa(0)∂uatK4πT(ρlogρ−ρ)]+..., where CI0a(1) is an integration constant.Using this expression a similar near horizon expression can be obtained for Θa(1) from (3.21) as follows(3.24)Θa(1)=Θ3ae2A(h)4πT+ip2ϵabBICIa(0)4πTZ(h)logρ+Θ4a4πTρ+..., where Θ3a and Θ4a are new integration constants. The fluctuation in gauge field at first order, aaI(1) at the near horizon limit follows from (3.22) and is given by(3.25)aaI(1)=aa0I(1)+ΣIJ(h)4πT[qJΘ3ae2A(h)4πT+CJ0a(1)]logρ+...., where we have introduced the constant term of integration as aa0I(1). Finally the τa at first order turns out to be(3.26)τa(1)=τ0a(1)−e−2A(h)4πTpZ(h)[ϵabBKCK0b(1)−i2(−e2A4πTΘ1a+4qIaa0I(0))+ipqIBIϵabτ0b(0)]logρ+....The constants of integration introduced at different orders can be determined by comparing with the near horizon behaviour with the full fledged expressions of the various fluctuations, obtained in (3.11). For that we need to consider the equations to the second order in ω.At the second order of ω we obtain the following equation for CIa(2)(3.27)∂uCIa(2)=qIe−2A2p2fZ[(e2A∂uΘa(0)+4qJaaJ(0))+2iϵab(CJb(1)−ipqJτb(0))BJ]−ΣIJf[aaJ(0)−i2(Θb(1)+ipτb(0))BJ] On the other hand for Θa(2) we get(3.28)∂u[e2Af2∂u(f−1Θa(2))]=4CIa(2)∂uatI−f∂u[12p2fZ[(e2A∂uΘa(0)+4qJaaJ(0))+2iϵab(CJb(1)−ipqJτb(0))BJ]+ipΣIJBIBJτa(1)−2iϵababI(1). aaI(2) can be obtained as usual, from(3.29)CIa(2)=f1/2eAΣIJ∂raaJ(2)−qIΘa(2).In order to compare to the boundary condition at horizon we need to find the leading order behaviour of the fields near the horizon. Substituting the expressions we have obtained for fields up to zeroth order and first order on right hand side of (3.27) one can easily find that the leading order terms of CIa(2) near horizon are of the order of logρ and (logρ)2. In particular, it does not have any 1/ρ in its expression near the horizon. It follows from equation for Θa(2) that the leading order expression of Θa(2) is given by(3.30)f−1Θa(2)=Θ6+Θ5∫due2Af2+Slogρ+..., where Θ5a and Θ6a are constants of integration and S is given by(3.31)S=12p2Z(h)[(−4πTe2AΘ1a+4qIaa0I)+2iϵab(CI0b(1)−ipqIτ0b(0))BI]+Θ2a(4πT)2.Collecting expressions of Θa at different orders of frequency together, we can write near horizon expression of Θa valid up to O(ω2) as(3.32)Θa=Θ2ae2A(h)4πT+4πTΘ1aρ+14πTe2A(h)[iωϵabCI0bBIp2Z(h)+2(−πTe2A(h)Θ1a+qIaa0I0)p2Z(h)ω2+ω2p3Z(h)ϵabτ0b(0)qIBI+ω2Θ2a(4πT)2]logρ+... In this equation, following [22] we have absorbed all the pertinent integration constants in Θ1a, Θ2a and CI0a, without any loss of generality, by redefining Θ2a, Θ1a and CI0a. Similarly the expression for the fluctuation in gauge field at near horizon limit is(3.33)aaI=aa0I+ΣIJ(h)4πT(CJ0a+qJΘ2ae2A(h)4πT)logρ+... where we have absorbed all the constants of integration in aa0I. Fluctuation in the axion τa at near horizon turns out to be(3.34)τap=τ0ap+14πTe2A(h)p2Z(h)[−ϵabBICI0b+2iω(−πTe2A(h)Θ1a+qIa0aI)+iω(qIBI)ϵabτ0b(0)p]logρ+... where constants are absorbed in τ0a.Comparing with the near horizon behaviour of Θa and ηaI=aaI+12BIϵabτbp as given in (3.11), we obtain(3.35)(ΣIJ(h)+BIBJ2p2Ze2A(h))CJ0a+ΣIJ(h)qJe2A(h)4πTΘ2a=iω{−(aa0I+12BIϵabτ0bp)−BIe2A(h)p2Z[ϵab(−πTe2A(h)Θ1b+qJa0bJ)−12(qJBJ)τ0ap]},Θ2a=−4πTp2Z(h)[ϵabBICI0b−2iω(−πTe2A(h)Θ1a+qIaa0I)−iω(qJBJ)ϵabτ0bp].From the two equations above (3.35) we can express CI0a and Θ2a in terms of other constants aa0I, Θ1a and τ0a in the following manner,(3.36)CI0a=iω(MIJ)ab{−[(ΣJK(h)+2qJqKp2Z(h)e2A(h))δbc+ΣJN(h)BNqKp2Z(h)e2A(h)ϵbc]ac0k+2πTp2Z(h)[qJδab+12ΣJN(h)BNϵbc]Θ1c−12[(ΣJK(h)+4qJqKp2Z(h)e2A(h))BKϵbc−(qMBM)ΣJK(h)BKp2Z(h)e2A(h)δbc]τ0cp},Θ2a=−4πTp2Z(h)ϵabBICI0b+iω4πTp2Z(h)[2(−πTe2A(h)Θ1a+qIa0aI)+(qIBI)ϵabτ0bp], up to leading order in ω, where we have introduced the matrix (MIJ)ab satisfying(3.37)[(δIJ+ΣIN(h)BNBJ2p2Z(h)e2A(h))δab−qIBJp2Z(h)e2A(h)ϵab](MJK)bc=δIKδac. In absence of magnetic field it reduces to δIJδab.In order to identify the operators in the boundary theory, we require the asymptotic solution of Θa, aaI and τa. It is sufficient to determine the asymptotic solution of the fields up to lowest order in frequency. From the linearised equations of motion of the fluctuations it is clear that magnetic field contributes at a higher order in frequency. Therefore, up to lowest order of frequency, expressions remain the same as those obtained in absence of magnetic field [22]. To this end we introduce(3.38)Ψ(v)=sgn(θ)∫vθ−3z−1dvF(v)2,Y1(v)=4sgn(θ)q12+z−θ(−vh2+z−θΨ(v)+sgn(θ)∫dvv−2z+1F−2),Y2(v)=4sgn(θ)q2θ−z(−vhθ−zΨ(v)+sgn(θ)∫dvv2θ−4z−1F−2). In terms of these functions we can write the asymptotic expansions of the solutions of the fields at small frequency(3.39)Θa(0)=v2(z−1)F(v)(Θ1a+Θ2aΨ(v)+4CIaYI(v)),aaI=aa0I−Θ1aatI−sgn(θ)Θ2qJ∫dvΣIJvz−3Ψ(v)−sgn(θ)∫dvΣIJv−z−1(F−1δKJ+4qJv2(z−1)YK(v))CKa.From (3.38) and (3.39) one can establish a relation between the parameters describing the asymptotic behaviour of the solutions and operators in the boundary theory. This relation has been discussed elaborately in [22] and we have included their discussion in the appendix. As explained there, a basis of symplectic variables that parametrize the asymptotic solutions can be identified from asymptotic behaviour of the generalised coordinates and momenta. To this end one considers the radial Hamiltonian formulation and express asymptotic solutions of the linear fluctuations of the fields Θa(0), aaI(0) and τa(0) and their conjugates in terms of the modes Θ1a, Θ2a, aa0I, CIa and τ0a. Then one makes a suitable canonical transformation, that can be realised by adding appropriate counterterms, leading to holographic renormalisation of the action. From the asymptotic behaviour of these transformed canonical variables the operators can be identified in terms of the modes parametrizing the asymptotic solution.Choice of the boundary condition turns out to play critical role in this identification. As explained in [22] adding an additional finite term in the renormalised on-shell action, the Dirichlet boundary condition can be imposed on the gauge field. In the case of electrically charged black hole as the background, it has been found that the expressions of the conductivities obtained using the near horizon method agrees with the Dirichlet boundary condition. In the present case, where we have magnetic field in addition, we are considering the Dirichlet boundary condition so as to compare the results already obtained using near horizon method. With the present set up generalising it to Neumann or mixed boundary condition is quite straightforward.In case of Dirichlet boundary condition, we are interested in energy operator Ea and current operator JIa as shown in [22]. Their expressions in terms of different modes are given by (A.10) and (A.11)(3.40)Ea=−12κ2(Θ2a+4μICI0a),JIa=−2κ2(CI0a−iωqIpτ0a),Xa=−2iωpκ2qIαaI, where αaI is obtained from the asymptotic behaviour for the renormalised variables as given in (A.12). From these expressions we can obtain the various correlation function, that leads to computation of the coefficients of thermoelectric conductivity.4Thermoelectric DC conductivitiesIn this section we obtain thermoelectric conductivities for the present model. In the last section we have derived Θ2a and CI0a in terms of other constants in (3.35). We substitute these expressions in the energy operator Ea given in (3.40), we get(4.1)Ea=−iω2κ2[{8πTp2ZqKδad−(−4πTp2ZϵabBI+4μIδab)(MIJ)bc[(ΣJK+2qJqKp2Ze2A)δcd+ΣJMBMqkp2Ze2Aϵcd}αd0K+(8πTp2Z(qKμK−πTe2A)δad+(−4πTp2ZϵabBI+4μIδab)(MIJ)bc{2πTp2Z(qJδcd+12ΣJMBMϵcd)−[(ΣJK+2qJqKp2Ze2A)μKδcd+ΣJMBMqKμKp2Ze2Aϵcd]})Θ1d+{−12(−4πTp2ZϵabBI+4μIδab)(MIJ)bc[(ΣJK+4qJqKp2Ze2A)BKϵcd−(qKBK)ΣJMBMp2Ze2Aδcd]+4πTp2Z(qKBK)ϵad}τ0dp, where we have used the asymptotic value of fluctuation in gauge field, αaI given in (A.12). In this section, to simplify the notation, unless otherwise mentioned A, ΣIJ and Z represents their respective values at the near horizon limit.Similarly, the current operator JIa turns out to be(4.2)JIa=2iωκ2{(MIJ)ab[(ΣJK+2qJqKp2Ze2A)δbc+ΣJMBMqKp2Ze2Aϵbc]αc0K−(MIJ)ab{2πTp2Z(qJδbc+12ΣJMBMϵbc)−[(ΣJK+2qJqKp2Ze2A)μKδbc+ΣJMBMqKμKp2Ze2Aϵbc]}Θ1c+{12(MIJ)ab[(ΣJK+2qJqKp2Ze2A)BKϵbc−(qNBN)ΣJKBKp2Ze2Aδbc]+qIδac}τ0cp},Xa=−2iωpκ2qIαIa, where the matrix (MIJ)ab is given by (3.37).From the above expressions one can obtain the following two-point functions(4.3)〈JIa(−ω)JJb(ω)〉=2iωκ2(MJK)bc[(ΣKI+2qKqIp2Ze2A)δca+ΣKMBMqIpZe2Aϵca],〈Ea(−ω)JIb(ω)〉=−2iωκ2(MIJ)bc{2πTp2Z(qJδca+12ΣJKBKϵca)−[(ΣJK+2qJqKp2Ze2A)δbc+ΣJMBMqKp2Ze2Aϵbc]μK},〈JIa(−ω)Eb(ω)〉=2iωκ2{−2πTp2ZqIδba+(−πTp2ZBJϵbc+μJδbc)(MJK)cd[(ΣKI+2qKqIp2Ze2A)δda+ΣKMBMqIp2Ze2Aϵda]},〈Ea(−ω)Eb(ω)〉=−2iωκ2[2πTp2Z(qKμK−πTe2A)δba+(−πTp2Ze2AϵbcBI+μIδbc)(MIJ)cd[(2πTp2ZqJ−(ΣJK+2qJqKp2Ze2A)μK]δda+ΣJMBM(πTp2Z−qKμKp2Ze2A)ϵda],〈Xa(−ω)JIb(ω)〉=2iωκ2[12(MIJ)bc[(ΣJK+4qJqKp2Ze2A)BKϵca−(qMBM)ΣJKBKp2Ze2Aδca]+qIδba,〈JIa(−ω)Xb(ω)〉=−2iωpκ2qIδab,with rest of the two point functions vanishing.Next following [22] we introduce the heat current(4.4)QDa=Ea−μIJIa. The two point function for heat current and electric currents are given by(4.5)〈QDa(−ω)QDb(ω)〉=2iωκ22(πTp2Z)2{p2Ze2Aδab+BMϵac(MMJ)cd[qJδda+12ΣJNBNϵda]},〈QDa(−ω)JIb(ω)〉=−2iωκ22πTp2Z[(MIJ)bc[qJδca+12ΣJKBKϵca],〈JIa(−ω)QDb(ω)〉=−2iωκ2{2πTp2ZqIδba+πTp2ZϵbcBJ(MJK)cd[(ΣKI+2qKqIp2Ze2A)δda+ΣKMBMqJp2Ze2Aϵda]}〈JIa(−ω)JJb(ω)〉=2iωκ2(MJK)bc[(ΣKI+2qKqIp2Ze2A)δca+ΣKMBMqIpZe2Aϵca],We obtain the thermoelectric conductivities from the above two point functions as follows.(4.6)σDDC=(TK¯abTα¯IabTαIabσIJab)=(〈QDa(−ω)QDb(ω)〉〈QDa(−ω)JIb(ω)〉〈JIa(−ω)QDb(ω)〉〈JIa(−ω)QDb(ω)〉). In order to obtain the following expressions for the components of the conductivity matrix in a compact form we have introduced the following parameters(4.7)rI=12ΣIJBJ,bI=BIp2Ze2A. In terms of these parameters the matrix (MIJ)ab is given from (3.37)(4.8)(MIJ)ab=δIJδab−[(1+r.b)rI+(q.b)qI]δab−[(1+r.b)qI−(q.b)rI]ϵab(1+r.b)2+(q.b)2bJ, where we have used (r.b)=rIbI, (q.b)=qIbI and △=(1+r.b)2+(q.b)2. With these expressions, components of conductivity matrix becomes(4.9)K¯ab=πsTκ2p2Z[(1+r.b)δba+(q.b)ϵba]△,α¯Iab=αIab=−4sTK¯bc(qIδca+rIϵca)],σIJab=2κ2ΣJIδba+16s2TK¯bc(qJδcd+rJϵcd)(qIδda+rIϵda), where we have used 4πe2A=s. All the components of the conductivity matrix reduce to the expressions of the same given in [22] for setting BI=0. It may be observed that both the U(1) gauge fields are on the same footing and that we have got α¯Iab=αIab.We have obtained the thermoelectric conductivities for the general case and in this form the symmetry between and electric and magnetic fields is also becomes apparent. We can apply this general result to the case of the black hole solution discussed in section 2. Substituting values of the various quantities in the above expressions we obtain the following forms for conductivities. For the solution we get △=(p2+B24v4z−6−θ)2+(2q2Bv2z−4)2 and using that we get,(4.10)K¯ab=8π2Tκ2p2vh2(z−θ)(p2+B24vh4z−6−θ)δba+2q2Bvh2z−4ϵba△,α1ab=−8πκ2vh2z−θ−2(p2+B24vh4z−6−θ)q1δba+2q1q2Bvh2z−4ϵba△,α2ab=−8πκ2vh2z−θ−2p2q2δba+[(p2+B24vh4z−6−θ)B8vh2z−2−θ+2q22Bvh2z−4]ϵba△,σ11ab=12κ2vhθ−4δba+8κ2q12vh2z−4(p2+B24vh4z−6−θ)δba+2q2Bvh2z−4ϵba△,σ12ab=8κ2q1q2p2δba+[2q22Bvh2z−4+B8vh2z−2−θ(p2+B24vh4z−6−θ)]ϵba△,σ22ab=p22κ2vh6z−8−2θB24+vh6−4z+θ(p2+16q22vhθ−2)△δba+q2Bκ2vh8z−12−2θB24+(2p2+16q22vhθ−2)vh−4z+6+θ△ϵba,Hall angle can be obtained from the above conductivities by taking the ratio of coefficients of ϵab and δab in the expression of σ. We get(4.11)ΘH=2q2Bp2vh2z−4[B24+vh−4z+6+θ(2p2+16q22vhθ−2)B24+vh−4z+6+θ(p2+16q22vhθ−2)]. As explained in [29] since the factor in the square bracket lies between 1 and 2 Hall coefficients can be approximated as(4.12)ΘH=2q2Bp2vh2z−4p2, these expressions, after setting θ=1−z, agree with the results obtained in [21] using the near horizon method.With the explicit expressions of various components of thermoelectric matrix we can study temperature dependence. For the analytic black hole solution the temperature is given by T=−sgn(θ)4πvhz+1F′(vh) which for the case of dyonic solution reduces to(4.13)T=−sgn(θ)4π[(z+2−θ)vhz−8q222−θvh2θ−z−2−p22−θvhθ−z−B24(2−z)vh3z−6. The expression of temperature is quite involved and it is difficult to obtain an analytic expression of the conductivities in terms of the temperature. Nevertheless, choosing appropriate limits of the quantities we can identify regimes, where one can discuss scaling behaviour of the coefficients with the temperature.We begin with θ<0, where the first term is positive while rest of the terms are negative in the expression of temperature. To identify a regime of large temperature, following [22] we consider q22vh2θ−z−2<<vhz, p2vhθ−z<<vhz and B2vh3(z−2)<<vhz. In this regime one can identify T≡8q124π(z−1)vhz. The behaviour of thermoelectric conductivity matrix will depend on the relative strengths of the different terms in the temperature. We have considered the following three regions of parameters. Apart from that one can also obtain the cases, where two terms are comparable, but there it is difficult to identify the scaling behaviour of the conductivities.We begin with the range of parameters where momentum dissipation is strong compared to charge and magnetic field, which is given by, B2vh3(z−2),q22vh2θ−z−2<<p2vhθ−z<<vhz. In this limit we obtain(4.14)Kab∼8π2Tκ2p4[T2(z−θ)zδba+2q2Bp2T4z−2θ−4zϵab],σ11ab∼8q12κ2p2[T2z−4zδba+2q2Bp2T4z−8zϵba],σ12ab∼q1κ2p2[8q2T2z−4zδba+BT4z−6−θzϵba],σ22ab∼12κ2[T2z−2−θzδba+4q2Bp2T4z−6−θzϵba],α1ab∼−8πq1κ2p2[T2z−θ−2zδba+2q2Bp2T2z−4zϵba],α2ab∼−8πκ2p2[q2T2z−θ−2zδba+B8T4z−2θ−4zϵba]. The Hall angle is θH∼T2z−4z. Since θ<0 we cannot get linear resistivity for σ22xx in this regime. Choosing z=1 we get θH∼1/T2 and σ22xx∼T−θ showing a positive power of T for conductivity. Instead if we choose, B2vh3(z−2)<<p2vhθ−z<<q22vh2θ−z−2<<vhz,p2>>2q2Bvh2z−4 all the coefficients will remain the same except σ22. It becomes(4.15)σ22ab=8q22κ2p2[T2z−4zδba+2q2Bp2T4z−8zϵba]. In this regime, σ22xx and Hall angle have similar temperature dependence. So for z=1 both scale as ∼T−2. Choosing z=4/3 one gets σ22xx∼T−1 implying linear resistivity. However, Hall angle also becomes θH∼T−1.Another scaling regime, that one may consider corresponds to the range where the charge is strong compared to momentum dissipation and magnetic field. That is given by B2vh3(z−2), p2vhθ−z<<q22vh2θ−z−2<<vhz and leads to the following conductivities:(4.16)Kab∼8π2Tκ2(2q2B)[12q2BT2(4−θ−z)zδba+1p2T2(2−θ)zϵab],forB2vh3(z−2)<<p2vhθ−z,α1ab∼−8πq1κ2[p24q22B2T6−2z−θzδba+12q2BT2−θzϵba]forB2vh3(z−2)<<p2vhθ−z,α1ab∼−8πq1κ2[4B2T2(z−θ)zδba+12q2BT2−θzϵba]forp2vhθ−z<<B2vh3(z−2),α2ab∼−8πκ2[p24q2B2T6−2z−θzδba+12BT2−θzϵba],σ11ab∼8q12p2κ2[T2z−4zδba+2q2Bp2T4z−8zϵba],forp2vhθ−z>>q2Bvhz+θ−4,∼8q12κ2[p2(2q2B)2T4−2zzδba+12q2Bϵba],forp2vhθ−z<<q2Bvhz+θ−4,σ12ab∼8q1q2κ2[p24q22B2T8−4zzδba+12q2BT4−2zzϵba],σ22ab∼12κ2[4p2B2T4−2z)zδba+8q2Bϵba]. In this regime, σ22xx and Hall angle have opposite temperature dependence. Choosing z=1 one gets temperature dependence to be T2 and T−2 respectively. For z=2, however both will be independent of temperature. Similarly one can consider the regime where magnetic field will be stronger compared to the momentum dissipation and charge. In that regime, σ22xx∼T(4−2z)z with Hall angle having opposite temperature dependence, once again.For small temperature, one can identify the following regions of parameters.B2vh3(z−2),q22vh2θ−z−2<<p2vhθ−z≲vhz, B2vh3(z−2),p2vhθ−z<<q22vh2θ−z−2≲vhz andp2vhθ−z,q22vh2θ−z−2<<B2vh3(z−2)≲vhz. However, obtaining an analytical expression for temperature for this region is difficult. The dependence on vh can be obtained from above by replacing T by vhz in (4.14) and (4.16) a respectively in the three regimes.For θ>0 first term is negative and so large temperature may corresponds to the regimes depending on whether p2vhθ−z, q22vh2θ−z−2 or B2vh3(z−2) dominates. In these regimes, temperature can be approximated by T≡p24π(2−θ)vhθ−z, T≡8q224π(2−θ)vh2θ−z−2 or T≡B2216π(2−z)vh3z−6, respectively. The scalings of conductivity matrix for various regimes will be as follows:For the parameter region corresponding to strong momentum dissipation, B2vh3(z−2), q22vh2θ−z−2<<p2vhθ−z we get(4.17)Kab∼8π2Tκ2p4[(Tp2)2(z−θ)θ−zδba+2q2Bp2(Tp2)4z−2θ−4θ−zϵab],α1ab∼−8πq1κ2p2[(Tp2)2z−θ−2θ−zδba+2q2Bp2(Tp2)2z−4θ−zϵba],α2ab∼−8πκ2p2[q2(Tp2)2z−θ−2θ−zδba+B8(Tp2)4z−2θ−4θ−zϵba],σ11ab∼8q12κ2p2[(Tp2)2z−4θ−zδba+2q2Bp2(Tp2)4z−8θ−zϵba],σ12ab∼q1κ2p2[8q2δba+B(Tp2)2z−2−θθ−zϵba],σ22ab∼12κ2[(Tp2)2z−2−θθ−zδba+4q2Bp2(Tp2)4z−6−θθ−zϵba. For z→2 σ22xx∼T−1, but Hall angle becomes independent of temperature.For the regime, where charge is strong compared to other two factors, given by B2vh3(z−2), p2vhθ−z<<q22vh2θ−z−2, conductivities turn out to beKab∼8π2Tκ2p2[p24q22B2(Tq22)8−2z−2θ2θ−z−2δba+12q2B(Tq22)2(2−θ)2θ−z−2ϵab],forB2vh3(z−2)<<p2vhθ−z,∼8π2Tκ2p2[116q22(Tq22)2z+2−3θ2θ−z−2δba+12q2B(Tq22)2(2−θ)2θ−z−2ϵab],forp2vhθ−z<<B2vh3(z−2),α1ab∼−8πq1κ2[p24q22B2(Tq22)6−2z−θ2θ−z−2δba+12q2B(Tq22)2−θ2θ−z−2ϵba]forB2vh3(z−2)<<p2vhθ−z,α1ab∼−8πq1κ2[116q22(Tq22)2(z−θ)2θ−z−2δba+12q2B(Tq22)2−θ2θ−z−2ϵba]forp2vhθ−z<<B2vh3(z−2),α2ab∼−8πκ2[p24q2B2(Tq22)6−2z−θ2θ−z−2δba+12B(Tq22)2−θ2θ−z−2ϵba].σ11ab∼8q12κ2[p24q22B2(Tq22)4−2z2θ−z−2δba+12q2Bϵba],forp2vhθ>>(2q2Bvhz+θ−4)2,∼12κ2[(Tq22)θ−42θ−z−2δba+12q2Bϵba],forp2vhθ<<(2q2Bvhz+θ−4)2,∼8q12κ2[116q22(Tq22)2z−2−θ2θ−z−2δba+12q2Bϵba],forp2vhθ−z<<B2vh3(z−2),(4.18)σ12ab∼8q1κ2[p24q2B2(Tq22)8−4z2θ−z−2δba+12B(Tq22)4−2z2θ−z−2ϵba],σ22ab∼12κ2[4p2B2(Tq22)4−2z2θ−z−2δba+8q2Bϵba]. As observed from above, σ22xx and Hall angle has opposite temperature dependence. For z=1 σ22xx∼T−1, but Hall angle becomes independent of time. Small temperature limit can be chosen in a similar way as in the case of θ<0. The behaviour will be similar to those obtained in the case of θ<0.We have seen the behaviour of the various thermoelectric coefficients depends on competing contributions from different terms. For high temperature limits we have discussed several regimes where the scaling with temperature can be identified. For small temperature, however, the dependence is quite involved and it is difficult to identify the behaviour with specific powers of temperature. In general, a numerical procedure can be used for obtaining temperature dependence.5ConclusionWe have used holographic techniques to analyze thermoelectric properties of systems dual to hyperscaling violating Lifshitz geometry. Considering a dyonically charged black hole as the background we have turned on necessary fluctuations in metrics and gauge fields. Solving the equations of motion of the fluctuations and imposing in-falling boundary condition at the horizon we have obtained the thermoelectric coefficients from the asymptotic behaviour of fluctuations in low frequency limit. Compared to the near horizon method, this method [22] has the advantage that it enables one to identify the boundary operators explicitly and is amenable to accommodate different boundary conditions.We have discussed the temperature dependence of various thermoelectric coefficients. Because of the background solution is too involved, we can analytically discuss only a few specific regimes. In one of the regimes, z=4/3 leads to linear resistivity but Hall angle goes as 1/T, though for z=1 it shows 1/T2 behaviour. Here we have explicitly considered the dyonic background. It may be interesting to obtain the result in the case of electrically charged background, by using mixed boundary condition on the gauge field. A natural extension of the present work is to explore AC conductivity using numerical techniques and study temperature dependence for intermediate frequencies. Another direction is to consider turning on mass for the bulk gauge field [30], which gives rise to additional exponents. The present method may also be applied to explore properties of the other models towards obtaining agreement with experimental observations.Appendix AIn order to determine the thermoelectric DC conductivities in this method we need to identify the operators in the boundary theory with the parameters describing the asymptotic behaviour of the solutions. These have been elaborated in [22] and in this appendix we include a brief review for convenience. First we will consider a new set of coordinates parametrizing “dual frame”, where radial coordinate is r¯, which is related to the Einstein frame radial coordinate r through the relation dr¯=−sgn(θ)eθ2μϕdr. The advantage of this dual coordinate is it allows both positive and negative values of θ and the UV boundary lies at r¯→∞.In order to identify the operators living in the boundary theory and the fields in the bulk theory one considers [19,22] the symplectic set of variables consisting generalised coordinates and its canonically conjugate momenta in the bulk Hamiltonian radial formalism. This enables one to identify the natural basis of symplectic variables that parametrize the space of asymptotic solutions.The metric in the Einstein or the dual frame can be decomposed in the following manner. ds2=dr2+γijdxidxj, where xi=t,xa. In the Hamiltonian formalism the metric and the gauge field can be decomposed as(A.1)ds2=(N2+NiNi)dr2+2Nidrdxi+γijdxidxj,AμIdxμ=ArIdr+AiIdxi, where N and Ni are the lapse and shift function and γij is the induced metric on radial slices at fixed values of r. Similarly Ar and Ai are transverse and longitudinal components of the gauge fields to the radial slices. We also write down the extrinsic curvature, which can be expressed as(A.2)Kij=12N(∂rγij−DiNj−DjNi), where Di is the covariant derivative with respect to the metric γij. We will use barred quantities for dual frame and unbarred one for Einstein frame.The Lagrangian in the dual frame, as obtained in [22] is given by(A.3)Lξ=12κ2∫d3x−γ¯N¯[(1+4ξ2αξ)K¯2−K¯ijK¯ij−αξN¯2(∂rϕ−N¯i∂iϕ−2ξαξN¯K¯)2−2N¯2ΣIJξ(ϕ)(FriI−N¯kFkiI)(FrJi−N¯lFlJi)−1N¯2Zξ(ϕ)(∂rχa−N¯i∂iχa)2+R[γ¯]−αξ∂iϕ∂¯iϕ−ΣIJξFijIFJij−Zξ∂iχa∂¯iχa−Vξ−2□γ¯]e2ξϕ.The canonical momenta in the dual frame can be obtained from the above Lagrangian as(A.4)π¯ij=δLδγ¯˙ij,π¯Ii=δLδA˙iI,π¯ϕ=δLδϕ˙,π¯χa=δLδχ˙a, with conjugate momenta of the non-dynamical fields, N¯, N¯i and Ar being zero.Expressing them in terms of quantities in the Einstein frame one gets(A.5)π¯ij=12κ2−γe2ξϕ(Kγij−Kij),π¯Ii=−2κ2−γΣIJγijFrjI,π¯ϕ=1κ2−γ(2ξK−α∂rϕ),π¯χa=−1κ2−γZ∂rχa.These expressions evaluated around the background in linearised order of perturbations in metric and other fields reduce to the following expressions.(A.6)πta=14κ2e2ξϕBe−3Af−1/2∂r(e4ASta),πIa=−2κ2eAf1/2ΣIJ(∂raaJ+f−1(∂ratJ)Sta),πχa=−1κ2e3Af1/2Z∂rτa.In order to make connection to the asymptotic expressions we will express the above equations in terms of Θa, aaI and τa. We will consider only the expression in zeroth order of ω. Furthermore, we will use the radial coordinate v instead of r. Substituting the background values of the fields and using dr=−sgn(θ)v−θ/2F−1/2(v)dvv we obtain,(A.7)πta=−sgn(θ)4κ2vθ−z−1∂v(v4−2θ(Θa(0)+iωpτa(0))),π1a=sgn(θ)2κ2[vz+θ−3F(v)∂vaa1(0)+4sgn(θ)q1(Θa(0)+iωpτa(0))],π2a=sgn(θ)2κ2[v3z+θ−1F(v)∂vaa2(0)+4sgn(θ)q2(Θa(0)+iωpτa(0))],πχa=iω2pκ2[−sgn(θ)v5−z−θ∂vΘa(0)−4qIaaI(0)]. Substituting the expressions for the fields in small frequency limit we can obtain the expressions of the canonical momenta. As has been explained in [22] the asymptotic expressions provide a map between the two sets. One set is given by the fluctuations, Θa(0), aaI(0), τa(0) along with their conjugate momenta. The other set consists of the modes Θ1a, Θ2a, aa0a, CIa and τa.The set of fluctuations should be identified with the local sources and operators in the boundary theory but with these expressions they will not be independent of radial variable v. In order to identify the local sources and operators one needs to consider holographic renormalisation of the action. Since our case is very similar to [22] we refer their analysis for details. This identification involves a canonical transformation among the fluctuations and their conjugate momenta, which can be realised by adding appropriate counterterms in the regularised action. The canonical transformation, in absence of magnetic field has been described elaborately In [22]. They have considered on shell regularised action for the model with the black hole solution as the background. In addition of counterterms at the boundary the variables πta, A1a and πχa undergo canonical transformations, keeping A2a and its canonical conjugate momentum unchanged.As has been mentioned earlier, since the effect of the magnetic field appears at the linear order in frequency or higher, small frequency expansion of the fluctuations Θa(0), aaI(0), τa(0) remain the same as in the case of zero magnetic field. However, there are differences in the expression of the blackening factor F(v) and so the counterterms will be modified in this case. In presence of magnetic field we are assuming one can make a similar canonical transformation through addition of counterterms and obtain the transformed variables which are appropriate to make identification of the local sources and operators on the boundary. A similar addition of counterterms will give rise to the following asymptotic expression of the transformed variables,(A.8)Πta=−14κ2v−2z(Θ2a+4μICIa)+...,aa1=aa01−μ1Θ1+...,Πχa=−2iωpκ2qIaaI+...,aa2=aa02−μ2Θ1+..., where the chemical potentials are given by(A.9)μ1=−4sgn(θ)q1vh2+z−θ2+z−θ,μ2=−4sgn(θ)q2vhθ−zθ−z. These transformed variables are related to the original symplectic variables through a canonical transformation. Following [22] we identify the asymptotic expressions of these transformed variables with the observables in the dual field theory as follows. One can define different holographically dual theory by imposing different boundary conditions. For Dirichlet boundary condition on A1a, which requires addition of an additional boundary term to the on shell action along with counterterms [22], the observables and the sources for energy flux are given by(A.10)Ea=2limr¯→∞e2zr¯Πta=−12κ2(Θ2a+4μICI0a),Θ1a=limr¯→∞e−2zr¯na, respectively where r¯ is related to r through r∼2|θ|e−θr¯2 and na is the shift function in the decomposition of the metric γ¯ij as γ¯ijdxidxj=−(n2−nana)dt2+2nadtdxa+σabdxadxb, a,b=1,2. Similarly the observable for U(1) currents and pseudoscalars are given by(A.11)JIa=limr¯→∞ΠIa=−2κ2(CI0a−iωqIpτ0a),Xa=limr¯→∞Πχa=−2iωpκ2qIαaI, respectively and αaI is given by(A.12)αaI=a0I−μIΘ1a.References[1]S.A.HartnollClass. Quantum Gravity26200922400210.1088/0264-9381/26/22/224002arXiv:0903.3246 [hep-th][2]C.P.HerzogJ. Phys. A42200934300110.1088/1751-8113/42/34/343001arXiv:0904.1975 [hep-th][3]G.T.HorowitzLect. Notes Phys.828201131310.1007/978-3-642-04864-7-10arXiv:1002.1722 [hep-th][4]G.T.HorowitzM.M.RobertsPhys. Rev. D78200812600810.1103/PhysRevD.78.126008arXiv:0810.1077 [hep-th][5]S.A.HartnollarXiv:1106.4324 [hep-th][6]S.SachdevAnnu. Rev. Condens. Matter Phys.32012910.1146/annurev-conmatphys-020911-125141arXiv:1108.1197 [cond-mat.str-el][7]A.G.GreenContemp. Phys.54120133310.1080/00107514.2013.779477arXiv:1304.5908 [cond-mat.str-el][8]S.A.HartnollA.LucasS.SachdevarXiv:1612.07324 [hep-th][9]J.M.MaldacenaInt. J. Theor. Phys.381999111310.1023/A:1026654312961Adv. Theor. Math. Phys.21998231arXiv:hep-th/9711200[10]S.S.GubserI.R.KlebanovA.M.PolyakovPhys. Lett. B428199810510.1016/S0370-2693(98)00377-3arXiv:hep-th/9802109[11]E.WittenAdv. Theor. Math. Phys.21998253arXiv:hep-th/9802150[12]S.KachruX.LiuM.MulliganPhys. Rev. D78200810600510.1103/PhysRevD.78.106005arXiv:0808.1725 [hep-th][13]M.TaylorarXiv:0812.0530 [hep-th][14]S.F.RossO.SaremiJ. High Energy Phys.0909200900910.1088/1126-6708/2009/09/009arXiv:0907.1846 [hep-th][15]S.F.RossClass. Quantum Gravity28201121501910.1088/0264-9381/28/21/215019arXiv:1107.4451 [hep-th][16]R.B.MannR.McNeesJ. High Energy Phys.1110201112910.1007/JHEP10(2011)129arXiv:1107.5792 [hep-th][17]M.BaggioJ.de BoerK.HolsheimerJ. High Energy Phys.1201201205810.1007/JHEP01(2012)058arXiv:1107.5562 [hep-th][18]T.GriffinP.HoravaC.M.Melby-ThompsonJ. High Energy Phys.1205201201010.1007/JHEP05(2012)010arXiv:1112.5660 [hep-th][19]W.ChemissanyI.PapadimitriouJ. High Energy Phys.1501201505210.1007/JHEP01(2015)052arXiv:1408.0795 [hep-th][20]S.CremoniniH.S.LiuH.LuC.N.PopeJ. High Energy Phys.1704201700910.1007/JHEP04(2017)009arXiv:1608.04394 [hep-th][21]N.BhatnagarS.SiwachInt. J. Mod. Phys. A33042018185002810.1142/S0217751X18500288arXiv:1707.04013 [hep-th][22]S.CremoniniM.CveticI.PapadimitriouJ. High Energy Phys.1804201809910.1007/JHEP04(2018)099arXiv:1801.04284 [hep-th][23]X.H.GeY.TianS.Y.WuS.F.WuS.F.WuJ. High Energy Phys.1611201612810.1007/JHEP11(2016)128arXiv:1606.07905 [hep-th][24]Z.N.ChenX.H.GeS.Y.WuG.H.YangH.S.ZhangNucl. Phys. B924201738710.1016/j.nuclphysb.2017.09.016arXiv:1709.08428 [hep-th][25]A.DonosJ.P.GauntlettJ. High Energy Phys.1404201404010.1007/JHEP04(2014)040arXiv:1311.3292 [hep-th][26]T.AndradeB.WithersJ. High Energy Phys.1405201410110.1007/JHEP05(2014)101arXiv:1311.5157 [hep-th][27]M.M.CaldarelliA.ChristodoulouI.PapadimitriouK.SkenderisJ. High Energy Phys.1704201700110.1007/JHEP04(2017)001arXiv:1612.07214 [hep-th][28]A.DonosJ.P.GauntlettJ. High Energy Phys.1411201408110.1007/JHEP11(2014)081arXiv:1406.4742 [hep-th][29]M.BlakeA.DonosPhys. Rev. Lett.1142201502160110.1103/PhysRevLett.114.021601arXiv:1406.1659 [hep-th][30]B.GouterauxE.KiritsisJ. High Energy Phys.1304201305310.1007/JHEP04(2013)053arXiv:1212.2625 [hep-th]