# Hyperscaling violating geometry with magnetic field and DC conductivity

Subir Mukhopadhyay and Chandrima Paul
Department of Physics, Sikkim University, 6th Mile, Gangtok 737102
subirkm<EMAIL_ADDRESS>
###### ABSTRACT

We consider dyonic black hole 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.

##  1 Introduction

Holographic techniques has been proved to be quite successful in analysing
strongly coupled systems arised in condensed matter [1, 2, 3, 4, 5, 6, 7, 8].
In the original proposal[9, 10, 11] it was for asymptotically anti de-Sitter
spacetime and thus are amenable to theories at the boundary characterised by
relativistic invariance at the boundary. Soon it transpires it can be
generalised to other asymptotic spacetimes as well [12, 13, 14, 15, 16, 17,
18, 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, 21, 22, 23, 24]. A four dimensional Einstein-Maxwell-Axion-
Dilaton theory gives rise to such geometries characterised by two parameters
$z$ and $\theta$, 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, 26, 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 dyonic black hole background 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 coditions 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 cunductivities 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 the 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.

##  2 Hyperscaling violating Lifshitz Black Hole

In 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
dyonically charged solution has been mentioned in [23, 21]. 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
$\Sigma_{IJ}$, $I,J=1,2$ which is a function of the dilaton $\phi$, having
positive eigenvalues. In addition, there are two axion fields, $\chi^{a}$,
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 $Z(\phi)$.

The four dimensional action is given by

$S=\int
d^{4}x\sqrt{-g}[R-\alpha(\partial\phi)^{2}-\Sigma_{IJ}F_{\mu\nu}^{I}F^{J\mu\nu}-Z(\phi)(\partial\chi^{a})^{2}-V(\phi)]+\frac{1}{2\kappa^{2}}\int\limits_{\mathcal{\partial}M}d^{3}x\sqrt{-\gamma}2K,$
(2.1)

where $\kappa^{2}$ in the second Gibbons-Hawking boundary term is given by
$8\pi G$. We consider two axion fields and two guage fields with $I=1,2$.
$V(\phi)$ is the potential, which is functions of dilaton fields.

From the action (2.1) we get the following equation of motion. The Einstein,
Maxwell and dilaton, axion equations are

$\begin{split}&R_{\mu\nu}=\alpha\partial_{\mu}\phi\partial_{\nu}\phi+\frac{1}{2}V(\phi)g_{\mu\nu}+Z(\phi)\partial_{\mu}\chi^{a}\partial_{\nu}\chi^{a}+2\Sigma_{IJ}(\phi)(F^{I}_{\mu\lambda}F^{J\lambda}_{\nu}-\frac{1}{4}g_{\mu\nu}F^{I}_{\rho\sigma}F^{J\rho\sigma}),\\\
&\nabla^{\mu}(\Sigma_{IJ}(\phi)F_{\mu\nu}^{J})=0,\\\
&\nabla^{\mu}(Z(\phi)\partial_{\mu}\chi^{a})=0\quad\text{and}\quad
2\alpha\Box\phi-V^{\prime}(\phi)=\Sigma_{IJ}^{\prime}(\phi)F^{I}_{\rho\sigma}F^{J\rho\sigma},\end{split}$
(2.2)

respectively.

In order to obtain asymptotically Lifshitz hyperscaling violating solution we
choose the following ans$\ddot{\text{a}}$tz for the metric, axion and the
gauge fields.

$\begin{split}ds_{B}^{2}&=\gamma_{\mu\nu}dx^{\mu}dx^{\nu}=dr^{2}+e^{2A}(-f(r)dt^{2}+dx^{a}dx^{a}),\\\
\chi^{a}_{B}&=px^{a},\quad\quad\phi_{B}=\phi_{B}(r),\quad\quad
A^{I}=a^{I}=a_{t}^{I}(r)dt+\frac{B^{I}}{4}\epsilon_{ab}x^{a}dx^{b},\end{split}$
(2.3)

Where $\gamma_{ab}$ 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 dyonic charge of the solution.
For the sake of generality, we have kept the constant magnetic field
$F^{I}_{ab}=\frac{1}{2}B^{I}\epsilon_{ab}$ associated with both the gauge
fields.

Substituting the ansatz (2.3) in the second equation of (LABEL:einstein)
implies the elctric charges
$q_{I}=-f^{-1/2}e^{A}\Sigma_{IJ}\partial_{r}a_{t}^{J}$ is constant. The first
and the last equation (LABEL:einstein), on substitution of the ansatz (2.3)
reduces to the following equations:

$\begin{split}\frac{f^{\prime\prime}}{2f}+3A^{\prime}\frac{f^{\prime}}{2f}-\frac{f^{\prime
2}}{4f^{2}}=p^{2}Z(\phi)e^{-2A}+2e^{-4A}(\Sigma^{IJ}(\phi)q_{I}q_{J}+\frac{1}{4}\Sigma_{IJ}(\phi)B^{I}B^{J}),\\\
A^{\prime\prime}+A^{\prime}(3A^{\prime}+\frac{f^{\prime}}{2f})+p^{2}Z(\phi)e^{-2A}+\frac{1}{2}V+e^{-4A}(\Sigma^{IJ}(\phi)q_{I}q_{J}+\frac{1}{4}\Sigma_{IJ}(\phi)B^{I}B^{J}),\\\
(6A^{\prime
2}+4A^{\prime}\frac{f^{\prime}}{2f})=\alpha(\partial_{r}\phi)^{2}-2p^{2}Z(\phi)e^{-2A}-V-2e^{-4A}(\Sigma^{IJ}(\phi)q_{I}q_{J}+\frac{1}{4}\Sigma_{IJ}(\phi)B^{I}B^{J}),\\\
2\alpha[\partial_{r}^{2}\phi+(3A^{\prime}+\frac{f^{\prime}}{2f})\partial_{r}\phi]-V^{\prime}(\phi)=2e^{-4A}(\Sigma^{IJ\prime}(\phi)(\phi)q_{I}q_{J}+\frac{1}{4}\Sigma_{IJ}^{\prime}(\phi)B^{I}B^{J}).\end{split}$
(2.4)

Given a form of $Z(\phi)$ and $\Sigma_{IJ}(\phi)$ 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
dyonic black hole solution [21, 23], which depends on two parameters $z$ and
$\theta$. We present the solution in radial coordinate $v$, which is
particularly suited for asymptotic behavior. The metric in terms of this
radial coordinate $v$ is given by

$ds^{2}=v^{-\theta}[-v^{2z}F(v)dt^{2}+\frac{dv^{2}}{v^{2}F(v)^{2}}+v^{2}(dx^{2}+dy^{2})],$
(2.5)

where in our ansatz (2.3) we set $e^{2A}=v^{2-\theta}$ and the blackening
factor $F(v)$ is given by

$F(v)=1+\frac{p^{2}}{(2-\theta)(z-2)v^{2z-\theta}}-\dfrac{m}{v^{2+z-\theta}}+\dfrac{8q_{2}^{2}}{(2-\theta)(z-\theta)v^{2(z+1-\theta)}}+\dfrac{B^{2}v^{2z-6}}{16(4+\theta-3z)(2-z)}.$
(2.6)

In terms of $v$ coordinate, the role of the blackening factor is played by
$F(v)$. This $v$ coordinate is related to $r$ through

$dr=-sgn(\theta)v^{-\theta/2}{\mathcal{F}}^{-1/2}(v)\frac{dv}{v}.$ (2.7)

Other fields and functions are given as follows: $\Sigma_{IJ}(\phi)$ and
$Z(\phi)$ are

$\Sigma_{11}(\phi)=\frac{1}{4}e^{[(\theta-4)/\mu]\phi},\quad\Sigma_{22}(\phi)=\frac{1}{4}e^{[(2z-2-\theta)/\mu]\phi},\quad\Sigma_{12}=0,\quad
Z(\phi)=\frac{1}{2}e^{[\mu/(\theta-2)]\phi},$ (2.8)

where $\alpha=1/2$ and $\mu$ is given by
$2\mu^{2}\alpha=(2-\theta)(2z-2-\theta)$. The dilaton, the axion and the gauge
fields are given by

$\phi=\mu\log v,\quad\chi^{a}=px^{a},\quad
a_{t}^{1}=\frac{4sgn(\theta)q_{1}}{2+z-\theta}(v^{2+z-\theta}-v_{h}^{2+z-\theta}),\quad
a_{t}^{2}=\frac{4sgn(\theta)q_{2}}{\theta-z}(v^{\theta-z}-v_{h}^{\theta-z}).$
(2.9)

The charge $q_{1}$ and the potential $V(\phi)$ are

$q_{1}^{2}=(2+z-\theta)(z-1)/8,\quad
V(\phi)=-(2+z-\theta)(1+z-\theta)e^{\theta\phi/\mu}-\frac{2z-2-\theta}{4(z-2)}B^{2}e^{(\theta+2z-6)(\phi/\mu)}.$
(2.10)

Unless otherwise mentioned we will keep our analysis general without commiting
to specific solution. The reason is as follows. For electrically charged case,
$B^{I}=0$ 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 dyonic 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.

##  3 Fluctuation

We 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.

$\gamma_{ij}=\gamma_{Bij}+h_{ij},\quad
A^{I}_{i}=A^{I}_{Bi}+a^{I}_{i},\quad\phi=\phi_{B}+\varphi,\quad\chi^{a}=\chi_{B}^{a}+\tau^{a},$
(3.1)

where $i,j$ takes values on $t$, $x$ and $y$. Defining
$S_{i}^{j}=\gamma^{jk}h_{ik}$, one can set
$S_{t}^{t}=S_{x}^{x}=S_{y}^{y}=S_{x}^{y}=0$ and $\varphi=a_{t}^{I}=0$
consistently, leaving nonzero fluctuations to be $S_{t}^{a}$, $S_{a}^{t}$,
$a_{a}^{I}$ and $\tau^{a}$. $S_{a}^{t}$ is related to $S_{t}^{a}$ 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:

$\begin{split}&[\partial_{r}^{2}+(3\partial_{r}A-\frac{\partial_{r}f}{2f})\partial_{r}-e^{-2A}(2p^{2}Z+e^{-2A}\Sigma_{IJ}B^{I}B^{J})]S_{t}^{a}=-2e^{-2A}[pZ(\partial_{t}\tau^{a})+2\Sigma_{IJ}(\partial_{r}a_{t}^{I})(\partial_{r}a_{a}^{J})\\\
&\hphantom{[\partial_{r}^{2}+(3\partial_{r}A-\frac{\partial_{r}f}{2f})\partial_{r}-e^{-2A}(2p^{2}Z+e^{-2A}\Sigma_{IJ}B^{I}B^{J})]S_{t}^{a}=}+e^{-2A}\Sigma_{IJ}(\partial_{t}a_{b}^{I})\epsilon_{ab}B^{J}],\\\
&\partial_{r}\partial_{t}S_{t}^{a}+2e^{-2A}\Sigma_{IJ}(\partial_{r}a_{t}^{I})B^{J}\epsilon_{ab}S^{b}_{t}=-2pfZ\partial_{r}\tau^{a}-4e^{-2A}\Sigma_{IJ}\partial_{r}a_{t}^{I}\partial_{t}a_{a}^{J}-2fe^{-2A}\Sigma_{IJ}B^{J}\epsilon_{ab}\partial_{r}a_{b}^{I}\\\
&\partial_{r}\\{\Sigma_{IJ}e^{A}f^{-1/2}[(\partial_{r}a_{t}^{J})S_{t}^{a}+f\partial_{r}a_{a}^{J}]\\}=f^{-1/2}e^{-A}\Sigma_{IJ}(\partial_{t}^{2}a_{a}^{J}+\frac{1}{2}\epsilon_{ab}\partial_{t}S^{b}_{t}B^{J},\\\
&\partial_{r}^{2}\tau^{a}+(3\partial_{r}A+\frac{\partial_{r}f}{2f}+\frac{\partial_{r}Z}{Z})\partial_{r}\tau^{a}-\frac{e^{-2A}}{f}\partial_{r}^{2}\tau^{a}=-f^{-1}e^{-2A}p\partial_{t}S_{t}^{a},\end{split}$
(3.2)

where we have not included equations for $S_{a}^{t}$, which follows from the
above set of equations. Considering the time dependence of the various
functions is given by $e^{i\omega t}$, the above set of equations reduce to
the following

$\begin{split}&[\partial_{r}^{2}+(3\partial_{r}A-\frac{\partial_{r}f}{2f})\partial_{r}-e^{-2A}(2p^{2}Z+e^{-2A}\Sigma_{IJ}B^{I}B^{J})]S_{t}^{a}=-2e^{-2A}[-i\omega
pZ\tau^{a}+2\Sigma_{IJ}(\partial_{r}a_{t}^{I})(\partial_{r}a_{a}^{J})\\\
&\hphantom{[\partial_{r}^{2}+(3\partial_{r}A-\frac{\partial
f}{2f})\partial_{r}-e^{-2A}(2p^{2}Z+e^{-2A}\Sigma_{IJ}B^{I}B^{J})]S_{t}^{a}=}+i\omega
e^{-2A}\Sigma_{IJ}a_{b}^{I}\epsilon_{ab}B^{J}],\\\
&i\omega\partial_{r}S_{t}^{a}+2e^{-2A}\Sigma_{IJ}(\partial_{r}a_{t}^{I})B^{J}\epsilon_{ab}S^{b}_{t}=-2pfZ\partial_{r}\tau^{a}-4i\omega
e^{-2A}\Sigma_{IJ}\partial_{r}a_{t}^{I}a_{a}^{J}-2fe^{-2A}\Sigma_{IJ}B^{J}\epsilon_{ab}\partial_{r}a_{b}^{I},\\\
&\partial_{r}\\{\Sigma_{IJ}e^{A}f^{-1/2}[(\partial_{r}a_{t}^{J})S_{t}^{a}+f\partial_{r}a_{a}^{J}]\\}=f^{-1/2}e^{-A}\Sigma_{IJ}(-\omega^{2}a_{a}^{J}+\frac{i\omega}{2}\epsilon_{ab}S^{b}_{t}B^{J}),\\\
&\partial_{r}[e^{3A}f^{1/2}Z\partial_{r}\tau^{a}]=-i\omega
pZe^{A}f^{-1/2}(S_{t}^{a}-\frac{i\omega}{p}\tau^{a}).\end{split}$ (3.3)

Following [22] we introduce new field

$\Theta^{a}=S_{t}^{a}-\frac{i\omega}{p}\tau^{a}.$ (3.4)

The boundary operator associated with $\Theta^{a}$ plays the role of energy
operator in the boundary theory. Introducing $\Omega=\omega^{2}-2p^{2}fZ$ we
write down the equations in terms of this new field $\Theta^{a}$. Some of the
terms, however, we have written in terms of $S_{t}^{a}$, which can be
expressed in terms of $\Theta^{a}$ and $\tau^{a}$.

$\begin{split}&\partial_{r}[2p^{2}fZ\Omega^{-1}(-f^{-1/2}e^{3A}\partial_{r}\Theta^{a}+4q_{I}a_{a}^{I})-2i\omega\Omega^{-1}B^{I}\epsilon_{ab}(q_{I}S_{t}^{b}-f^{1/2}e^{A}\Sigma_{IJ}\partial_{r}a_{b}^{J})]\\\
&-e^{A}f^{-1/2}(2p^{2}Z+e^{-2A}\Sigma_{IJ}B^{I}B^{J})\Theta^{a}=\frac{i\omega}{p}e^{-A}f^{-1/2}\Sigma_{IJ}B^{I}B^{J}\tau^{a}-2i\omega
e^{-A}f^{-1/2}\epsilon_{ab}\Sigma_{IJ}B^{J}a_{b}^{I},\\\
&f^{-1/2}e^{A}\partial_{r}(f^{1/2}e^{A}\Sigma_{IJ}\partial_{r}a_{a}^{J}-2p^{2}fZ\Omega^{-1}q_{I}\Theta^{a})-\frac{2p^{2}\omega^{2}}{\Omega^{2}}f^{-1/2}e^{A}\partial_{r}(fZ)q_{I}\Theta^{a}+\omega^{2}f^{-1}(\Sigma_{IJ}\\\
&-4\Omega^{-1}e^{-2A}fq_{I}q_{J})a_{a}^{J}+2i\omega\Omega^{-1}e^{-2A}\epsilon_{ab}q_{I}B^{J}(q_{J}S_{t}^{b}-e^{A}f^{1/2}\Sigma_{JK}\partial_{r}a_{b}^{K})-\frac{i\omega}{2}f^{-1}\Sigma_{IJ}\epsilon_{ab}S_{t}^{b}B^{J}=0,\\\
&\partial_{r}S_{t}^{a}+4e^{-2A}\Sigma_{IJ}(\partial_{r}a_{t}^{I})a_{a}^{J}=\frac{2i}{\omega}e^{-2A}\Sigma_{IJ}B^{J}\epsilon_{ab}(\partial_{r}a_{t}^{J}S_{t}^{b}+f\partial_{r}a_{b}^{J})+\frac{2ipfZ}{\omega}\partial_{r}\tau^{a},\\\
&\partial_{r}[e^{3A}f^{1/2}Z\partial_{r}\tau^{a}]=-i\omega
pZe^{A}f^{-1/2}\Theta^{a}.\end{split}$ (3.5)

In order to obtain near horizon limit, we will use another radial coordinate
$u$, which is related to $r$ through $du=-f(r)^{1/2}e^{-A(r)}dr$. In terms of
$u$ the metric becomes

$ds^{2}=e^{2A(u)}(-f(u)dt^{2}+\frac{du^{2}}{f(u)}+dx^{a}dx^{a}).$ (3.6)

The derivative in $u$ is related to that in $r$ through

$\partial_{r}=-\sqrt{f}e^{-A}\partial_{u},\quad\partial_{u}=-f^{-1/2}e^{A}\partial_{r}$
(3.7)

$u$ is related to $v$ through the relation $du=sgn(\theta)v^{z-3}dv$ where $z$
and $\theta$ are parameters determining behavior of the metric. The horizon of
the black hole solution is given by $u=u_{h}$, where $f(u_{h})=0$ and at the
near horizon limit $f(r)\equiv 4\pi T\rho+{\mathcal{O}}(\rho^{2})$, where
$\rho=u_{h}-u$. $A$, $Z$ and $\Sigma_{IJ}$ approaches constant values at the
near horizon limit. The near horizon limit of the four equations can be
arranged in the following manner.

$\begin{split}&\frac{2p^{2}Z}{\omega^{2}}[f\partial_{u}(f\partial_{u}(e^{2A}\Theta^{a}))]+2p^{2}Ze^{2A}\Theta^{a}-\frac{2i}{\omega}\epsilon_{ab}\Sigma_{IJ}B^{I}[f\partial_{u}(f\partial_{u}a_{b}^{J})+\omega^{2}a_{b}^{J}]+\frac{8p^{2}Z}{\omega^{2}}q_{I}f\partial_{u}(fa_{a}^{I})\\\
&-\frac{2i}{\omega}q_{I}B^{I}\epsilon_{ab}\partial_{u}S_{t}^{b}+\Sigma_{IJ}B^{I}B^{J}S_{t}^{a}=0,\\\
&\Sigma_{IJ}[f\partial_{u}(f\partial_{u}a_{a}^{J})+\omega^{2}a_{a}^{J}]-\frac{2p^{2}Zq_{I}}{\omega^{2}}f^{2}\partial_{u}\Theta^{a}-4e^{-2A}q_{I}q_{J}fa_{a}^{J}+\frac{2i}{\omega}e^{-2A}q_{I}B^{J}\epsilon_{ab}\Sigma_{JK}f^{2}\partial_{u}a_{b}^{K}\\\
&+\frac{2i}{\omega}e^{-2A}\epsilon_{ab}q_{I}q_{J}B^{J}fS_{t}^{b}-\frac{i\omega}{2}\Sigma_{IJ}\epsilon_{ab}S_{t}^{b}B^{J}=0,\\\
&\partial_{u}S_{t}^{a}-\frac{2i}{\omega}e^{-2A}\epsilon_{ab}q_{I}B^{I}S_{t}^{b}+4e^{-2A}q_{I}a_{a}^{I}-\frac{2i}{\omega}e^{-2A}\epsilon_{ab}\Sigma_{IJ}B^{J}f\partial_{u}a_{b}^{J}-\frac{2ip}{\omega}f\partial_{u}\tau^{a}=0,\\\
&f\partial_{u}(fZ\partial_{u}e^{2A}\tau^{a})=-i\omega
pZe^{2A}\Theta^{a}.\end{split}$ (3.8)

Considering the terms contributing in leading order of $\rho$ we obtain

$\begin{split}&\frac{2p^{2}Z}{\omega^{2}}[f\partial_{u}(f\partial_{u}(e^{2A}\Theta^{a}))]+2p^{2}Ze^{2A}\Theta^{a}-\frac{2i}{\omega}\epsilon_{ab}\Sigma_{IJ}B^{I}[f\partial_{u}(f\partial_{u}a_{b}^{J})+\omega^{2}a_{b}^{J}]+\Sigma_{IJ}B^{I}B^{J}S_{t}^{a}=0,\\\
&\Sigma_{IJ}[f\partial_{u}(f\partial_{u}a_{a}^{J})+\omega^{2}a_{a}^{J}]-\frac{i\omega}{2}\Sigma_{IJ}\epsilon_{ab}S_{t}^{b}B^{J}=0.\end{split}$
(3.9)

Introducing

$\eta_{a}^{I}=a_{a}^{I}+\frac{1}{2p}e^{2A}B^{I}\epsilon_{ab}\tau^{b},$ (3.10)

and choosing the in-falling behaviour, we obtain the following near horizon
behaviour

$e^{2A}\Theta^{a}\sim\rho^{\frac{-i\omega}{4\pi
T}},\quad\quad\eta_{a}^{I}\sim\rho^{\frac{-i\omega}{4\pi T}}.$ (3.11)

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
$\Theta^{a}$, $a_{a}^{I}$ and $\tau^{a}$. 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

$\Theta^{a}=\Theta^{a(0)}+\omega\Theta^{a(1)}+\omega^{2}\Theta^{a(2)}+...,a_{a}^{I}=a_{a}^{I(0)}+\omega
a_{a}^{I(1)}+\omega^{2}a_{a}^{(2)}+...,\tau^{a}=\tau^{a(0)}+\omega\tau^{a(1)}+\omega^{2}\tau^{a(2)}+....$
(3.12)

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 (LABEL:eom2) we obtain from the
second equation in (LABEL:eom2)

$\partial_{r}(f^{1/2}e^{A}\Sigma_{IJ}\partial_{r}a_{a}^{J(0)}-q_{I}\Theta^{a(0)})=0,$
(3.13)

which suggests it is convenient to define a new function

$C_{I}^{a}=f^{1/2}e^{A}\Sigma_{IJ}\partial_{r}a_{a}^{J}-q_{I}\Theta^{a}.$
(3.14)

Then (3.13) implies $C_{I}^{a(0)}$ is a constant. From the first equation in
(LABEL:eom2)) we get

$\partial_{r}[e^{3A}f^{3/2}\partial_{r}(f^{-1}\Theta^{a(0)})+4a_{t}^{I}C_{I}^{a(0)}]=0,$
(3.15)

where we have used the equation of background fields (2.4). From the third
equation of (LABEL:eom2) one obtains for axion

$\partial_{u}\tau^{a(0)}=\epsilon_{ab}\frac{C_{I}^{b(0)}B^{I}}{e^{2A}Z}f^{-1}.$
(3.16)

From (3.13) and (3.15) we write the solutions in terms of integrals

$\begin{split}\Theta^{a(0)}&=f\Theta_{1}^{a}+f\Theta_{2}^{a}\int\frac{du}{e^{2A}f^{2}}-4fC_{I}^{a(0)}\int\frac{a_{t}^{I}du}{e^{2A}f^{2}},\\\
a_{a}^{I(0)}&=a^{I(0)}_{a0}-C_{I}^{a(0)}\int\frac{\Sigma^{IJ}}{f}du-
q_{J}\Theta_{1}^{a}\int\Sigma^{IJ}du-
q_{J}\Theta_{2}^{a}\int\Sigma^{IJ}\int\frac{du}{e^{2A}f^{2}}\\\
&-4q_{J}C_{K}^{a(0)}\int du\Sigma^{IJ}\int\frac{a_{t}^{K}du}{e^{2A}f^{2}},\\\
\tau^{a(0)}&=\tau^{a(0)}_{0}+\epsilon_{ab}C_{I}^{b(0)}B^{I}\int\frac{du}{e^{2A}fZ},\end{split}$
(3.17)

where $\Theta^{a}_{1}$, $\Theta^{a}_{2}$, $a^{I(0)}_{a0}$ and
$\tau^{a(0)}_{0}$ are constants of integration.

At the near horizon limit, $A$, $Z$ and $\Sigma_{IJ}$ are approaching constant
value $A(h)$, $Z(h)$ and $\Sigma_{IJ}(h)$. Behaviour of $f(u)$ near
$u\rightarrow u_{h}$ is $f\sim 4\pi T\rho$ and
$a^{I}_{t}\sim{\mathcal{O}}(\rho)$, which leads to

$\begin{split}\Theta^{a(0)}=(4\pi T\rho)\Theta_{1}+\frac{\Theta_{2}^{a}}{4\pi
Te^{2A(h)}}-\frac{4C_{I}^{a(0)}\partial_{u}a_{t}^{I}}{4\pi
Te^{2A(h)}}\rho\log\rho,\\\
a_{a}^{I(0)}=a_{a0}^{I(0)}+(\frac{q_{J}\Theta_{2}^{a}}{4\pi
Te^{2A(h)}}+C_{J}^{a(0)})\frac{\Sigma^{IJ}(h)}{4\pi
T}\log\rho+q_{J}\Theta_{1}^{a}\Sigma^{IJ}(h)\rho,\\\
\tau^{a(0)}=\tau^{a(0)}_{0}-\epsilon_{ab}\frac{C_{I}^{b(0)}B^{I}}{4\pi
Te^{2A(h)}Z(h)}\log\rho.\end{split}$ (3.18)

The equations at the zeroeth order of frequency are very much similar to that
obtained in absence of magnetic field[22] as in the equations $B^{I}$ appears
at the first order of $\omega$.

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 $\omega$, by using solutions obtained in the
lower orders. Substituting the (3.12)in the second equation in (LABEL:eom2))
we get

$\begin{split}&f^{-1/2}e^{A}\partial_{r}(f^{1/2}e^{A}\Sigma_{IJ}\partial_{r}a_{a}^{J(1)}-q_{I}\Theta^{a(1)})-\frac{2i}{2p^{2}fZ}e^{-2A}\epsilon_{ab}q_{I}B^{J}(q_{J}\Theta^{b(0)}-e^{A}f^{1/2}\Sigma_{JK}\partial_{r}a_{b}^{K(0)})\\\
&-\frac{i}{2}f^{-1}\Sigma_{IJ}\epsilon_{ab}\Theta^{b(0)}B^{J}=0,\end{split}$
(3.19)

which leads to

$\begin{split}\partial_{u}C_{I}^{a(1)}=-if^{-1}\epsilon_{ab}B^{J}(-\frac{q_{I}C_{J}^{b(0)}}{p^{2}Ze^{2A}}+\Sigma_{IJ}\frac{\Theta^{b(0)}}{2}).\end{split}$
(3.20)

By integrating (3.20) we can write $C_{I}^{a(1)}$ in terms of the zeroeth
order terms. Similarly, $\Theta^{a(1)}$ and $\tau^{a(1)}$ satisfy

$\begin{split}&\partial_{u}[e^{2A}f^{2}\partial_{u}(f^{-1}\Theta^{a(1)})]-4C_{I}^{a(1)}\partial_{u}a_{t}^{I}+2i\epsilon_{ab}\Sigma_{IJ}B^{J}a_{b}^{I(0)}\\\
&+\frac{i}{p^{2}}\epsilon_{ab}B^{I}fC_{I}^{a(0)}\partial_{u}(\frac{1}{fZ})-\frac{i}{p}\Sigma_{IJ}B^{I}B^{J}\tau^{a(0)}=0,\\\
&\partial_{u}[e^{2A}fZ\partial_{u}\tau^{a(1)}]=ipZe^{2A}f^{-1}\Theta^{a(0)},\end{split}$
(3.21)

while $a_{a}^{I(1)}$ can be obtained from

$\partial_{u}a_{a}^{I(1)}=-f^{-1}\Sigma^{IJ}C_{J}^{a(1)}-q_{I}f^{-1}\Theta^{a(1)}.$
(3.22)

Like $C_{I}^{a(1)}$, all these equations can be integrated to obtain
expressions at first order in terms of the zeroeth 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 $\Sigma$ and using the expressions obtained for the
zeroeth order fields. For $C_{I}^{a(1)}$ we obtain,

$\begin{split}C_{I}^{a(1)}=&C_{I0}^{a(1)}+\frac{i\epsilon_{ab}B^{J}}{e^{2A(h)}4\pi
T}[\left(-\frac{q_{I}C_{J}^{b(0)}}{p^{2}Z(h)}+\frac{\Theta_{2}^{b}\Sigma_{IJ}(h)}{8\pi
T}\right)\log\rho+\frac{1}{2}\Theta_{1}^{a}\Sigma_{IJ}(h)e^{2A(h)}4\pi
T\rho\\\ &+\frac{2\Sigma_{IJ}C_{K}^{a(0)}\partial_{u}a_{t}^{K}}{4\pi
T}(\rho\log\rho-\rho)]+...,\end{split}$ (3.23)

where $C_{I0}^{a(1)}$ is an integration constant.

Using this expression a similar near horizon expression can be obtained for
$\Theta^{a(1)}$ from (LABEL:theta1-eq) as follows

$\begin{split}\Theta^{a(1)}=\frac{\Theta^{a}_{3}}{e^{2A(h)}4\pi
T}+\frac{i}{p^{2}}\frac{\epsilon_{ab}B^{I}C_{I}^{a(0)}}{Z(h)}\log\rho+\Theta_{4}^{a}4\pi
T\rho+...,\end{split}$ (3.24)

where $\Theta_{3}^{a}$ and $\Theta_{4}^{a}$ are new integration constants. The
fluctuation in gauge field at first order, $a_{a}^{I(1)}$ at the near horizon
limit follows from (3.22) and is given by

$a_{a}^{I(1)}=a_{a0}^{I(1)}+\frac{\Sigma^{IJ}(h)}{4\pi
T}\left[\frac{q_{J}\Theta_{3}^{a}}{e^{2A(h)}4\pi
T}+C_{J0}^{a(1)}\right]\log\rho+....,$ (3.25)

where we have introduced the constant term of integration as $a_{a0}^{I(1)}$.
Finally the $\tau^{a}$ at first order turns out to be

$\begin{split}\tau^{a(1)}=\tau^{a(1)}_{0}-\frac{e^{-2A(h)}}{4\pi
TpZ(h)}\left[\epsilon_{ab}B^{K}C_{K0}^{b(1)}-\frac{i}{2}(e^{2A}4\pi
T\Theta_{1}^{a}+4q_{i}a_{a0}^{I(0)})+\frac{i}{p}q_{I}B^{I}\epsilon_{ab}\tau^{b(0)}_{0}\right]\log\rho+....,\end{split}$
(3.26)

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 $\omega$.

At the second order of $\omega$ we obtain the following equation for
$C_{I}^{a(2)}$

$\begin{split}\partial_{u}C_{I}^{a(2)}=&\frac{q_{I}e^{-2A}}{2p^{2}fZ}[(e^{2A}\partial_{u}\Theta^{a(0)}+4q_{J}a_{a}^{J(0)})+2i\epsilon_{ab}(C_{J}^{b(1)}-\frac{i}{p}q_{J}\tau^{b(0)})B^{J}]\\\
&-\frac{\Sigma_{IJ}}{f}[a_{a}^{J(0)}-\frac{i}{2}(\Theta^{b(1)}+\frac{i}{p}\tau^{b(0)})B^{J}]\end{split}$
(3.27)

On the other hand for $\Theta^{a(2)}$ we get

$\begin{split}\partial_{u}[e^{2A}f^{2}\partial_{u}(f^{-1}\Theta^{a(2)})]=&4C_{I}^{a(2)}\partial_{u}a_{t}^{I}-f\partial_{u}[\frac{1}{2p^{2}fZ}[(e^{2A}\partial_{u}\Theta^{a(0)}+4q_{J}a_{a}^{J(0)})\\\
+2i\epsilon_{ab}(C_{J}^{b(1)}-&\frac{i}{p}q_{J}\tau^{b(0)})B^{J}]+\frac{i}{p}\Sigma_{IJ}B^{I}B^{J}\tau^{a(1)}-2i\epsilon_{ab}a_{b}^{I(1)}.\end{split}$
(3.28)

$a_{a}^{I(2)}$ can be obtained as usual, from

$C_{I}^{a(2)}=f^{1/2}e^{A}\Sigma_{IJ}\partial_{r}a_{a}^{J(2)}-q_{I}\Theta^{a(2)}.$
(3.29)

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 upto zeroeth order and first order on
right hand side of (3.27) one can easily find that the leading order terms of
$C_{I}^{a(2)}$ near horizon are of the order of $\log\rho$ and
$(\log\rho)^{2}$. In particular, it does not have any $1/\rho$ in its
expression near the horizon. It follows from equation for $\Theta^{a(2)}$ that
the leading order expression of $\Theta^{a(2)}$ is given by

$f^{-1}\Theta^{a(2)}=\Theta_{6}+\Theta_{5}\int\frac{du}{e^{2A}f^{2}}+S\log\rho+...,$
(3.30)

where $\Theta_{5}^{a}$ and $\Theta_{6}^{a}$ are constants of integration and
$S$ is given by

$S=\frac{1}{2p^{2}Z(h)}[(-4\pi
Te^{2A}\Theta_{1}^{a}+4q_{I}a_{a0}^{I})+2i\epsilon_{ab}(C^{b(1)}_{I0}-\frac{i}{p}q_{I}\tau^{b(0)}_{0})B^{I}]+\frac{\Theta_{2}^{a}}{(4\pi
T)^{2}}.$ (3.31)

Collecting expressions of $\Theta^{a}$ at different orders of frequency
together, we can write near horizon expression of $\Theta^{a}$ valid upto
${\mathcal{O}}(\omega^{2})$ as

$\begin{split}\Theta^{a}=&\frac{\Theta_{2}^{a}}{e^{2A(h)}4\pi T}+4\pi
T\Theta_{1}^{a}\rho+(\frac{1}{4\pi
Te^{2A(h)}}[\frac{i\omega\epsilon_{ab}C_{I0}^{b}}{p^{2}Z(h)}+\frac{2(-\pi
Te^{2A(h)}\Theta_{1}^{a}+q_{I}a_{a0}^{I0})}{p^{2}Z(h)}\omega^{2}\\\
&+\frac{\omega^{2}}{p^{3}Z(h)}\epsilon_{ab}\tau^{b(0)}_{0}q_{I}B^{I}]+\omega^{2}\frac{\Theta_{2}^{a}}{(4\pi
T)^{2}})\log\rho+...\end{split}$ (3.32)

In this equation, following [22] we have absorbed all the pertinent
integration constants in $\Theta_{1}^{a}$, $\Theta_{2}^{a}$ and $C_{I0}^{a}$,
without any loss of generality, by redefining $\Theta_{2}^{a}$,
$\Theta_{1}^{a}$ and $C_{I0}^{a}$. Similarly the expression for the
fluctuation in gauge field at near horizon limit is

$a_{a}^{I}=a_{a0}^{I}+\frac{\Sigma^{IJ}(h)}{4\pi
T}(C_{J0}^{a}+\frac{q_{J}\Theta_{2}^{a}}{e^{2A(h)}4\pi T})\log\rho+...$ (3.33)

where we have absorbed all the constants of integration in $a_{a0}^{I}$.
Fluctuation in the axion $\tau^{a}$ at near horizon turns out to be

$\frac{\tau^{a}}{p}=\frac{\tau^{a}_{0}}{p}+\frac{1}{4\pi
Te^{2A(h)}p^{2}Z(h)}[-\epsilon_{ab}B^{I}C_{I0}^{b}+2i\omega(-\pi
Te^{2A}(h)\Theta_{1}^{a}+q_{I}a_{0a}^{I})+i\omega(q_{I}B^{I})\epsilon_{ab}\frac{\tau^{b(0)}_{0}}{p}]\log\rho+...$
(3.34)

where constants are absorbed in $\tau^{a}_{0}$.

Comparing with the near horizon behaviour of $\Theta^{a}$ and
$\eta_{a}^{I}=a_{a}^{I}+\frac{1}{2}B^{I}\epsilon_{ab}\frac{\tau^{b}}{p}$ as
given in (3.11), we obtain

$\begin{split}(\Sigma^{IJ}(h)+\frac{B^{I}B^{J}}{2p^{2}Ze^{2A(h)}})C_{J0}^{a}+\frac{\Sigma^{IJ}(h)q_{J}}{e^{2A(h)}4\pi
T}\Theta_{2}^{a}&=i\omega\\{(a_{a0}^{I}+\frac{1}{2}B^{I}\epsilon_{ab}\frac{\tau^{b}_{0}}{p})-\frac{B^{I}}{e^{2A(h)}p^{2}Z}[\epsilon_{ab}\\\
&(-\pi
Te^{2A(h)}\Theta_{1}^{b}+q_{I}a^{I}_{0b})+\frac{1}{2}(q_{J}B^{J})\frac{\tau^{a}_{0}}{p}\\},\\\
\Theta_{2}^{a}=-\frac{4\pi
T}{p^{2}Z(h)}[\epsilon_{ab}B^{I}C_{I0}^{b}-2i\omega(-\pi
Te^{2A(h)}&\Theta_{1}^{a}+q_{I}a_{a0}^{I})-i\omega(q_{J}B^{J})\epsilon_{ab}\frac{\tau^{b}_{0}}{p}].\end{split}$
(3.35)

From the two equations above (3.35) we can express $C_{I0}^{a}$ and
$\Theta_{2}^{a}$ in terms of other constants $a^{I}_{a0}$, $\Theta_{1}^{a}$
and $\tau^{a}_{0}$ in the following manner,

$\begin{split}C_{I0}^{a}&=i\omega(M_{I}^{~{}J})_{ab}\\{-[(\Sigma_{JK}(h)+\frac{2q_{J}q_{K}}{p^{2}Z(h)e^{2A(h)}})\delta_{bc}+\frac{\Sigma_{JN}(h)B^{N}q_{K}}{p^{2}Z(h)e^{2A(h)}}\epsilon_{bc}]a^{k}_{c0}+\frac{2\pi
T}{p^{2}Z(h)}[q_{J}\delta_{ab}\\\
&+\frac{1}{2}\Sigma_{JN}(h)B^{N}\epsilon_{bc}]\Theta_{1}^{c}-\frac{1}{2}[(\Sigma_{JK}(h)+\frac{4q_{J}q_{K}}{p^{2}Z(h)e^{2A(h)}})B^{K}\epsilon_{bc}-(q_{M}B^{M})\frac{\Sigma_{JK}(h)B^{K}}{p^{2}Z(h)e^{2A(h)}}\delta_{bc}]\frac{\tau^{c}_{0}}{p}\\},\\\
\Theta_{2}^{a}&=-\frac{4\pi
T}{p^{2}Z(h)}\epsilon_{ab}B^{I}C_{I0}^{b}+i\omega\frac{4\pi
T}{p^{2}Z(h)}[2(-\pi
Te^{2A(h)}\Theta_{1}^{a}+q_{I}a_{0a}^{I})+(q_{I}B^{I})\epsilon_{ab}\frac{\tau_{0}^{b}}{p}],\end{split}$
(3.36)

upto leading order in $\omega$, where we have introduced the matrix
$(M_{I}^{J})_{ab}$ satisfying

$[(\delta_{I}^{J}+\frac{\Sigma_{IN}(h)B^{N}B^{J}}{2p^{2}Z(h)e^{2A(h)}})\delta_{ab}-\frac{q_{I}B^{J}}{p^{2}Z(h)e^{2A(h)}}\epsilon_{ab}](M_{J}^{~{}K})_{bc}=\delta_{I}^{K}\delta_{ac}.$
(3.37)

In absence of magnetic field it reduces to $\delta_{I}^{J}\delta_{ab}$.

In order to identify the operators in the boundary theory, we require the
asymptotic solution of $\Theta^{a}$, $a_{a}^{I}$ and $\tau^{a}$. It is
sufficient to determine the asymptotic solution of the fields upto lowest
order in frequency. From the linearised equations of motion of the
fluctutations it is clear that magnetic field contributes at a higher order in
frequency. Therefore, upto lowest order of frequency, expressions remain the
same as those obtained in absence of magnetic field [22]. To this end we
introduce

$\begin{split}\Psi(v)&=sgn(\theta)\int\frac{v^{\theta-3z-1}dv}{F(v)^{2}},\\\
Y^{1}(v)&=\frac{4sgn(\theta)q_{1}}{2+z-\theta}(-v_{h}^{2+z-\theta}\Psi(v)+sgn(\theta)\int
dvv^{-2z+1}F^{-2}),\\\
Y^{2}(v)&=\frac{4sgn(\theta)q_{2}}{\theta-z}(-v_{h}^{\theta-z}\Psi(v)+sgn(\theta)\int
dvv^{2\theta-4z-1}F^{-2}).\\\ \end{split}$ (3.38)

In terms of these functions we can write the asymptotic expansions of the
solutions of the fields at small frequency

$\begin{split}\Theta^{a(0)}&=v^{2(z-1)}F(v)(\Theta_{1}^{a}+\Theta_{2}^{a}\Psi(v)+4C_{I}^{a}Y^{I}(v)),\\\
a_{a}^{I}&=a_{a0}^{I}-\Theta_{1}^{a}a^{I}_{t}-sgn(\theta)\Theta_{2}q_{J}\int
dv\Sigma^{IJ}v^{z-3}\Psi(v)\\\ &-sgn(\theta)\int
dv\Sigma^{IJ}v^{-z-1}(F^{-1}\delta_{K}^{J}+4q_{J}v^{2(z-1)}Y^{K}(v))C_{K}^{a}.\end{split}$
(3.39)

From (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 $\Theta^{a(0)}$,
$a_{a}^{I(0)}$ and $\tau^{a(0)}$ and their conjugates in terms of the modes
$\Theta_{1}^{a}$, $\Theta_{2}^{a}$, $a_{a0}^{I}$, $C_{I}^{a}$ and
$\tau^{a}_{0}$. 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 magenteic 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
${\mathcal{E}}^{a}$ and current operator ${\mathcal{J}}_{I}^{a}$ as shown in
[22]. Their expressions in terms of different modes are given by (A.10) and
(A.11)

$\begin{split}{\mathcal{E}}^{a}=-\frac{1}{2\kappa^{2}}(\Theta_{2}^{a}+4\mu^{I}C_{I0}^{a}),\quad{\mathcal{J}}_{I}^{a}=-\frac{2}{\kappa^{2}}(C_{I0}^{a}-\frac{i\omega
q_{I}}{p}\tau_{0}^{a}),\quad{\mathcal{X}}_{a}=-\frac{2i\omega}{p\kappa^{2}}q_{I}\alpha_{a}^{I}.\end{split}$
(3.40)

where $\alpha_{a}^{I}$ 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.

##  4 Thermoelectric DC conductivities

In this section we obtain thermoelectric conductivities for the present model.
In the last section we have derived $\Theta_{2}^{a}$ and $C_{I0}^{a}$ in terms
of other constants in (3.35). We substitute these expressions in the energy
operator ${\mathcal{E}}^{a}$ given in (3.40), we get

$\begin{split}&{\mathcal{E}}^{a}=-\frac{i\omega}{2\kappa^{2}}[\\{\frac{8\pi
T}{p^{2}Z}q_{K}\delta_{ad}-(-\frac{4\pi
T}{p^{2}Z}\epsilon_{ab}B^{I}+4\mu^{I}\delta_{ab})(M_{I}^{~{}J})_{bc}[(\Sigma_{JK}+\frac{4q_{J}q_{K}}{p^{2}Ze^{2A}})\delta_{cd}+\frac{\Sigma_{JM}B^{M}q_{k}}{p^{2}Ze^{2A}}\epsilon_{cd}\\}\alpha_{d0}^{K}\\\
&+(\frac{8\pi T}{p^{2}Z}(q_{K}\mu^{K}-\pi Te^{2A})\delta_{ad}+(-\frac{4\pi
T}{p^{2}Z}\epsilon_{ab}B^{I}+4\mu^{I}\delta_{ab})(M_{I}^{J})_{bc}\\{\frac{2\pi
T}{2p^{2}Z}(q_{J}\delta_{cd}+\frac{1}{2}\Sigma_{JM}B^{M}\epsilon_{cd})\\\
&-[(\Sigma_{JK}+\frac{2q_{J}q_{K}}{p^{2}Ze^{2A}})\mu^{K}\delta_{cd}+\frac{\Sigma_{JM}B^{M}q_{K}\mu^{K}}{p^{2}Ze^{2A}}\epsilon_{cd}]\\})\Theta_{1}^{d}+\\{-\frac{1}{2}(-\frac{4\pi
T}{p^{2}Z}\epsilon_{ab}B^{I}+4\mu^{I}\delta_{ab})(M_{I}^{~{}J})_{bc}\\\
&[(\Sigma_{JK}+\frac{4q_{J}q_{K}}{p^{2}Ze^{2A}})B^{K}\epsilon_{cd}-\frac{(q_{K}B^{K})\Sigma_{JM}B^{M}}{p^{2}Ze^{2A}}\delta_{cd}]+\frac{4\pi
T}{p^{2}Z}(q_{K}B^{K})\epsilon_{ad}\\}\frac{\tau_{0}^{d}}{p}.\end{split}$
(4.1)

where we have used the asymptotic value of fluctuation in gauge field,
$\alpha_{a}^{I}$ givn in (A.12). In this section, to simplify the notation,
unless otherwise mentioned $A$, $\Sigma_{IJ}$ and $Z$ represents their
respective values at the near horizon limit.

Similarly, the current operator ${\mathcal{J}}_{I}^{a}$ turns out to be

$\begin{split}&J_{I}^{a}=\frac{2i\omega}{\kappa^{2}}[(M_{I}^{~{}J})_{ab}[(\Sigma_{JK}+\frac{2q_{J}q_{K}}{p^{2}Ze^{2A}})\delta_{bc}+\frac{\Sigma_{JM}B^{M}q_{K}}{p^{2}Ze^{2A}}\epsilon_{bc}]\alpha_{c0}^{K}\\\
&-(M_{I}^{~{}J})_{ab}\\{\frac{2\pi
T}{2p^{2}Z}(q_{J}\delta_{bc}+\frac{1}{2}\Sigma_{JM}B^{M}\epsilon_{bc})\\\
&-[(\Sigma_{JK}+\frac{2q_{J}q_{K}}{p^{2}Ze^{2A}})\mu^{K}\delta_{bc}+\frac{\Sigma_{JM}B^{M}q_{K}\mu^{K}}{p^{2}Ze^{2A}}\epsilon_{bc}]\\}\Theta_{1}^{c}\\\
&+\\{\frac{1}{2}(M_{I}^{~{}J})_{ab}[(\Sigma_{JK}+\frac{2q_{J}q_{K}}{p^{2}Ze^{2A}})B^{K}\epsilon_{bc}-(q_{N}B^{N})\frac{\Sigma_{JK}B^{K}}{p^{2}Ze^{2A}}\delta_{bc}]+q_{I}\delta_{ac}\\}\frac{\tau_{0}^{c}}{p},\\\
&{\mathcal{X}}_{a}=-\frac{2i\omega}{p\kappa^{2}}q_{I}\alpha^{I}_{a},\end{split}$
(4.2)

where the matrix $(M_{I}^{J})_{ab}$ is given by (3.37).

From the above expressions one can obtain the following two-point functions

$\begin{split}\langle{\mathcal{J}}_{I}^{a}(-\omega){\mathcal{J}}_{J}^{b}(\omega)\rangle&=\frac{2i\omega}{\kappa^{2}}(M_{J}^{~{}K})_{bc}[(\Sigma_{KI}+\frac{2q_{K}q_{I}}{p^{2}Ze^{2A}})\delta_{ca}+\frac{\Sigma_{KM}B^{M}q_{I}}{p^{Z}e^{2A}}\epsilon_{ca}],\\\
\langle{\mathcal{E}}^{a}(-\omega){\mathcal{J}}_{I}^{b}(\omega)\rangle&=-\frac{2i\omega}{\kappa^{2}}(M_{I}^{J})_{bc}\\{\frac{2\pi
T}{p^{2}Z}(q_{J}\delta_{ca}+\frac{1}{2}\Sigma_{JK}B^{K}\epsilon_{ca})\\\
&-[(\Sigma_{JK}+\frac{2q_{J}q_{K}}{p^{2}Ze^{2A}})\delta_{bc}+\frac{\Sigma_{JM}B^{M}q_{K}}{p^{2}Ze^{2A}}\epsilon_{bc}]\mu^{K}\\},\\\
\langle{\mathcal{J}}_{I}^{a}(-\omega){\mathcal{E}}^{b}(\omega)\rangle&=\frac{2i\omega}{\kappa^{2}}[(-\frac{2\pi
T}{p^{2}Z}q_{I}\delta_{ba}-(-\frac{\pi
T}{p^{2}Z}B^{J}\epsilon_{bc}+\mu^{J}\delta_{bc})(M_{J}^{~{}K})_{cd}\\\
&[(\Sigma_{KI}+\frac{2q_{K}q_{I}}{p^{2}Ze^{2A}})\delta_{da}+\frac{\Sigma_{KM}B^{M}q_{I}}{p^{2}Ze^{2A}}\epsilon_{da}]\\},\\\
\langle{\mathcal{E}}^{a}(-\omega){\mathcal{E}}^{b}(\omega)\rangle&=\frac{2i\omega}{\kappa^{2}}[\frac{2\pi
T}{p^{2}Z}(q_{K}\mu^{K}-\pi Te^{2A})\delta_{ba}+(-\frac{\pi
T}{p^{2}Ze^{2A}}\epsilon^{bc}B^{I}+4\mu^{I}\delta_{bc})\\\
&(M_{I}^{J})_{cd}[(\frac{2\pi
T}{p^{2}Z}q_{J}-(\Sigma_{JK}+\frac{2q_{J}q_{K}}{p^{2}Ze^{2A}})\mu^{K}]\delta_{da}+\Sigma_{JM}B^{M}(\frac{\pi
T}{p^{2}Z}-\frac{q_{K}\mu^{K}}{p^{2}Ze^{2A}})\epsilon_{da}],\\\
\langle{\mathcal{X}}^{a}(-\omega){\mathcal{J}}_{I}^{b}(\omega)\rangle&=\frac{2i\omega}{\kappa^{2}}[\frac{1}{2}(M_{I}^{J})_{bc}[(\Sigma_{JK}+\frac{4q_{J}q_{K}}{p^{2}Ze^{2A}})B^{K}\epsilon_{ca}-(q_{J}B^{J})\frac{\Sigma_{JK}B^{K}}{p^{2}Ze^{2A}}\delta^{ca}]+q_{I}\delta_{ba},\\\
\langle{\mathcal{J}}_{I}^{a}(-\omega){\mathcal{X}}^{b}(\omega)\rangle&=-\frac{2i\omega}{p\kappa^{2}}q_{I}\delta^{ab},\end{split}$
(4.3)

with rest of the two point functions vanishing.

Next following [22] we introduce the heat current

${\mathcal{Q}}_{D}^{a}={\mathcal{E}}^{a}-\mu^{I}{\mathcal{J}}_{I}^{a}.$ (4.4)

The two point function for heat current and electric currents are given by

$\begin{split}\langle{\mathcal{Q}}_{D}^{a}(-\omega){\mathcal{Q}}_{D}^{b}(\omega)\rangle&=\frac{2i\omega}{\kappa^{2}}2(\frac{\pi
T}{p^{2}Z})^{2}\\{p^{2}Ze^{2A}\delta^{ab}+B^{M}\epsilon_{ac}(M_{M}^{~{}J})_{cd}[q_{J}\delta_{da}+\frac{1}{2}\Sigma_{JN}B^{N}\epsilon_{da}]\\},\\\
\langle{\mathcal{Q}}_{D}^{a}(-\omega){\mathcal{J}}_{I}^{b}(\omega)\rangle&=-\frac{2i\omega}{\kappa^{2}}\frac{2\pi
T}{p^{2}Z}[(M_{I}^{~{}J})_{bc}[q_{J}\delta_{ca}+\frac{1}{2}\Sigma_{JK}B^{K}\epsilon_{ca}],\\\
\langle{\mathcal{J}}_{I}^{a}(-\omega){\mathcal{Q}}_{D}^{b}(\omega)\rangle&=-\frac{2i\omega}{\kappa^{2}}\\{\frac{2\pi
T}{p^{2}Z}q_{I}\delta_{ba}+\frac{\pi
T}{p^{2}Z}\epsilon_{bc}B^{J}(M_{J}^{~{}K})^{cd}[(\Sigma_{KI}+\frac{2q_{K}q_{I}}{p^{2}Ze^{2A}})\delta_{da}+\frac{\Sigma_{KM}B^{M}q_{J}}{p^{2}ZE^{2}A}\epsilon_{da}]\\}\\\
\langle{\mathcal{J}}_{I}^{a}(-\omega){\mathcal{J}}_{J}^{b}(\omega)\rangle&=\frac{2i\omega}{\kappa^{2}}(M_{J}^{~{}K})_{bc}[(\Sigma_{KI}+\frac{2q_{K}q_{I}}{p^{2}Ze^{2A}})\delta_{ca}+\frac{\Sigma_{KM}B^{M}q_{I}}{p^{Z}e^{2A}}\epsilon_{ca}],\end{split}$
(4.5)

We obtain the thermoelectric conductivities from the above two point functions
as follows.

${\mathcal{\sigma}}_{D}^{DC}=\left(\begin{array}[]{cc}T\bar{\mathbb{K}}^{ab}&T\bar{\mathcal{\alpha}}_{I}^{ab}\\\
T{\mathcal{\alpha}}_{I}^{ab}&{\mathcal{\sigma}}_{IJ}^{ab}\end{array}\right)=\left(\begin{array}[]{cc}\langle{\mathcal{Q}}_{D}^{a}(-\omega){\mathcal{Q}}_{D}^{b}(\omega)\rangle&\langle{\mathcal{Q}}_{D}^{a}(-\omega){\mathcal{J}}_{I}^{b}(\omega)\rangle\\\
\langle{\mathcal{J}}_{I}^{a}(-\omega){\mathcal{Q}}_{D}^{b}(\omega)\rangle&\langle{\mathcal{J}}_{I}^{a}(-\omega){\mathcal{Q}}_{D}^{b}(\omega)\rangle\end{array}\right).$
(4.6)

In order to obtain the following expressions for the components of the
conductivity matrix in a compact form we have introduced the following
parameters

$r_{I}=\frac{1}{2}\Sigma_{IJ}B^{J},\quad\quad
b^{I}=\frac{B^{I}}{p^{2}Ze^{2A}}.$ (4.7)

In terms of these parameters the matrix $(M_{I}^{~{}J})_{ab}$ is given from
(3.37)

$(M_{I}^{~{}J})_{ab}=\delta_{I}^{J}\delta_{ab}-\frac{[(1+r.b)r_{I}+(q.b)q_{I}]\delta_{ab}-[(1+r.b)q_{I}-(q.b)r_{I}]\epsilon_{ab}}{(1+r.b)^{2}+(q.b)^{2}}b^{J}.$
(4.8)

where we have used $(r.b)=r_{I}b^{I}$, $(q.b)=q_{I}b^{I}$ and
$\triangle=(1+r.b)^{2}+(q.b)^{2}$. With these expressions, components of
conductivity matrix becomes

$\begin{split}\bar{\mathbb{K}}^{ab}&=\frac{\pi
sT}{\kappa^{2}p^{2}Z}\frac{[(1+r.b)\delta_{ba}+(q.b)\epsilon_{ba}]}{\triangle},\\\
\bar{\alpha}_{I}^{ab}&=\alpha_{I}^{ab}=-\frac{4}{sT}\bar{\mathbb{K}}^{bc}(q_{I}\delta_{ca}+r_{I}\epsilon_{ca})],\\\
{\mathcal{\sigma}}_{IJ}^{ab}&=\frac{2}{\kappa^{2}}\Sigma_{JI}\delta^{ba}+\frac{16}{s^{2}T}\bar{\mathbb{K}}^{bc}(q_{J}\delta_{cd}+r_{J}\epsilon_{cd})(q_{I}\delta_{da}+r_{I}\epsilon_{da}),\end{split}$
(4.9)

where we have used $4\pi e^{2A}=s$. All the components of the conductivity
matrix reduce to the expressions of the same given in [22] for setting
$B^{I}=0$ It may be observed that both the $U(1)$ gauge fields are on the same
footing and that we have got $\bar{\alpha}_{I}^{ab}=\alpha_{I}^{ab}$.

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 dyonic black
hole 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
$\triangle=(p^{2}+\frac{B^{2}}{4}v^{4z-6-\theta})^{2}+(2q_{2}Bv^{2z-4})^{2}$
and using that we get,

$\begin{split}\bar{\mathbb{K}}^{ab}&=\frac{8\pi^{2}T}{\kappa^{2}p^{2}}v_{h}^{2(z-\theta)}\frac{(p^{2}+\frac{B^{2}}{4}v_{h}^{4z-6-\theta})\delta_{ba}+2q_{2}Bv_{h}^{2z-4}\epsilon_{ba}}{\triangle},\\\
{\mathbb{\alpha}}_{1}^{ab}&=-\frac{8\pi}{\kappa^{2}}v_{h}^{2z-\theta-2}\frac{(p^{2}+\frac{B^{2}}{4}v_{h}^{4z-6-\theta})q_{1}\delta_{ba}+2q_{1}q_{2}Bv_{h}^{2z-4}\epsilon_{ba}}{\triangle},\\\
{\mathbb{\alpha}}_{2}^{ab}&=-\frac{8\pi}{\kappa^{2}}v_{h}^{2z-\theta-2}\frac{p^{2}q_{2}\delta_{ba}+[(p^{2}+\frac{B^{2}}{4}v_{h}^{4z-6-\theta})\frac{B}{8}v_{h}^{2z-2-\theta}+2q_{2}^{2}Bv_{h}^{2z-4}]\epsilon_{ba}}{\triangle},\\\
{\mathcal{\sigma}}_{11}^{ab}&=\frac{1}{2\kappa^{2}}v_{h}^{\theta-4}\delta_{ba}+\frac{8}{\kappa^{2}}q_{1}^{2}v_{h}^{2z-4}\frac{(p^{2}+\frac{B^{2}}{4}v_{h}^{4z-6-\theta})\delta_{ba}+2q_{2}Bv_{h}^{2z-4}\epsilon_{ba}}{\triangle},\\\
{\mathcal{\sigma}}_{12}^{ab}&=\frac{8}{\kappa^{2}}q_{1}\frac{q_{2}p^{2}\delta_{ba}+[2q_{2}^{2}Bv_{h}^{2z-4}+\frac{B}{8}v_{h}^{2z-2-\theta}(p^{2}+\frac{B^{2}}{4}v_{h}^{4z-6-\theta})]\epsilon_{ba}}{\triangle},\\\
{\mathcal{\sigma}}_{22}^{ab}&=\frac{p^{2}}{2\kappa^{2}}v_{h}^{6z-8-2\theta}\frac{\frac{B^{2}}{4}+v_{h}^{6-4z+\theta}(p^{2}+16q_{2}^{2}v_{h}^{\theta-2})}{\triangle}\delta_{ba}+\frac{q_{2}B}{\kappa^{2}}v_{h}^{8z-12-2\theta}\frac{\frac{B^{2}}{4}+(2p^{2}+16q_{2}^{2}v_{h}^{\theta-2})v_{h}^{-4z+6+\theta}}{\triangle}\epsilon_{ba},\end{split}$
(4.10)

Hall angle can be obtained from the above conductivities by taking the ratio
of coefficients of $\epsilon_{ab}$ and $\delta_{ab}$ in the expression of
${\mathcal{\sigma}}$. We get

$\Theta_{H}=\frac{2q_{2}B}{p^{2}}v_{h}^{2z-4}[\frac{\frac{B^{2}}{4}+v_{h}^{-4z+6+\theta}(2p^{2}+16q_{2}^{2}v_{h}^{\theta-2})}{\frac{B^{2}}{4}+v_{h}^{-4z+6+\theta}(p^{2}+16q_{2}^{2}v_{h}^{\theta-2})}].$
(4.11)

As explained in [29] since the factor in the square bracket lies between 1 and
2 Hall coefiicient can be approximated as

$\Theta_{H}=\frac{2q_{2}B}{p^{2}}v_{h}^{2z-4}{p^{2}}.$ (4.12)

these expressions,after setting $\theta=1-z$, 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
$T=-\frac{sgn(\theta)}{4\pi}v_{h}^{z+1}F^{\prime}(v_{h})$ which for the case
of dyonic solution reduces to

$T=-\frac{sgn(\theta)}{4\pi}[(z+2-\theta)v_{h}^{z}-\frac{8q_{2}^{2}}{2-\theta}v_{h}^{2\theta-z-2}-\frac{p^{2}}{2-\theta}v_{h}^{\theta-z}-\frac{B^{2}}{4(2-z)}v_{h}^{3z-6}.$
(4.13)

The expression of temperature is quite involved and it is difficult to obtan
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 $\theta<0$, 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
$q_{2}^{2}v_{h}^{2\theta-z-2}<<v_{h}^{z}$, $p^{2}v_{h}^{\theta-z}<<v_{h}^{z}$
and $B^{2}v_{h}^{3(z-2)}<<v_{h}^{z}$. In this regime one can identify
$T\equiv\frac{8q_{1}^{2}}{4\pi(z-1)}v_{h}^{z}$. 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 dissipaion is strong
compared to charge and magnetic field, which is given by,
$B^{2}v_{h}^{3(z-2)},q_{2}^{2}v_{h}^{2\theta-z-2}<<p^{2}v_{h}^{\theta-z}<<v_{h}^{z}$.
In this limit we obtain

$\begin{split}{\mathbb{K}}^{ab}&\sim\frac{8\pi^{2}T}{\kappa^{2}p^{4}}[T^{\frac{2(z-\theta)}{z}}\delta_{ba}+2q_{2}BT^{\frac{4z-2\theta-4}{z}}\epsilon_{ab}],\\\
{\mathcal{\sigma}}_{11}^{ab}&\sim\frac{8q_{1}^{2}}{\kappa^{2}p^{2}}[T^{\frac{2z-4}{z}}\delta_{ba}+\frac{2q_{2}B}{p^{2}}T^{\frac{4z-8}{z}}\epsilon_{ba}],\\\
{\mathcal{\sigma}}_{12}^{ab}&\sim\frac{q_{1}}{\kappa^{2}p^{2}}[8q_{2}\delta_{ba}+BT^{\frac{4z-6-\theta}{z}}\epsilon_{ba}]\\\
{\mathcal{\sigma}}_{22}^{ab}&\sim\frac{1}{2\kappa^{2}}[T^{\frac{2z-2-\theta}{z}}\delta_{ba}+\frac{4q_{2}B}{p^{2}}T^{\frac{4z-6-\theta}{z}}\epsilon_{ba}],\\\
{\mathcal{\alpha}}_{1}^{ab}&\sim-\frac{8\pi
q_{1}}{\kappa^{2}p^{2}}[T^{\frac{2z-\theta-2}{z}}\delta_{ba}+\frac{2q_{2}B}{p^{2}}T^{\frac{2z-4}{z}}\epsilon_{ba}],\\\
{\mathcal{\alpha}}_{2}^{ab}&\sim-\frac{8\pi}{\kappa^{2}p^{2}}[q_{2}T^{\frac{2z-\theta-2}{z}}\delta_{ba}+\frac{B}{8}T^{\frac{4z-2\theta-4}{z}}\epsilon_{ba}].\end{split}$
(4.14)

The Hall angle is $\theta_{H}\sim T^{\frac{2z-4}{z}}$. Since $\theta<0$ we
cannot get linear resistivity for $\sigma_{22}^{xx}$ in this regime. Choosing
$z=1$ we get $\theta_{H}\sim 1/T^{2}$ and $\sigma_{22}^{xx}\sim T^{-\theta}$
showing a positive power of $T$ for conductivity. Instead if we choose,
$B^{2}v_{h}^{3(z-2)}<<p^{2}v_{h}^{\theta-z}<<q_{2}^{2}v_{h}^{2\theta-z-2}<<v_{h}^{z},p^{2}>>2q_{2}Bv_{h}^{2z-4}$
all the coefficients will remain the same except $\sigma_{22}$. It becomes

$\sigma_{22}^{ab}=\frac{8q_{2}^{2}}{\kappa^{2}p^{2}}[T^{\frac{2z-4}{z}}\delta_{ba}+\frac{2q_{2}B}{p^{2}}T^{\frac{4z-8}{z}}\epsilon_{ba}].$
(4.15)

In this regime, $\sigma_{22}^{xx}$ and Hall angle have similar temperature
dependence. So for $z=1$ both scale as $\sim T^{-2}$. Choosing $z=4/3$ one
gets $\sigma_{22}^{xx}\sim T^{-1}$ implying linear resistivity. However, Hall
angle also becomes $\theta_{H}\sim T^{-1}$.

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
$B^{2}v_{h}^{3(z-2)},p^{2}v_{h}^{\theta-z}<<q_{2}^{2}v_{h}^{2\theta-z-2}<<v_{h}^{z}$
and leads to the following conductivities:

$\begin{split}{\mathbb{K}}^{ab}&\sim\frac{8\pi^{2}T}{\kappa^{2}(2q_{2}B)}[\frac{1}{2q_{2}B}T^{\frac{2(4-\theta-z)}{z}}\delta_{ba}+\frac{1}{p^{2}}T^{\frac{2(2-\theta)}{z}}\epsilon_{ab}],\quad\text{for}\quad
B^{2}v_{h}^{3(z-2)}<<p^{2}v_{h}^{\theta-z},\\\
{\mathcal{\alpha}}_{1}^{ab}&\sim-\frac{8\pi
q_{1}}{\kappa^{2}}[\frac{p^{2}}{4q_{2}^{2}B^{2}}T^{\frac{6-2z-\theta}{z}}\delta_{ba}+\frac{1}{2q_{2}B}T^{\frac{2-\theta}{z}}\epsilon_{ba}]\quad\text{for}\quad
B^{2}v_{h}^{3(z-2)}<<p^{2}v_{h}^{\theta-z},\\\
{\mathcal{\alpha}}_{1}^{ab}&\sim-\frac{8\pi
q_{1}}{\kappa^{2}}[\frac{4}{B^{2}}T^{\frac{2(z-\theta)}{z}}\delta_{ba}+\frac{1}{2q_{2}B}T^{\frac{2-\theta}{z}}\epsilon_{ba}]\quad\text{for}\quad
p^{2}v_{h}^{\theta-z}<<B^{2}v_{h}^{3(z-2)},\\\
{\mathcal{\alpha}}_{2}^{ab}&\sim-\frac{8\pi}{\kappa^{2}}[\frac{p^{2}}{4q_{2}B^{2}}T^{\frac{6-2z-\theta}{z}}\delta_{ba}+\frac{1}{2B}T^{\frac{2-\theta}{z}}\epsilon_{ba}],\\\
{\mathcal{\sigma}}_{11}^{ab}&\sim\frac{q_{1}^{2}}{2\kappa^{2}}[\frac{4p^{2}}{q_{2}^{2}B^{2}}T^{\frac{4-2z}{z}}\delta_{ba}+\frac{16}{2q_{2}B}\epsilon_{ba}],\quad\text{for}\quad
p^{2}v_{h}^{\theta}>>(q_{2}Bv_{h}^{z+\theta-4})^{2},\\\
&\sim\frac{q_{1}^{2}}{2\kappa^{2}}[T^{\frac{\theta-4}{z}}\delta_{ba}+\frac{16}{2q_{2}B}\epsilon_{ba}],\quad\text{for}\quad
p^{2}v_{h}^{\theta}<<(q_{2}Bv_{h}^{z+\theta-4})^{2},\\\
{\mathcal{\sigma}}_{12}^{ab}&\sim\frac{8q_{1}q_{2}}{\kappa^{2}}[\frac{p^{2}}{4q_{2}^{2}B^{2}}T^{\frac{8-4z}{z}}\delta_{ba}+\frac{1}{2q_{2}B}T^{\frac{4-2z}{z}}\epsilon_{ba}],\\\
{\mathcal{\sigma}}_{22}^{ab}&\sim\frac{1}{2\kappa^{2}}[\frac{4p^{2}}{B^{2}}T^{\frac{4-2z)}{z}}\delta_{ba}+\frac{8q_{2}}{B}\epsilon_{ba}].\end{split}$
(4.16)

In this regime, $\sigma_{22}^{xx}$ and Hall angle have opposite temperature
dependence. Choosing $z=1$ one gets temperature dependence to be $T^{2}$ 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,
$\sigma_{22}^{xx}\sim T^{\frac{(4-2z)}{z}}$ with Hall angle having opposite
temperature dependence, once again.

For small temperature, one can identify the following regions of parameters.
$B^{2}v_{h}^{3(z-2)},q_{2}^{2}v_{h}^{2\theta-z-2}<<p^{2}v_{h}^{\theta-z}\lesssim
v_{h}^{z}$,
$B^{2}v_{h}^{3(z-2)},p^{2}v_{h}^{\theta-z}<<q_{2}^{2}v_{h}^{2\theta-z-2}\lesssim
v_{h}^{z}$ and
$p^{2}v_{h}^{\theta-z},q_{2}^{2}v_{h}^{2\theta-z-2}<<B^{2}v_{h}^{3(z-2)}\lesssim
v_{h}^{z}$ . However, obtaining an analytical expression for temperature for
this region is difficult. The dependence on $v_{h}$ can be obtained from above
by replacing $T$ by $v_{h}^{z}$ in (4.14) and (4) a respectively in the three
regimes.

For $\theta>0$ first term is negative and so large temperature may corresponds
to the regimes depending on whether $p^{2}v_{h}^{\theta-z}$,
$q_{2}^{2}v_{h}^{2\theta-z-2}$ or $B^{2}v_{h}^{3(z-2)}$ dominates. In these
regimes, temperature can be approximated by
$T\equiv\frac{p^{2}}{4\pi(2-\theta)}v_{h}^{\theta-z}$,
$T\equiv\frac{8q_{2}^{2}}{4\pi(2-\theta)}v_{h}^{2\theta-z-2}$ or
$T\equiv\frac{B_{2}^{2}}{16\pi(2-z)}v_{h}^{3z-6}$, respectively. The scalings
of conductivity matrix for various regimes will be as follows:

For the parameter region corresponding to strong momentum dissipation,
$B^{2}v_{h}^{3(z-2)}$ , $q_{2}^{2}v_{h}^{2\theta-z-2}<<p^{2}v_{h}^{\theta-z}$
we get

$\begin{split}{\mathbb{K}}^{ab}&\sim\frac{8\pi^{2}T}{\kappa^{2}p^{4}}[\left(\frac{T}{p^{2}}\right)^{\frac{2(z-\theta)}{\theta-z}}\delta_{ba}+2\frac{q_{2}B}{p^{2}}\left(\frac{T}{p^{2}}\right)^{\frac{4z-2\theta-4}{\theta-z}}\epsilon_{ab}],\\\
{\mathcal{\alpha}}_{1}^{ab}&\sim-\frac{8\pi
q_{1}}{\kappa^{2}p^{2}}[\left(\frac{T}{p^{2}}\right)^{\frac{2z-\theta-2}{\theta-z}}\delta_{ba}+\frac{2q_{2}B}{p^{2}}\left(\frac{T}{p^{2}}\right)^{\frac{2z-4}{\theta-z}}\epsilon_{ba}],\\\
{\mathcal{\alpha}}_{2}^{ab}&\sim-\frac{8\pi}{\kappa^{2}p^{2}}[q_{2}\left(\frac{T}{p^{2}}\right)^{\frac{2z-\theta-2}{\theta-z}}\delta_{ba}+\frac{B}{8}\left(\frac{T}{p^{2}}\right)^{\frac{4z-2\theta-4}{\theta-z}}\epsilon_{ba}],\\\
{\mathcal{\sigma}}_{11}^{ab}&\sim\frac{8q_{1}^{2}}{\kappa^{2}p^{2}}[\left(\frac{T}{p^{2}}\right)^{\frac{2z-4}{\theta-z}}\delta_{ba}+\frac{2q_{2}B}{p^{2}}\left(\frac{T}{p^{2}}\right)^{\frac{4z-8}{\theta-z}}\epsilon_{ba}],\\\
{\mathcal{\sigma}}_{12}^{ab}&\sim\frac{q_{1}}{\kappa^{2}p^{2}}[8q_{2}\delta_{ba}+B\left(\frac{T}{p^{2}}\right)^{\frac{2z-2-\theta}{\theta-z}}\epsilon_{ba}],\\\
{\mathcal{\sigma}}_{22}^{ab}&\sim\frac{1}{2\kappa^{2}}[\left(\frac{T}{p^{2}}\right)^{\frac{2z-2-\theta}{\theta-z}}\delta_{ba}+\frac{4q_{2}B}{p^{2}}\left(\frac{T}{p^{2}}\right)^{\frac{4z-6-\theta}{\theta-z}}\epsilon_{ba}.\end{split}$
(4.17)

For $z\rightarrow 2$ $\sigma_{22}^{xx}\sim T^{-1}$, but Hall angle becomes
independent of temperature.

For the regime, where charge is strong compared to other two factors, given by
$B^{2}v_{h}^{3(z-2)}$ , $p^{2}v_{h}^{\theta-z}<<q_{2}^{2}v_{h}^{2\theta-z-2}$,
conductivities turn out to be

$\begin{split}{\mathbb{K}}^{ab}&\sim\frac{8\pi^{2}T}{\kappa^{2}p^{2}}[\frac{p^{2}}{4q_{2}^{2}B^{2}}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{8-2z-2\theta}{2\theta-z-2}}\delta_{ba}+\frac{1}{2q_{2}B}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{2(2-\theta)}{2\theta-z-2}}\epsilon_{ab}],\quad\text{for}\quad
B^{2}v_{h}^{3(z-2)}<<p^{2}v_{h}^{\theta-z},\\\
&\sim\frac{8\pi^{2}T}{\kappa^{2}p^{2}}[\frac{1}{16q_{2}^{2}}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{2z+2-3\theta}{2\theta-z-2}}\delta_{ba}+\frac{1}{2q_{2}B}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{2(2-\theta)}{2\theta-z-2}}\epsilon_{ab}],\quad\text{for}\quad
p^{2}v_{h}^{\theta-z}<<B^{2}v_{h}^{3(z-2)},\\\
{\mathcal{\alpha}}_{1}^{ab}&\sim-\frac{8\pi
q_{1}}{\kappa^{2}}[\frac{p^{2}}{4q_{2}^{2}B^{2}}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{6-2z-\theta}{2\theta-z-2}}\delta_{ba}+\frac{1}{2q_{2}B}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{2-\theta}{2\theta-z-2}}\epsilon_{ba}]\quad\text{for}\quad
B^{2}v_{h}^{3(z-2)}<<p^{2}v_{h}^{\theta-z},\\\
{\mathcal{\alpha}}_{1}^{ab}&\sim-\frac{8\pi
q_{1}}{\kappa^{2}}[\frac{1}{16q_{2}^{2}}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{2(z-\theta)}{2\theta-z-2}}\delta_{ba}+\frac{1}{2q_{2}B}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{2-\theta}{2\theta-z-2}}\epsilon_{ba}]\quad\text{for}\quad
p^{2}v_{h}^{\theta-z}<<B^{2}v_{h}^{3(z-2)},\\\
{\mathcal{\alpha}}_{2}^{ab}&\sim-\frac{8\pi}{\kappa^{2}}[\frac{p^{2}}{4q_{2}B^{2}}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{6-2z-\theta}{2\theta-z-2}}\delta_{ba}+\frac{1}{2B}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{2-\theta}{2\theta-z-2}}\epsilon_{ba}].\\\
{\mathcal{\sigma}}_{11}^{ab}&\sim\frac{8q_{1}^{2}}{\kappa^{2}}[\frac{p^{2}}{4q_{2}^{2}B^{2}}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{4-2z}{2\theta-z-2}}\delta_{ba}+\frac{1}{2q_{2}B}\epsilon_{ba}],\quad\text{for}\quad
p^{2}v_{h}^{\theta}>>(2q_{2}Bv_{h}^{z+\theta-4})^{2},\\\
&\sim\frac{1}{2\kappa^{2}}[\left(\frac{T}{q_{2}^{2}}\right)^{\frac{\theta-4}{2\theta-z-2}}\delta_{ba}+\frac{1}{2q_{2}B}\epsilon_{ba}],\quad\text{for}\quad
p^{2}v_{h}^{\theta}<<(2q_{2}Bv_{h}^{z+\theta-4})^{2},\\\
&\sim\frac{8q_{1}^{2}}{\kappa^{2}}[\frac{1}{16q_{2}^{2}}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{2z-2-\theta}{2\theta-z-2}}\delta_{ba}+\frac{1}{2q_{2}B}\epsilon_{ba}],\quad\text{for}\quad
p^{2}v_{h}^{\theta-z}<<B^{2}v_{h}^{3(z-2)},\end{split}$
$\begin{split}{\mathcal{\sigma}}_{12}^{ab}&\sim\frac{8q_{1}}{\kappa^{2}}[\frac{p^{2}}{4q_{2}B^{2}}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{8-4z}{2\theta-z-2}}\delta_{ba}+\frac{1}{2B}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{4-2z}{2\theta-z-2}}\epsilon_{ba}],\\\
{\mathcal{\sigma}}_{22}^{ab}&\sim\frac{1}{2\kappa^{2}}[\frac{4p^{2}}{B^{2}}\left(\frac{T}{q_{2}^{2}}\right)^{\frac{4-2z}{2\theta-z-2}}\delta_{ba}+\frac{8q_{2}}{B}\epsilon_{ba}].\end{split}$
(4.18)

As observed from above, $\sigma_{22}^{xx}$ and Hall angle has opposite
temperature dependence. for For $z=1$ $\sigma_{22}^{xx}\sim T^{-1}$, but Hall
angle becomes independent of time. Small temperature limit can be chosen in a
similar way as in the case of $\theta<0$. The bahaviour will be similar to
those obtained in the case of $\theta<0$.

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.

##  5 Conclusion

We 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 $1/T^{2}$ behaviour. Here we have explicitly consider 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 A Appendix

In 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 has 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 ${\bar{r}}$, which is related to the
Einstein frame radial coordinate $r$ through the relation
$d\bar{r}=-sgn(\theta)e^{\frac{\theta}{2\mu}\phi}dr$. The advantage of this
dual coordinate is it allows both positive and negative values of $\theta$ and
the UV boundary lies at ${\bar{r}}\rightarrow\infty$.

In order to identify the operators living in the boundary theory and the
fields in the bulk theory one considers [22, 19] 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. $ds^{2}=dr^{2}+\gamma_{ij}dx^{i}dx^{j}$, where
$x^{i}={t,x^{a}}$. In the Hamiltonian formalism the metric and the gauge field
can be decomposed as

$ds^{2}=(N^{2}+N_{i}N^{i})dr^{2}+2N_{i}drdx^{i}+\gamma_{ij}dx^{i}dx^{j},\quad
A^{I}_{\mu}dx^{\mu}=A^{I}_{r}dr+A^{I}_{i}dx^{i},$ (A.1)

where $N$ and $N_{i}$ are the lapse and shift function and $\gamma_{ij}$ is
the induced metric on radial slices at fixed values of $r$. Similarly $A_{r}$
and $A_{i}$ 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

$K_{ij}=\frac{1}{2N}(\partial_{r}\gamma_{ij}-D_{i}N_{j}-D_{j}N_{i}),$ (A.2)

where $D_{i}$ is the covariant derivative with respect to the metric
$\gamma_{ij}$. 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

$\begin{split}L_{\xi}&=\frac{1}{2\kappa^{2}}\int
d^{3}x\sqrt{-\bar{\gamma}}\bar{N}[(1+\frac{4\xi^{2}}{\alpha_{\xi}})\bar{K}^{2}-\bar{K}^{ij}\bar{K}_{ij}-\frac{\alpha_{\xi}}{\bar{N}^{2}}(\partial_{r}\phi-\bar{N}^{i}\partial_{i}\phi-\frac{2\xi}{\alpha_{\xi}}\bar{N}\bar{K})^{2}\\\
&-\frac{2}{\bar{N}^{2}}\Sigma^{\xi}_{IJ}(\phi)(F^{I}_{ri}-\bar{N}^{k}F^{I}_{ki})(F^{Ji}_{r}-\bar{N}^{l}F^{Ji}_{l})-\frac{1}{\bar{N}^{2}}Z_{\xi}(\phi)(\partial_{r}\chi^{a}-\bar{N}^{i}\partial_{i}\chi^{a})^{2}\\\
&+R[\bar{\gamma}]-\alpha_{\xi}\partial_{i}\phi\bar{\partial}^{i}\phi-\Sigma^{\xi}_{IJ}F^{I}_{ij}F^{Jij}-Z_{\xi}\partial_{i}\chi^{a}\bar{\partial}^{i}\chi^{a}-V_{\xi}-2\Box_{\bar{\gamma}}]e^{2\xi\phi},\end{split}$
(A.3)

The canonical momenta in the dual frame can be obtained from the above
lagrangian as

$\bar{\pi}^{ij}=\frac{\delta
L}{\delta\dot{\bar{\gamma}}_{ij}},\quad\bar{\pi}^{i}_{I}=\frac{\delta
L}{\delta\dot{A}_{i}^{I}},\quad\bar{\pi}_{\phi}=\frac{\delta
L}{\delta\dot{\phi}},\quad\bar{\pi}_{\chi^{a}}=\frac{\delta
L}{\delta\dot{\chi}^{a}},$ (A.4)

with conjugate momenta of the non-dynamical fields, $\bar{N}$, $\bar{N}_{i}$
and $A_{r}$ being zero.

Expressing them in terms of quantities in the Einstein frame one gets

$\begin{split}\bar{\pi}^{ij}&=\frac{1}{2\kappa^{2}}\sqrt{-\gamma}e^{2\xi\phi}(K\gamma^{ij}-K^{ij}),\quad\bar{\pi}^{i}_{I}=-\frac{2}{\kappa^{2}}\sqrt{-\gamma}\Sigma_{IJ}\gamma^{ij}F_{rj}^{I},\\\
\bar{\pi}_{\phi}&=\frac{1}{\kappa^{2}}\sqrt{-\gamma}(2\xi
K-\alpha\partial_{r}\phi),\quad\bar{\pi}_{\chi^{a}}=-\frac{1}{\kappa^{2}}\sqrt{-\gamma}Z\partial_{r}\chi^{a}.\end{split}$
(A.5)

These expressions evaluated around the background in linearised order of
perturbations in metric and other fields reduce to the following expressions.

$\begin{split}\pi^{ta}=\frac{1}{4\kappa^{2}}e^{2\xi\phi_{B}}e^{-3A}f^{-1/2}\partial_{r}(e^{4A}S_{t}^{a}),\\\
\pi^{a}_{I}=-\frac{2}{\kappa^{2}}e^{A}f^{1/2}\Sigma_{IJ}(\partial_{r}a_{a}^{J}+f^{-1}(\partial_{r}a_{t}^{J})S_{t}^{a}),\\\
\pi_{\chi^{a}}=-\frac{1}{\kappa^{2}}e^{3A}f^{1/2}Z\partial_{r}\tau^{a}.\end{split}$
(A.6)

In order to make connection to the asymptotic expressions we will express the
above equations in terms of $\Theta^{a}$, $a_{a}^{I}$ and $\tau^{a}$. We will
consider only the expression in zeroeth order of $\omega$. Furthermore, we
will use the radial coordinate $v$ instead of $r$. Substituting the background
values of the fields and using
$dr=-sgn(\theta)v^{-\theta/2}{\mathcal{F}}^{-1/2}(v)\frac{dv}{v}$ we obtain,

$\begin{split}\pi^{ta}&=-\frac{sgn(\theta)}{4\kappa^{2}}v^{\theta-z-1}\partial_{v}(v^{4-2\theta}(\Theta^{a(0)}+\frac{i\omega}{p}\tau^{a(0)})),\\\
\pi_{1}^{a}&=\frac{sgn(\theta)}{2\kappa^{2}}[v^{z+\theta-3}F(v)\partial_{v}a_{a}^{1(0)}+4sgn(\theta)q_{1}(\Theta^{a(0)}+\frac{i\omega}{p}\tau^{a(0)})],\\\
\pi_{2}^{a}&=\frac{sgn(\theta)}{2\kappa^{2}}[v^{3z+\theta-1}F(v)\partial_{v}a_{a}^{2(0)}+4sgn(\theta)q_{2}(\Theta^{a(0)}+\frac{i\omega}{p}\tau^{a(0)})],\\\
\pi_{\chi^{a}}&=\frac{i\omega}{2p\kappa^{2}}[-sgn(\theta)v^{5-z-\theta}\partial_{v}\Theta^{a(0)}-4q_{I}a_{a}^{I(0)}].\end{split}$
(A.7)

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, $\Theta^{a(0)}$, $a_{a}^{I(0)}$, $\tau^{a(0)}$
along with their conjugate momenta. The other set consists of the modes
$\Theta_{1}^{a}$, $\Theta_{2}^{a}$, $a_{a0}^{a}$, $C_{I}^{a}$ and $\tau^{a}$.

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. Through addition of counterterms at the boundary
the variables $\pi^{ta}$, $A_{1}^{a}$ and $\pi^{\chi^{a}}$ undergo canonical
transformations, keeping $A_{2}^{a}$ 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 $\Theta^{a(0)}$, $a_{a}^{I(0)}$, $\tau^{a(0)}$ remain the same as
in the case of zero magnetic field. However, there are differences in the
expression of the blackening factor ${\mathcal{F}}(v)$ 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,

$\begin{split}\Pi^{ta}=-\frac{1}{4\kappa^{2}}v^{-2z}(\Theta_{2}^{a}+4\mu^{I}C_{I}^{a})+...,\quad\quad{\mathfrak{a}}_{a}^{1}=a_{a0}^{1}-\mu^{1}\Theta_{1}+...,\\\
\Pi_{\chi^{a}}=\frac{-2i\omega}{p\kappa^{2}}q_{I}{\mathfrak{a}}_{a}^{I}+...,\quad\quad{\mathfrak{a}}_{a}^{2}=a_{a0}^{2}-\mu^{2}\Theta_{1}+...,\end{split}$
(A.8)

where the chemical potentials are given by

$\mu^{1}=-\frac{4sgn(\theta)q_{1}v_{h}^{2+z-\theta}}{2+z-\theta},\quad\mu^{2}=-\frac{4sgn(\theta)q_{2}v_{h}^{\theta-z}}{\theta-z}.$
(A.9)

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
$A_{1}^{a}$, 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

$\begin{split}{\mathcal{E}}^{a}=2\lim\limits_{\bar{r}\rightarrow\infty}e^{2z\bar{r}}\Pi^{ta}=-\frac{1}{2\kappa^{2}}(\Theta_{2}^{a}+4\mu^{I}C_{I0}^{a}),\quad\quad\Theta^{a}_{1}=\lim\limits_{\bar{r}\rightarrow\infty}e^{-2z\bar{r}}n_{a},\end{split}$
(A.10)

respectively where $\bar{r}$ is related to $r$ through
$r\sim\frac{2}{|\theta|}e^{-}\frac{\theta\bar{r}}{2}$ and $n_{a}$ is the shift
function in the decomposition of the metric $\bar{\gamma}_{ij}$ as
$\bar{\gamma}_{ij}dx^{i}dx^{j}=-(n^{2}-n_{a}n^{a})dt^{2}+2n_{a}dtdx^{a}+\sigma_{ab}dx^{a}dx^{b}$,
$a,b=1,2$. Similarly the observable for $U(1)$ currents and pseudoscalars are
given by

$\begin{split}{\mathcal{J}}_{I}^{a}=\lim\limits_{\bar{r}\rightarrow\infty}\Pi^{a}_{I}=-\frac{2}{\kappa^{2}}(C_{I0}^{a}-\frac{i\omega
q_{I}}{p}\tau_{0}^{a}),\quad\quad{\mathcal{X}}_{a}=\lim\limits_{\bar{r}\rightarrow\infty}\Pi_{\chi^{a}}=-\frac{2i\omega}{p\kappa^{2}}q_{I}\alpha_{a}^{I},\end{split}$
(A.11)

respectively and $\alpha_{a}^{I}$ is given by

$\alpha_{a}^{I}=a_{0}^{I}-\mu^{I}\Theta_{1}^{a}.$ (A.12)

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