# Holographic DC Conductivity for Backreacted NLED in Massive Gravity

Shihao Bi a<EMAIL_ADDRESS>Jun Taoa<EMAIL_ADDRESS>aCenter for
Theoretical Physics, College of Physics, Sichuan University, Chengdu, 610065,
China

###### Abstract

In this work a holographic model with the charge current dual to a general
nonlinear electrodynamics (NLED) is discussed in the framework of massive
gravity. Massive graviton can breaks the diffeomorphism invariance in the bulk
and generates momentum dissipation in the dual boundary theory. The expression
of DC conductivities in a finite magnetic field are obtained, with the
backreaction of NLED field on the background geometry. General transport
properties in various limits are presented, and then we turn to the three of
specific NLED models: the conventional Maxwell electrodynamics, the Maxwell-
Chern-Simons electrodynamics, and the Born-Infeld electrodynamics, to study
the parameter-dependence of in-plane resistivity. Two mechanisms leading to
the Mott-insulating behaviors and negative magneto-resistivity are revealed at
zero temperature, and the role played by the massive gravity coupling
parameters are discussed.

††preprint: CTP-SCU/2021001

##  I Introduction

The discovery of gauge/gravity duality makes it possible to deal with the
strongly-coupled gauge theories on the boundary from the classical
gravitational theories in the higher dimensional bulk Susskind1997:PRD ;
Maldacena1997:IJTP ; Witten1998:ATMP ; Ammon2015:CUP ; Eleftherios2011:LNP ;
Natsuume2016:LNP . And the well-known prediction on the ratio of the shear
viscosity to the entropy density for $\mathcal{N}=4$ super Yang-Mills (SYM)
theory was found to be close to the experimental results of real quark-gluon
plasma (QGP) Policastro2001:PRL ; Buchel2004:PRL ; Kovtun2005:PRL ;
Benincasa2006:JHEP , making it more convincing to physical community. In
recent years the idea of gauge/gravity duality has been applied to
hydrodynamics DTSon2006:JHEP ; DTSon2007:ARNPS ; Kovtun2007:JPA ;
Rangamani2009:CQG , quantum chromodynamics (QCD) Adams2001:PRD ;
Brodsky2004:PLB ; Brodsky2005:PRL ; Brodsky2009:PRL ; Erlich2005:PRL ;
Rold20005:NPB ; Zayakin2008:JHEP ; Edelstein2009:AIP ; Gursoy2011:QCD ;
Alfimov2015:JHEP , nuclear physics Bergman2007:JHEP ; Baldino2017:PRD , and
strongly-coupled condensed matter systems Hartnoll2008:PRL ; Hartnoll2009:CQG
; Herzog2009:JPA ; Herzog2010:PRD ; Mcgreevy2010:AHEP ; Nishioka2010:JHEP ;
Cubrovic2009:Sci ; Liu2011:PRD ; Iqbal2012:NFL ; Faulkner2011:JHEP ;
Cai2015:SCP , etc, and new insights are brought into these physical branches.

The establishment of quantum many-body theory AGD2012:QFT ; Landau:vol9  in a
fairly low energy scale compared with high energy physics is another
magnificent and fascinating story apart from that of the standard model, which
have profoundly extended and deepen our understanding of realistic matters.
Many characterization methods, based on the linear response theory or quantum
transport theory, have been developed in the scattering or transport
experiments, to reveal the electronic or lattice structures, surface
topography, defects and disorder, and study the transport properties of the
material samples. In the framework of the energy band theory the materials are
roughly divided into three categories: metals, semiconductors, and insulators,
which differs in their electrical conductivities. And it is known to all of us
the classical electron motion in the presence of the magnetic field has been
studied since 1879, which is the famous classical Hall effect Hall1879 . Its
quantum version, known as the quantum Hall effect, was first observed in 1980
by von Klitzing Klitzing1980:PRL , has aroused a wide research enthusiasm in
the last two decades and witnessed impressive theoretical and experimental
breakthroughs Qi2011:RMP ; Hasan2010:RMP . The new quantum phases of matter,
termed topological insulators, exhibit another kind of bulk-boundary
correspondence between the gapped insulated bulk and gapless metallic edge
states on the boundary. Its exotic transport properties have also attracted
lots of research interests, among which the conductivity behavior in the
presence of magnetic field is of vital importance, for it provides a promising
way on controlling the electrical properties with the help of external fields.
For normal metals the resistivity always increases with the magnetic field
strength Wannier1972:PRB , showing the positive magneto-resistivity. However,
the experimental measurements Kim2013:PRL ; Xiong2015:Sci ; Li2016:NC ;
Zhang2016:NC ; Zhao2016:SR  in topological materials, such as Dirac or Weyl
semimetals, etc., demonstrated the presence of negative magneto-resistivity or
crossover for positive to negative, which are mainly attributed to chiral
anomaly DTSon2013:PRB ; Qi2013:CRP ; Burkov2014:PRL ; Burkov2015:PRB ;
Lu2017:FP . In addition, the negative magneto-resistivity behaviors are also
found in holographic chiral anomalous systems Jimenez2014:PRD ;
Jimenez2015:JHEP ; Landsteiner2015:JHEP ; Sun2016:JHEP .

Mott insulators, which are believed to be parent materials of cuprate high
$T_{c}$ superconductors Wen2006:RMP , is another example beyond the framework
of energy band theory due to the strong Coulomb repulsive interaction. The
strong electron-electron interaction would prevent the available charge
carriers to efficiently transport charges as if suffers the electronic traffic
jam. The holographic construction of Mott insulators has been a heated topic
and much progress has been made Edalati2011:PRL ; Edalati2011:PRD ;
Wu2012:JHEP ; Wu2014:JHEP ; Wu2015:JHEP ; Wu2016:JHEP ; Fujita2015:JHEP ;
Nishioka2010:JHEP ; Kiritsis2015:JHEP . In Ref. Baggioli2016:JHEP , the
holographic model coupled with a particular NLED named iDBI was proposed to
study the strong electron-electron interaction by introducing the self-
interaction of NLED field, and the Mott-insulating behaviors appear for large
enough self-interaction strength.

In this paper, we construct a holographic model coupled with a general NLED
field to investigate the magneto-transport of the strongly-interacting system
on the boundary from the perspective of gauge/gravity duality. The
backreaction of NLED field on the bulk geometry is taken into consideration as
in Ref. Cremonini2017:JHEP . The massive gravity, with massive potentials
associated with the graviton mass, is adopted to break the diffeomorphism
invariance in the bulk producing momentum relaxation in the dual boundary
theory Vegh2013 .

This paper is organized as follows. In Sec. II we establish our holographic
model of massive gravity with NLED field. Then the DC conductivities with non-
zero magnetic field are derived as the function of $\rho$, $h$, $T$, and
related massive gravity coupling parameters in Sec. III, with various limit
situation being discussed. Then in Sec. IV we present detailed investigation
on the in-plane resistivity in the framework of conventional Maxwell
electrodynamics, the CP-violating Maxwell-Chern-Simons electrodynamics, and
Born-Infeld electrodynamics. Finally, we make our conclusion in Sec. V. We
will use the unit $\hslash=G=k_{B}=l=1$.

##  II Holographic Setup

The 4-dimensional massive gravity Vegh2013  with a negative cosmological
constant $\Lambda$ coupled to a nonlinear electromagnetic field $A_{\mu}$ we
are considering is given by

$S=\frac{1}{16\pi}\int\mathrm{d}^{4}x\sqrt{-g}\left[R-2\Lambda+m^{2}\sum_{j=1}^{4}c_{j}\mathcal{U}_{j}(g,f)+\mathcal{L}(s,p)\right],$
(1)

where $\Lambda=-3/l^{2}$, $m$ is the mass parameter, $f$ is a fixed symmetric
tensor called the reference metric, $c_{j}$ are coupling constants 111To make
the massive gravity theory self-consistent, all the $c_{j}$ will be set to be
negative., and $\mathcal{U}_{j}(g,f)$ denotes the symmetric polynomials of the
eigenvalue of the $4\times 4$ matrix
$\mathcal{K}_{\nu}^{\mu}=\sqrt{g^{\mu\lambda}f_{\lambda\nu}}$ given as

$\displaystyle\mathcal{U}_{1}$ $\displaystyle=[\mathcal{K}],$ (2)
$\displaystyle\mathcal{U}_{2}$
$\displaystyle=[\mathcal{K}]^{2}-\left[\mathcal{K}^{2}\right],$
$\displaystyle\mathcal{U}_{3}$
$\displaystyle=[\mathcal{K}]^{3}-3[\mathcal{K}]\left[\mathcal{K}^{2}\right]+2\left[\mathcal{K}^{3}\right],$
$\displaystyle\mathcal{U}_{4}$
$\displaystyle=[\mathcal{K}]^{4}-6[\mathcal{K}]^{2}\left[\mathcal{K}^{2}\right]+8[\mathcal{K}]\left[\mathcal{K}^{3}\right]+3\left[\mathcal{K}^{2}\right]^{2}-6\left[\mathcal{K}^{4}\right].$

The square root in $\mathcal{K}$ means
$\left(\sqrt{\mathcal{K}}\right)_{\lambda}^{\mu}\left(\sqrt{\mathcal{K}}\right)_{\nu}^{\lambda}=\mathscr{\mathcal{K}}_{\nu}^{\mu}$
and $\left[\mathcal{K}\right]\equiv\mathcal{K}_{\mu}^{\mu}$. The NLED
Lagrangian $\mathcal{L}(s,p)$ in Eq. 1 is constructed as the function of two
independent nontrivial scalar using the field strength tensor
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$

$\displaystyle s=$ $\displaystyle-\frac{1}{4}F^{\mu\nu}F_{\mu\nu},$ (3a)
$\displaystyle p=$
$\displaystyle-\frac{1}{8}\varepsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$
(3b)

Here $\epsilon^{abcd}\equiv-\left[a\;b\;c\;d\right]/\sqrt{-g}$ is a totally
antisymmetric Lorentz tensor, and $\left[a\;b\;c\;d\right]$ is the permutation
symbol. In the weak field limit we assume the NLED reduces to the Maxwell-
Chern-Simons Lagrangian $\mathcal{L}\left(s,p\right)\approx s+\theta p$ with
$\theta$ defined as $\mathcal{L}^{\left(0,1\right)}\left(0,0\right)$ for later
convenience.

Varying the action with respect to $g^{\mu\nu}$ and $A^{\mu}$ we obtain the
equations of motion

$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}=$
$\displaystyle\frac{1}{2}T_{\mu\nu},$ (4a)
$\displaystyle\nabla_{\mu}G^{\mu\nu}=$ $\displaystyle 0.$ (4b)

where the energy-momentum tensor is

$T_{\mu\nu}=g_{\mu\nu}\left(\mathcal{L}(s,p)-p\frac{\partial\mathcal{L}(s,p)}{\partial
p}\right)+\frac{\partial\mathcal{L}(s,p)}{\partial
s}F_{\mu}^{\lambda}F_{\nu\lambda}+m^{2}\chi_{\mu\nu},$ (5)

with

$\displaystyle\chi_{\mu\nu}=$ $\displaystyle
c_{1}\left(\mathcal{U}_{1}g_{\mu\nu}-\mathcal{K}_{\mu\nu}\right)+c_{2}\left(\mathcal{U}_{2}g_{\mu\nu}-2\mathcal{U}_{1}\mathcal{K}_{\mu\nu}+2\mathcal{K}_{\mu\nu}^{2}\right)$
(6)
$\displaystyle+c_{3}\left(\mathcal{U}_{3}g_{\mu\nu}-3\mathcal{U}_{2}\mathcal{K}_{\mu\nu}+6\mathcal{U}_{1}\mathcal{K}_{\mu\nu}^{2}-6\mathcal{K}_{\mu\nu}^{3}\right)$
$\displaystyle+c_{4}\left(\mathcal{U}_{4}g_{\mu\nu}-4\mathcal{U}_{3}\mathcal{K}_{\mu\nu}+12\mathcal{U}_{2}\mathcal{K}_{\mu\nu}^{2}-24\mathcal{U}_{1}\mathcal{K}_{\mu\nu}^{3}+24\mathcal{K}_{\mu\nu}^{4}\right).$

And we introduce

$G^{\mu\nu}=-\frac{\partial\mathcal{L}(s,p)}{\partial
F_{\mu\nu}}=\frac{\partial\mathcal{L}(s,p)}{\partial
s}F^{\mu\nu}+\frac{1}{2}\frac{\partial\mathcal{L}(s,p)}{\partial
p}\varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}.$ (7)

We are looking for a black brane solution with asymptotic AdS spacetime by
taking the following ansatz Vegh2013 ; Cai2015:PRD  for the metric and the
NLED field, and the reference metric:

$\displaystyle\mathrm{d}s^{2}=$
$\displaystyle-{}f(r)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f(r)}+r^{2}\left(\mathrm{d}x^{2}+\mathrm{d}y^{2}\right),$
(8a) $\displaystyle A=$ $\displaystyle
A_{t}(r)\mathrm{d}t+\frac{h}{2}\left(x\mathrm{d}y-y\mathrm{d}x\right),$ (8b)
$\displaystyle f_{\mu\nu}=$
$\displaystyle\operatorname{diag}\left(0,0,\alpha^{2},\alpha^{2}\right).$ (8c)

where $h$ is the magnetic field strength. From Eqs. 8a and 8b we find the
nontrivial scalars Eq. 3b are

$\displaystyle s=$
$\displaystyle\frac{1}{2}\left(A_{t}^{\prime}(r)^{2}-\frac{h^{2}}{r^{4}}\right),$
(9a) $\displaystyle p=$ $\displaystyle-\frac{hA_{t}^{\prime}(r)}{r^{2}}.$ (9b)

The equations of motion then are obtained as

$\displaystyle rf^{\prime}(r)+f(r)-3r^{2}=$ $\displaystyle c_{1}\alpha
m^{2}r+c_{2}\alpha^{2}m^{2}+\frac{r^{2}}{2}\left(A_{t}^{\prime}(r)G^{rt}+\mathcal{L}(s,p)\right),$
(10a) $\displaystyle rf^{\prime\prime}(r)+2f^{\prime}(r)-6r=$ $\displaystyle
c_{1}\alpha m^{2}+r\left(\mathcal{L}(s,p)+hG^{xy}\right),$ (10b)
$\displaystyle\left[r^{2}G^{rt}\right]^{\prime}=$ $\displaystyle 0,$ (10c)

where the non-vanishing components $G^{\mu\nu}$ are

$\displaystyle G^{rt}=$ $\displaystyle\frac{\partial\mathcal{L}}{\partial
p}\frac{h}{r^{2}}-\frac{\partial\mathcal{L}}{\partial s}A_{t}^{\prime}(r),$
(11a) $\displaystyle G^{xy}=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial
s}\frac{h}{r^{4}}+\frac{\partial\mathcal{L}}{\partial
p}\frac{A_{t}^{\prime}(r)}{r^{2}}.$ (11b)

Eq. 10c leads to $G^{tr}=\rho/r^{2}$ with $\rho$ being a constant. The event
horizon $r_{h}$ is the root of $f(r)$, i.e., $f(r_{h})=0$, and the Hawking
temperature of the black brane is given by

$T=\frac{f^{\prime}(r_{h})}{4\pi}.$ (12)

Then at $r=r_{h}$ Eq. 10a reduces to

$4\pi Tr_{h}-3r_{h}^{2}=c_{1}\alpha
m^{2}r_{h}+c_{2}\alpha^{2}m^{2}+\frac{r_{h}^{2}}{2}\left(A_{t}^{\prime}(r_{h})G_{h}^{rt}+\mathcal{L}(s_{h},p_{h})\right),$
(13)

where

$\displaystyle s_{h}=$
$\displaystyle\frac{1}{2}\left(A_{t}^{\prime}(r_{h})^{2}-\frac{h^{2}}{r_{h}^{4}}\right),$
(14a) $\displaystyle p_{h}=$
$\displaystyle-\frac{hA_{t}^{\prime}(r_{h})}{r_{h}^{2}},$ (14b) $\displaystyle
G_{h}^{rt}=$
$\displaystyle\mathcal{L}^{\left(0,1\right)}\left(s_{h},p_{h}\right)\frac{h}{r_{h}^{2}}-\mathcal{L}^{\left(1,0\right)}\left(s_{h},p_{h}\right)A_{t}^{\prime}\left(r_{h}\right)$
(14c)

##  III DC Conductivity

From the perspective of gauge/gravity duality, the black brane solution Eqs.
8a and 8b in the bulk can describe an equilibrium state at finite temperature
$T$ given by Eq. 13. And the conserved current $\mathcal{J}^{\mu}$ in the
boundary theory is connected with the conjugate momentum of the NLED field in
the bulk, which allows us calculate the DC conductivity in the framework of
linear response theory Donos2014:JHEP ; Blake2014:PRL .

###  III.1 Derivation of DC Conductivity

The following perturbations on the metric and the NLED field are applied to
derive the DC conductivity:

$\displaystyle\delta g_{ti}=$ $\displaystyle r^{2}h_{ti}(r),$ (15a)
$\displaystyle\delta g_{ri}=$ $\displaystyle r^{2}h_{ri}(r),$ (15b)
$\displaystyle\delta A_{i}=$ $\displaystyle-E_{i}t+a_{i}(r),$ (15c)

where $i=x,y$. We first consider the $t$ component. The absence of $A_{t}(r)$
in the Eq. 9b leads to the radially independent conjugate momentum
$\partial_{r}\Pi^{i}=0$, with

$\Pi^{t}=\frac{\partial\mathcal{L}\left(s,p\right)}{\partial\left(A_{t}^{\prime}\left(r\right)\right)}.$
(16)

Then the expectation value of $\mathcal{J}^{t}$ in the dual boundary field
theory is given by

$\left\langle\mathcal{J}^{t}\right\rangle=\Pi^{t}.$ (17)

In the linear level we have $\left\langle\mathcal{J}^{t}\right\rangle=\rho$,
which indicate that $\rho$ can be interpreted as the charge density in the
dual field theory. At the event horizon $r=r_{h}$ the charge density $\rho$ is
given by

$\rho=\mathcal{L}^{\left(1,0\right)}\left(s_{h},p_{h}\right)r_{h}^{2}A_{t}^{\prime}\left(r_{h}\right)-\mathcal{L}^{\left(0,1\right)}\left(s_{h},p_{h}\right)h.$
(18)

Then we consider the planar components. The NLED is explicitly independent of
$a_{i}(r)$, making the conjugate momentum of the field $a_{i}(r)$

$\Pi^{i}=\frac{\partial\mathcal{L}\left(s,p\right)}{\partial\left(a_{i}^{\prime}(r)\right)}=\frac{\partial\mathcal{L}\left(s,p\right)}{\partial\left(\partial_{r}A_{i}\right)}=\sqrt{-g}G^{ir},$
(19)

being radially independent as well. And the charge currents in the dual field
theory are given by $\left\langle\mathcal{J}^{i}\right\rangle=\Pi^{i}$, and
can be expressed with the perturbed metric and field components $h_{ti}$,
$h_{ri}$, $a^{\prime}_{i}$ and $E_{i}$:

$\displaystyle\left\langle\mathcal{J}^{x}\right\rangle$
$\displaystyle=-\mathcal{L}^{\left(1,0\right)}\left(s,p\right)\left[f\left(r\right)a_{x}^{\prime}\left(r\right)+hf\left(r\right)h_{ry}\left(r\right)+r^{2}A_{t}^{\prime}\left(r\right)h_{tx}\left(r\right)\right]-\mathcal{L}^{\left(0,1\right)}\left(s,p\right)E_{y},$
(20a) $\displaystyle\left\langle\mathcal{J}^{y}\right\rangle$
$\displaystyle=-\mathcal{L}^{\left(1,0\right)}\left(s,p\right)\left[f\left(r\right)a_{y}^{\prime}\left(r\right)-hf\left(r\right)h_{rx}\left(r\right)+r^{2}A_{t}^{\prime}\left(r\right)h_{ty}\left(r\right)\right]+\mathcal{L}^{\left(0,1\right)}\left(s,p\right)E_{x}.$
(20b)

The perturbed metric components $h_{ti}$ and $h_{ri}$ are coupled to $E_{i}$
in gravitational field equation and can be reduced. Thus we take the first
order of the perturbed Einstein’s equations of $tx$ and $ty$ components and we
get:

$\displaystyle\left(\frac{h^{2}}{r^{2}}-\frac{\alpha
m^{2}c_{1}r}{\mathcal{L}^{\left(1,0\right)}(s,p)}\right)h_{tx}(r)-hA_{t}^{\prime}(r)f(r)h_{ry}(r)=$
$\displaystyle A_{t}^{\prime}(r)f(r)a^{\prime}_{x}(r)-\frac{E_{y}h}{r^{2}},$
(21a) $\displaystyle\left(\frac{h^{2}}{r^{2}}-\frac{\alpha
m^{2}c_{1}r}{\mathcal{L}^{\left(1,0\right)}(s,p)}\right)h_{ty}(r)+hA_{t}^{\prime}(r)f(r)h_{rx}(r)=$
$\displaystyle A_{t}^{\prime}(r)f(r)a^{\prime}_{y}(r)+\frac{E_{x}h}{r^{2}}.$
(21b)

Next by using the regularity constraints of the metric and field near the
event horizon Blake2014:PRL :

$\displaystyle f\left(r\right)$ $\displaystyle=4\pi
T\left(r-r_{h}\right)+\cdots,$ (22) $\displaystyle A_{t}\left(r\right)$
$\displaystyle=A_{t}^{\prime}\left(r_{h}\right)\left(r-r_{h}\right)+\cdots,$
$\displaystyle a_{i}\left(r\right)$ $\displaystyle=-\frac{E_{i}}{4\pi
T}\ln\left(r-r_{h}\right)+\cdots,$ $\displaystyle h_{ri}\left(r\right)$
$\displaystyle=\frac{h_{ti}\left(r\right)}{f\left(r\right)}+\cdots,$

the Eqs. 21a and 21b at the event horizon $r=r_{h}$ are

$\displaystyle
hA_{t}^{\prime}(r_{h})h_{ty}(r_{h})-\left(\frac{h^{2}}{r_{h}^{2}}-\frac{\alpha
m^{2}c_{1}r_{h}}{\mathcal{L}^{\left(1,0\right)}(s_{h},p_{h})}\right)h_{tx}(r_{h})=A_{t}^{\prime}(r_{h})E{}_{x}+\frac{h}{r_{h}^{2}}E_{y},$
(23a) $\displaystyle\left(\frac{h^{2}}{r_{h}^{2}}-\frac{\alpha
m^{2}c_{1}r_{h}}{\mathcal{L}^{\left(1,0\right)}(s_{h},p_{h})}\right)h_{ty}(r_{h})+hA_{t}^{\prime}(r_{h})h_{tx}(r_{h})=\frac{h}{r_{h}^{2}}E_{x}-A_{t}^{\prime}(r_{h})E{}_{y}.$
(23b)

Solving Eqs. 23a and 23b for $h_{ti}(r_{h})$ in terms of $E_{i}$ and inserting
into Eqs. 20a and 20b, one can evaluate the current
$\left\langle\mathcal{J}^{i}\right\rangle$ to the electric fields $E_{i}$ at
the event horizon $r=r_{h}$ via
$\left\langle\mathcal{J}^{i}\right\rangle=\sigma_{ij}E_{j}$, where the DC
conductivities are given by

$\displaystyle\sigma_{xx}=$ $\displaystyle\sigma_{yy}=\dfrac{\alpha
m^{2}c_{1}r_{h}\left(\dfrac{\alpha
m^{2}c_{1}r_{h}}{\mathcal{L}^{\left(1,0\right)}(s_{h},p_{h})}-\dfrac{h^{2}}{r_{h}^{2}}+r_{h}^{2}A^{\prime}_{t}{}^{2}(r_{h})\right)}{\left(\dfrac{\alpha
m^{2}c_{1}r_{h}}{\mathcal{L}^{\left(1,0\right)}(s_{h},p_{h})}-\dfrac{h^{2}}{r_{h}^{2}}\right)^{2}+h^{2}A^{\prime}_{t}{}^{2}(r_{h})},$
(24a) $\displaystyle\sigma_{xy}=$
$\displaystyle-\sigma_{yx}=\dfrac{\mathcal{L}^{\left(1,0\right)}(s_{h},p_{h})r_{h}^{2}A^{\prime}_{t}(r_{h})}{h}\left(1-\dfrac{\left(\dfrac{\alpha
m^{2}c_{1}r_{h}}{\mathcal{L}^{\left(1,0\right)}(s_{h},p_{h})}\right)^{2}}{\left(\dfrac{\alpha
m^{2}c_{1}r_{h}}{\mathcal{L}^{\left(1,0\right)}(s_{h},p_{h})}-\dfrac{h^{2}}{r_{h}^{2}}\right)^{2}+h^{2}A^{\prime}_{t}{}^{2}(r_{h})}\right)-\mathcal{L}^{\left(0,1\right)}(s_{h},p_{h}).$
(24b)

The resistivity matrix is the inverse of the conductivity matrix:

$R_{xx}=R_{yy}=\frac{\sigma_{xx}}{\sigma_{xx}^{2}+\sigma_{xy}^{2}}\quad\text{and}\quad
R_{xy}=-R_{yx}=\frac{\sigma_{xy}}{\sigma_{xx}^{2}+\sigma_{xy}^{2}}.$ (25)

The event horizon radius $r_{h}$ is the solution of Eq. 13 and will be
affected by the temperature $T$, the charge density $\rho$, the magnetic field
$h$ and the parameters $c_{1,2}$, $m$ and $\alpha$. For a given Lagrangian
$\mathcal{L}(s,p)$ one first solve $A_{t}^{\prime}(r_{h})$ from Eq. 18, and
then plug it into Eq. 13 to solve $r_{h}$, which will be brought into Eqs. 24a
and 24b to obtain the DC conductivity as a complicated function of the
parameters mentioned above.

###  III.2 Various Limits

The concrete examples of NLED will be discussed in Sec. IV and the properties
of DC resistivity are discussed in detail. Before focus on the specific NLED
model we first consider some general properties of the DC conductivity in
various limit.

####  III.2.1 Massless and Massive Limits

The massless limit corresponds to $m\to 0$ and the system will restore the
Lorentz invariance. It has been shown that in a Lorentz invariant theory that
the DC conductivities in the presence of a magnetic field were $\sigma_{xx}=0$
and $\sigma_{xy}=\rho/h$ Hartnoll2007:PRD . For comparison, in the massless
limit the DC conductivities in Eqs. 24a and 24b become

$\sigma_{xx}=\frac{\alpha
m^{2}\left|c_{1}\right|r_{h}^{3}}{h^{2}}+\mathcal{O}(m^{4})\quad\text{and}\quad\sigma_{xy}=\frac{\rho}{h}+\mathcal{O}(m^{4}).$
(26)

And in the massive limit $m^{-1}\to+\infty$, the DC conductivities have the
asymptotic behaviors

$\sigma_{xx}=\mathcal{L}^{\left(1,0\right)}(s_{h},p_{h})+\mathcal{O}(m^{-2})\quad\text{and}\quad\sigma_{xy}=-\mathcal{L}^{\left(0,1\right)}(s_{h},p_{h})+\mathcal{O}(m^{-2}).$
(27)

which agree with the calculation by treating the NLED field as a probe one
Guo2017:PRD , for the geometry is dominated by the massive terms.

####  III.2.2 Zero Field and Charge Density Limits

In the zero field limit $h=0$ case, the DC conductivities become

$\sigma_{xx}=\mathcal{L}^{\left(1,0\right)}\left(\frac{A_{t}^{\prime
2}\left(r_{h}\right)}{2},0\right)-\frac{\rho^{2}}{\alpha
m^{2}c_{1}r_{h}^{3}}\quad\text{and}\quad\sigma_{xy}=-\mathcal{L}^{\left(0,1\right)}\left(\frac{A_{t}^{\prime
2}\left(r_{h}\right)}{2},0\right).$ (28)

where $A_{t}^{\prime}\left(r_{h}\right)$ is obtained by solving

$\rho=\mathcal{L}^{\left(1,0\right)}\left(\frac{A_{t}^{\prime
2}\left(r_{h}\right)}{2},0\right)r_{h}^{2}A_{t}^{\prime}\left(r_{h}\right).$
(29)

At zero charge density $\rho=0$, the DC conductivities become

$\sigma_{xx}^{-1}=\frac{1}{\mathcal{L}^{\left(1,0\right)}\left(-\dfrac{h^{2}}{2r_{h}^{4}},0\right)}-\dfrac{h^{2}}{\alpha
m^{2}c_{1}r_{h}^{3}}\quad\text{and}\quad\sigma_{xy}=-\mathcal{L}^{\left(0,1\right)}\left(-\frac{h^{2}}{2r_{h}^{4}},0\right).$
(30)

We find that the DC conductivities are in general non-zero and can be
interpreted as incoherent contributions Davison2015:JHEP , known as the charge
conjugation symmetric contribution $\sigma_{\text{ccs}}$. There is another
contribution from explicit charge density relaxed by some momentum
dissipation, $\sigma_{\text{diss}}$, depending on the charge density $\rho$.
The results show that, for a general NLED model, the DC conductivities usually
depend on $\sigma_{\text{diss}}$ and $\sigma_{\text{ccs}}$ in a nontrivial
way.

####  III.2.3 High Temperature Limit

Finally, we consider the high temperature limit
$T\gg\max\left\\{\sqrt{h},\sqrt{\rho},|c_{1}|\alpha m^{2},\sqrt{|c_{2}|}\alpha
m\right\\}$. In this limit, Eq. 13 gives $T\approx\dfrac{3}{4\pi}r_{h}$. The
longitudinal resistivity then reduces to

$R_{xx}=\frac{1}{1+\theta^{2}}\left\\{1+\frac{27}{64\pi^{3}\alpha\left|c_{1}\right|T^{3}}\left[h^{2}\left(1+\theta^{2}\right)+2h\theta\rho-\frac{1-\theta^{2}}{1+\theta^{2}}\rho^{2}\right]\right\\}+\mathcal{O}\left(T^{-6}\right).$
(31)

The nonlinear effect will be suppressed by the temperature and we only keep
the leading order. Usually the metal and insulator have different temperature
dependence. For the metallic materials the phonon scattering enlarges the
resistivity, while the thermal excitation of carriers in insulating materials
can promotes the conductivity. And the metal-insulator transition can occur
when the coefficient of $1/T^{2}$ changes the sign, which is shown in Fig. 1
in the $h/\rho-\theta$ parameter space. The $\theta$ term can break the
$\left(\rho,h\right)\to\left(\rho,-h\right)$ or
$\left(\rho,h\right)\to\left(-\rho,h\right)$ symmetries for $\sigma_{ij}$ or
$R_{ij}$, but $\sigma_{ij}$ or $R_{ij}$ are invariant under
$\left(\rho,h\right)\to\left(-\rho,-h\right)$. And the phases are central
symmetric in the parameter plane.

Fig. 1: Left panel: The metal-insulator phase diagram in the high temperature
limit. The red region has positive temperature derivative of $R_{xx}$ and thus
describes a metal, while the blue region has negative temperature derivative,
and hence an insulator. The black solid lines are the phase boundaries. Right
panel: The Mott-insulating region (blue) and the negative magneto-resistivity
(red) in the high temperature limit

The Mott-insulating and magneto-resistance behaviors are also presented in
Fig. 1. The Mott-insulating region is where $\partial_{|\rho|}R_{xx}>0$ in the
parameter plane. The strong electron-electron interaction prevents the charge
carriers from transporting. From Eq. 31 we see that as long as $\theta^{2}>1$
the Mott-insulating emerges in the absence of the magnetic field. The magneto-
resistance is defined as

$MR_{xx}=\frac{R_{xx}\left(h\right)-R_{xx}\left(0\right)}{R_{xx}\left(0\right)},$
(32)

and we see that in the Fig. 1 the negative magneto-resistivity can only occur
with non-zero $\theta$. For $\theta=0$, Eq. 31 reduces to

$R_{xx}=1+\frac{27\left(h^{2}-\rho^{2}\right)}{64\pi^{3}\alpha\left|c_{1}\right|T^{3}}+\mathcal{O}\left(T^{-6}\right),$
(33)

which gives metallic behavior for $h|/\rho|<1$ and insulating behavior for
$|/\rho|>1$. And one always has $\partial_{|\rho|}R_{xx}<0$ and
$\partial_{|h|}R_{xx}>0$, indicating the absence of Mott-insulating behavior
and negative magneto-resistivity.

##  IV Various NLED Models

In this section, we will use Eqs. 13, 18, 24a and 24a to study the dependence
of the in-plane resistivity $R_{xx}$ on the temperature $T$, the charge
density $\rho$ and the magnetic field $h$ in some specific NLED models. For
convenience we rescale the $c_{i}$’s as $c_{1}\sim\alpha m^{2}c_{1}$ and
$c_{2}\sim\alpha^{2}m^{2}c_{2}$. The conventional Maxwell electrodynamics is
firstly presented, with a detail discussion on the $R_{xx}$ and $R_{xy}$’s
dependence on involved massive gravity coupling parameters $c_{1}$ and
$c_{2}$, the charge density $\rho$, the magnetic field $h$ and the temperature
$T$. Then the Chern-Simons $\theta$ term as an extension is introduced to
investigate the CP-violating effect. Finally, we discuss the Born-Infeld
electrodynamics and the influence of non-linear effect on the DC resistivity.
The high temperature behaviors have been discussed in Sec. III.2.3, so we will
mainly focus on the behavior of $R_{xx}$ around $T=0$ in this section.

###  IV.1 Maxwell Electrodynamics

We first consider the Maxwell electrodynamics, in which $\mathcal{L}(s,p)=s$.
From Eq. 18 we can solve out that

$A_{t}^{\prime}\left(r_{h}\right)=\frac{\rho}{r_{h}^{2}},$ (34)

and bring it into Eq. 13 we get the equation

$4\pi Tr_{h}-3r_{h}^{2}-c_{1}r_{h}-c_{2}+\frac{\rho^{2}+h^{2}}{4r_{h}^{2}}=0.$
(35)

It is notice that $c_{1}$ offers an effective temperature correction, and make
the solution complicated even at $T=0$. Although the $c_{2}$ is a constant
playing a similar role as the momentum dissipation strength in Ref.
Peng2018:EPJC , the effect on the resistivity is quite different. The effect
of $c_{1}$ and $c_{2}$ on $R_{xx}$ and $R_{xy}$ at zero temperature is
presented in Fig. 2. For $c_{1}=0$ the $R_{xx}=0$ and $R_{xy}$ is constant,
and is independent of $c_{2}$. For non-zero $c_{1}$, the $R_{xx}$ increases
and saturates, while the $R_{xy}$ decreases to zero as $c_{1}$ becomes more
negative. And a larger $|c_{2}|$ makes the surface steeper.

Fig. 2: The influence of $c_{1}$ and $c_{2}$ on $R_{xx}$ and $R_{xy}$. We take
$T=0$ and $\rho=0.3$,$h=0.2$ as an example. Fig. 3: Upper left: $R_{xx}$ as a
function of $\rho$ and $h$. Upper right: Contour plot of $R_{xx}$. Lower left:
$R_{xx}$ vs $\rho$ with $h=0,0.5,1,1.5,2$. Lower right: $MR_{xx}$ vs $h$ with
$\rho=0,0.1,0.2,0.3,0.4,0.5$. We set $T=0$, $c_{1}=-1$, and $c_{2}=0$.

In the following Figs. 3, 4 and 5 we separately investigate the resistivities
as functions of $\rho$ and $h$ under different $T$ and $c_{2}$. Because
$c_{1}$ acts as an effective temperature modification, we set $c_{1}=-1$.

First, we consider zero temperature $T=0$, and $c_{2}=0$. No Mott-insulating
behaviors and positive magneto-resistivity is observed in Fig. 3. In addition,
non-zero charge density can suppress the resistivity as more charge carriers
are introduced.

Fig. 4: Upper left: $R_{xx}$ as a function of $\rho$ and $h$. Upper right:
Contour plot of $R_{xx}$. Lower left: $R_{xx}$ vs $\rho$ with
$h=0,0.5,1,1.5,2$. Lower right: $MR_{xx}$ vs $h$ with
$\rho=0,0.1,0.2,0.3,0.4,0.5$. We set $T=0.4$, $c_{1}=-1$, and $c_{2}=0$.

In Fig. 5 the effect of $c_{2}$ at zero temperature is presented. The figures
show similar feature as those in Fig. 4. The larger $c_{2}$ also suppress the
resistivity, while the influence is not so obvious as the temperature. All the
pictures presented here do not exhibit the Mott-insulating behaviors and
negative magneto-resistivities. Then the finite temperature situation with
$T=0.4$ and $c_{2}=0$ is studied in Fig. 4, where the saddle surface is
relatively flatter than that in Fig. 3. And the magneto-resistivity is greatly
suppressed by the temperature and the differences induced by charge density
$\rho$ is smoothen out.

Fig. 5: Upper left: $R_{xx}$ as a function of $\rho$ and $h$. Upper right:
Contour plot of $R_{xx}$. Lower left: $R_{xx}$ vs $\rho$ with
$h=0,0.5,1,1.5,2$. Lower right: $MR_{xx}$ vs $h$ with
$\rho=0,0.1,0.2,0.3,0.4,0.5$. We set $T=0$, $c_{1}=-1$, and $c_{2}=-4$.

Then we present the dependence of $R_{xx}$ on $h/\rho$ and $T/\sqrt{\rho}$ for
$c_{1}=-1$. The effect of different $c_{1}$ and $c_{2}$ can be deduced by
change the temperature correspondingly, based on the analysis above. For
$h<\rho$, Fig. 6 shows that the temperature dependence of $R_{xx}$ is
monotonic, and corresponds to metallic behavior. For $h>\rho$, the $R_{xx}$
increases first and then decreases monotonically after reaching a maximum. The
insulating behavior appears at high temperatures in this case. Moreover, if we
take larger $c_{i}$, the metallic behaviors at $h>\rho$ would disappear and
the $R_{xx}$ decreases monotonically with increasing temperature. So we come
to the conclusion that increasing the magnetic field would induce a finite-
temperature transition or crossover from metallic to insulating behavior. In
the last sub-figure of Fig. 6 the influence of temperature on the magneto-
resistivity is shown. As temperature increases, the magneto-resistivity is
remarkably suppressed.

Fig. 6: Upper left: $R_{xx}$ as a function of $h/\rho$ and $T/\sqrt{\rho}$.
Upper right: Contour plot of $R_{xx}$. Lower left: $R_{xx}$ vs $T/\sqrt{\rho}$
with $h/\rho=0,0.5,1,1.5,2$. Lower right: $MR_{xx}$ vs $h$ with
$T/\sqrt{\rho}=0,0.1,0.2,0.3,0.4,0.5$. We set $T=0$, $c_{1}=-1$, and
$c_{2}=0$.

At the end of the discussion of Maxwell electrodynamics, the Hall angle
$\theta_{H}$, defined as

$\theta_{H}=\arctan\frac{\sigma_{xy}}{\sigma_{xx}},$ (36)

is shown in Fig. 7. At zero temperature for large $h$ or $\rho$ the Hall angle
get saturated, with the sign depend on those of $h$ or $\rho$. The surface is
anti-symmetric under the transformation
$\left(\rho,h\right)\to\left(\rho,-h\right)$ and
$\left(\rho,h\right)\to\left(-\rho,h\right)$. As we expect, the temperature
would evidently suppress the Hall angle.

Fig. 7: Hall angel at $T=0$ (left panel) and $T=0.4$ (right panel). $c_{1}=-1$
and $c_{2}=0$.

###  IV.2 Maxwell-Chern-Simons Electrodynamics

The Lorentz and gauge invariance do not forbid the appearance of CP-violating
Chern-Simons $\theta$ term in the electrodynamics Lagrangian

$\mathcal{L}(s,p)=s+\theta p,$ (37)

and the Chern-Simons theory is of vital importance for the both integer and
fractional quantum Hall effects in condensed matter physics Zhang1989:PRL ;
Fradkin1991:PRB ; Zhang1992:IJMPB ; DavidTong:QHE . And the value of $\theta$
can be related to the Hall conductivity in the unit of $e^{2}/\hslash$. After
taking the Chern-Simons term into consideration, Eq. 18 gives

$A_{t}^{\prime}\left(r_{h}\right)=\frac{\rho+\theta h}{r_{h}^{2}},$ (38)

and the Eq. 13 becomes

$4\pi
Tr_{h}-3r_{h}^{2}-c_{1}r_{h}-c_{2}+\frac{\left(1+\theta^{2}\right)h^{2}+2\theta
h\rho+\rho^{2}}{4r_{h}^{2}}=0.$ (39)

We then reinvestigate the dependence of DC resistivity $R_{xx}$ on $\rho$ and
$h$. As one expects, the similar saddle surface to that of Maxwell
electrodynamics is found, and the reflection asymmetry and central symmetry
due to the $\theta$ term is shown in the upper panel of Fig. 8. The more
surprising things are the appearance of Mott-insulating behavior
$\partial_{|\rho|}R_{xx}>0$ and negative magneto-resistivity $MR_{xx}<0$. At
zero field $R_{xx}$ is the even function of $\rho$, and as $|\rho|$ increases,
the value of $R_{xx}$ increases and reaches a maximum, showing the feature of
Mott-insulating behavior, and then decreases monotonically. While for finite
magnetic field, the reflection symmetry for positive and negative $\rho$ is
broken. Along the positive $\rho$ direction the behavior of $R_{xx}$ is
similar, however, for negative $\rho$ it is more complicated. as $|\rho|$
increases, the value of $R_{xx}$ decreases and reaches a minimum, and then
share similar behavior as the positive $\rho$ case.

Fig. 8: Upper left: $R_{xx}$ as a function of $h$ and $\rho$. Upper right:
Contour plot of $R_{xx}$. Lower left: $R_{xx}$ vs $\rho$ with
$h=0,0.05,0.1,0.2,0.3$. Lower right: $MR_{xx}$ vs $h$ with
$\rho=-0.6,-0.3,0,0.3,0.6$. We set $T=0$, $c_{1}=-1.5$, $c_{2}=0$ and
$\theta=2$.

We then further study how the value of $\theta$ affect the Mott-insulating
behavior in Fig. 9. To clearly show the region $\partial_{|\rho|}R_{xx}>0$ we
set other region’s value to be zero. Similar results have been obtained as in
Ref. Peng2018:EPJC . At zero field there is no Mott-insulating behavior for
$\theta=1$. A larger $\theta$ makes it possible even with $h=0$. On the other
hand, the Mott-insulating region becomes larger but $\partial_{|\rho|}R_{xx}$
becomes smaller as $\theta$ increases.

Fig. 9: Mott-insulating region for $\theta=1,2,3$ in the parameter plane for
Maxwell-Chern-Simons electrodynamics at zero temperature. We set $c_{1}=-1.5$
and $c_{2}=0$.

The negative magneto-resistivity revealed in Fig. 8 is also shown in the
$\rho-h$ parameter plane in Fig. 10, from which we can see that the negative
magneto-resistivity emerges in the finite interval of $h/\rho$ of about
$\left[-1.2,0\right]$. The positive magneto-resistivity region’s value is set
to be zero as well. At larger magnetic field, the negative magneto-resistivity
phenomenon disappears, and the magneto-resistivity increases almost linearly
with $h$. For zero density the negative magneto-resistivity does not occur.

Fig. 10: Negative magneto-resistivity region for $\theta=3$ in the parameter
plane for Maxwell-Chern-Simons electrodynamics at zero temperature. We set
$c_{1}=-1.5$ and $c_{2}=0$.

Finally we show the behavior of Hall resistivity $R_{xy}$ and Hall
conductivity $\sigma_{xy}$ in Fig. 11. The negative Hall resistivity comes
from the negative transverse conductivity $\sigma_{xy}$ depending on the
direction of magnetic field $h$ and the type of charge carriers. From the
figure we see that at zero field the transverse conductivity $\sigma_{xy}$ is
exactly $-\theta$ in regardless of charge density $\rho$, which agrees with
the scenario in quantum Hall physics Zhang1992:IJMPB ; Qi2011:RMP . At finite
magnetic field, for positive $\rho$ the Hall conductivity will decrease to
zero, and then changes its sign, while for negative $\rho$ the Hall
conductivity remains negative. And we can also study the impact of magnetic
field on the Hall conductivity with fixed $\rho$. All the transverse
conductivity for various $\rho$ as strong field tend to be zero as the result
of localization. For positive $\rho$, in the positive magnetic field, the Hall
conductivity decreases to zero, then changes its sign and increases to a
maximum, and finally decreases monotonically to zero from the positive side.
And in the negative magnetic field, the Hall conductivity increases to a
negative maximum, and then decreases to zero from the negative side. The case
of negative $\rho$ can be known by simply making transformation
$\left(\rho,h\right)\to\left(-\rho,-h\right)$ in the discussion above.

Fig. 11: Upper left: $R_{xy}$ as a function of $h$ and $\rho$. Upper right:
Contour plot of $R_{xy}$. Lower left: $\sigma_{xy}$ vs $\rho$ with
$h=0,0.05,0.1,0.2,0.3$. Lower right: $\sigma_{xy}$ vs $h$ with
$\rho=-6,-3,0,3,6$. We set $T=0$, $c_{1}=-1.5$, $c_{2}=0$ and $\theta=2$.

###  IV.3 Born-Infeld Electrodynamics

The Born-Infeld electrodynamics is described by a square-root Lagrangian
Born1933:Nat ; Born1934:PRSLA

$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^{2}p^{2}}\right),$ (40)

where the coupling parameter $a=\left(2\pi\alpha^{\prime}\right)^{2}$ relates
to the Regge slope $\alpha^{\prime}$. It is believed that such a NLED governs
the dynamics of electromagnetic fields on D-branes. If we take the zero-slope
limit $\alpha^{\prime}\to 0$, the Maxwell Lagrangian is recovered

$\mathcal{L}(s,p)=s+\mathcal{O}(a).$ (41)

The Born-Infeld electrodynamics takes the advantage of eliminating the
divergence of electrostatic self-energy and incorporating maximal electric
fields zwiebach2004:CUP . This can be seen from the solution of Eq. 18

$A_{t}^{\prime}\left(r\right)=\frac{\rho}{\sqrt{a\left(\rho^{2}+h^{2}\right)+r^{4}}},$
(42)

which is finite when $r\to 0$. The $r_{h}$ is solved form

$4\pi
r_{h}T-3r_{h}^{2}-c_{1}r_{h}-c_{2}+\frac{\rho^{2}}{2\sqrt{a\left(h^{2}+\rho^{2}\right)+r_{h}^{4}}}+\frac{1}{2a}\left(\sqrt{\frac{\left(ah^{2}+r_{h}^{4}\right)^{2}}{a\left(h^{2}+\rho^{2}\right)+r_{h}^{4}}}-r_{h}^{2}\right)=0.$
(43)

Fig. 12: $R_{xx}$ as a function of $h$ and $\rho$ and some intersection curves
for fixed $h$ and $\rho$ for $a=-0.4$ (upper panel) and $a=-1$ (lower panel).
The surface ends at some place where $h$ and $\rho$ touch the upper bound.

The DC resistivity is then studied in the framework of Born-Infeld
electrodynamics and the results are shown in Fig. 13. For positive $a$ the
behavior of $R_{xx}$ is similar to the Maxwell case Peng2018:EPJC ;
Cremonini2017:JHEP , and the negative $a$ can bring in more interesting
phenomenon. However, for negative $a$ the Eq. 42 suffers a singularity
$r=r_{s}\equiv\sqrt[1/4]{\left|a\right|\left(h^{2}+\rho^{2}\right)}$, and the
physical solution requires that $r_{h}>r_{s}$, setting an upper bound for
$h^{2}+\rho^{2}$ Peng2018:EPJC . We then present two cases of $a=-0.4$ and
$a=-1$, respectively. For $a=-0.4$, We the saddle surface is similar to the
previous results, and there is no Mott-insulating behaviors and negative
magneto-resistivity. However, if one increases the absolute value of the
negative $a$, the region admitting physical solution for $R_{xx}$ shrinks and
is truncated at the upper bound of $h^{2}+\rho^{2}$. For zero and finite small
magnetic field, the Mott-insulating behavior is absent, while for a larger
field the Mott-insulating behavior emerges. And for a relatively small $\rho$,
the negative magneto-resistivity is present, and increasing $\rho$ can destroy
it. To expose the role of $a$ we expand the interaction between electrons to
the first order Peng2018:EPJC

$F(a)=\rho
A_{t}^{\prime}\left(r\right)\sim\frac{\rho^{2}}{r^{2}}\left[1-\frac{a}{2r^{4}}\left(h^{2}+\rho^{2}\right)\right]+\mathcal{O}(a^{2}),$
(44)

in which the leading order is the familiar Coulomb interaction, and the
nonlinearity parameter serves as an effective modification. A positive $a$
suppress the interaction and we do not expect different phenomenon compared
with the Maxwell electrodynamics. However, for a negative $a$ the interaction
is enhanced at $r_{h}$

$\frac{F(a)}{F(0)}=1+\frac{1}{2}\left(\frac{r_{s}}{r_{h}}\right)^{4},$ (45)

and we expect that the NLED model with negative $a$ can grasp some features of
strongly correlated systems. Besides, in the leading order expansion Eq. 41
the Chern-Simons term does not appear, and can be deduced from Eq. 33 that the
system will not exhibit Mott-insulating behavior and negative magneto-
resistivity. Thus we conclude that the Born-Infeld electrodynamics provides a
new mechanism different from the Chern-Simons theory to give rise to the Mott-
insulating behavior and negative magneto-resistivity, and the temperature can
induce a transition at finite temperature.

One can also consider another kind of construction of square-root Lagrangian
zwiebach2004:CUP

$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as}\right),$ (46)

which has the same leading order expansion as Eq. 41, and will re-derive the
results of Born-Infeld electrodynamics in the presence of zero field. The
solution of Eq. 18 gives

$A_{t}^{\prime}\left(r\right)=\frac{\rho}{r^{2}}\sqrt{\frac{ah^{2}+r^{4}}{a\rho^{2}+r^{4}}},$
(47)

which can be deduced that the effective interaction is

$F=\rho
A_{t}^{\prime}\left(r\right)\sim\frac{\rho^{2}}{r^{2}}\left[1-\frac{a}{2r^{4}}\left(\rho^{2}-h^{2}\right)\right]+\mathcal{O}(a^{2}).$
(48)

The radius equation is

$4\pi
r_{h}T-3r_{h}^{2}-c_{1}r_{h}-c_{2}+\frac{h^{2}}{2r_{h}^{2}}+\frac{r_{h}^{2}}{2a}\left(\sqrt{\frac{a\rho^{2}+r_{h}^{4}}{ah^{2}+r_{h}^{4}}}-1\right)=0.$
(49)

We re-check the $R_{xx}$ in the square-root electrodynamics for negative $a$.
For $a=-0.4$ the saddle surface is similar to the previous results, and both
the zero field and finite field no Mott insulating behaviors are shown. And
for fixed $\rho$, it is surprising that for larger $\rho$, the transition from
positive magneto-resistivity to negative magneto-resistivity is observed for
about $\rho>h$. We then studied the case of larger negative $a$. To our
surprise, the saddle surface changes its direction and looks as if it rotates
$90^{\circ}$. We find that the Mott-insulating behavior appears and becomes
significant at large $\rho$, and negative magneto-resistivity is observed for
various $\rho$.

Fig. 13: $R_{xy}$ as a function of $h$ and $\rho$ and some intersection curves
for fixed $h$ and $\rho$ for $a=-0.4$ (upper panel) and $a=-1$ (lower panel).

##  V Conclusion

In this work the black brane solution of four-dimensional massive gravity with
backreacted NLED is obtained, and with the dictionary of gauge/gravity
duality, the transport properties of the strongly correlated systems in the
presence of finite magnetic field in 2+1-dimensional boundary is studied. In
our holographic setup the bulk geometry and NLED field are perturbed, and the
DC conductivities are obtained in the linear response regime. Then some
general properties are obtained in various limit, which agrees well with the
previous work. To make it concrete, we present the study of the conventional
Maxwell electrodynamics, the topological non-trivial Maxwell-Chern-Simons
electrodynamics, and the Born-Infeld electrodynamics with string-theoretical
correction taken into consideration. We concentrate on two interesting
phenomena, i.e., the Mott-insulating behavior and negative magneto-
resistivity, and results at zero temperature are summarized in Tab. 1.

| Lagrangian | Parameter | Mott-insulating behavior | Negative magneto-resistivity
---|---|---|---|---
Maxwell | $s$ |  | No | No
Maxwell- | $s+\theta p$ | $\theta$ | See Fig. 9 | See Fig. 10
Chern-
Simons
Born-Infeld | $\dfrac{1}{a}\left(1-\sqrt{1-2as-a^{2}p^{2}}\right)$ | $a>0$ | No | No
$a=-0.4$ | No | No
$a=-1$ | Finite $h$ and small $\rho$. | Finite $h$ and small $\rho$.
See Fig. 12 | See Fig. 12
Square | $\dfrac{1}{a}\left(1-\sqrt{1-2as}\right)$ | $a>0$ | No | No
$a=-0.4$ | No | Larger $\rho$. See Fig. 13
$a=-1$ | Larger $\rho$. See Fig. 13 | Yes
Tab. 1: Conclusion of Mott-insulating behavior and negative magneto-
resistivity for various NLED models at zero temperature $T=0$.

The massive gravity coupling parameters’ influence on the in-plane resistivity
is compared with the temperature, where we find the $c_{1}$ behaves as the
effective temperature correction, and non-zero one can lead to the DC
conductivity. While the $c_{2}$ has less significant effect on $R_{xx}$ than
$c_{1}$. Moreover, the dependence on $\rho$ and $h$ is shown and we find the
field can induce the metal-insulator transition or Mott-insulating behavior
and negative magneto-resistivity. Two different mechanism, the Chern-Simons
term, and the negative nonlinearity parameter are proved that can give rise to
Mott-insulating behavior and negative magneto-resistivity. We hope our work
could explain some experimental phenomenon in strongly correlated systems.

## Acknowledgment

We are grateful to thank Peng Wang for useful discussions. This work is
supported by NSFC (Grant No.11947408).

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