Negative magnetoresistivity in holography
Revised: June
magnetoresistivity in holography
YaWen Sun 0 2
Qing Yang 0 1 2
0 C/ Nicolas Cabrera 13
1 Institute of Theoretical Physics, Chinese Academy of Sciences
2 Instituto de F sica Teorica UAM/CSIC, Universidad Autonoma de Madrid
3 15 , Cantoblanco, 28049 Madrid , Spain
Negative magnetoresistivity is a special magnetotransport property associated with chiral anomaly in four dimensional chiral anomalous systems, which refers to the transport behavior that the DC longitudinal magnetoresistivity decreases with increasing magnetic eld. We calculate the longitudinal magnetoconductivity in the presence of backreactions of the magnetic eld to gravity in holographic zero charge and axial charge density systems with and without axial charge dissipation. In the absence of axial charge dissipation, we nd that the quantum critical conductivity grows with increasing magnetic eld when the backreaction strength is larger than a critical value, in contrast to the monotonically decreasing behavior of quantum critical conductivity in the probe limit. With axial charge dissipation, we nd the negative magnetoresistivity behavior. The DC longitudinal magnetoconductivity scales as B in the large magnetic eld limit, which deviates from the exact B2 scaling of the probe limit result. In both cases, the small frequency longitudinal magnetoconductivity still agrees with the formula obtained from the hydrodynamic linear response theory, even in the large magnetic eld limit. condensed matter physics (AdS/CMT), Anomalies in Field and String Theories
AdSCFT Correspondence; Gaugegravity correspondence; Holography and

Negative
HJEP09(216)
1 Introduction
3
4
2.1
2.2
2.3
3.1
3.2
Background solution at nite temperature
Longitudinal magnetoconductivity
Zero temperature
Adding axial charge dissipation
Background solutions at nite temperature
Longitudinal magnetoconductivity
Conclusion and discussion 2
U(1)A holographic system with magnetic
without axial charge dissipation
A Zero temperature background solutions with axial charge dissipation
several associated anomalous transport behaviors, including negative magnetoresistivity [3],
anomalous Hall e ect, chiral magnetic e ect [4], chiral vortical e ect [5, 6], etc. Chiral
magnetic and vortical e ects have been studied extensively in holographic chiral anomalous
systems [5{13] via the gauge/gravity duality (see [14{16] for recent reviews). Meanwhile,
anomalous Hall e ect was proposed as an order parameter in the realization of a holographic
quantum phase transition between a topological and a trivial semimetal state [17, 18].
Negative magnetoresistivity refers to the anomalous transport behavior of the
longitudinal DC magnetoresistivity decreasing with increasing magnetic eld, or the longitudinal
DC magnetoconductivity increasing with increasing magnetic eld in the presence of chiral
anomaly, in contrast to the positive magnetoresistivity behavior for normal metal [19].
Negative magnetoresistivity has been observed in several experiments during the last several
years, including [20{26].
Using a linear response theory in the hydrodynamic regime [27, 28], it was shown
in [29] (and later generalized to Lifshitz spacetime in [30]) that the longitudinal DC
magnetoconductivity in the presence of chiral anomaly is divergent, even in the zero density
limit. Energy, momentum and axial charge dissipations are all needed to make it nite.
At the zero charge and axial charge density limit, only axia charge dissipation is needed
{ 1 {
to have a nite DC longitudinal magnetoconductivity. At weak coupling, from the kinetic
theory [3, 31{33], it was calculated that at small B
toconductivity has a B2 behavior while at large B
T 2, the DC longitudinal magne
T 2, it goes linearly in B. Negative
magnetoresistivity behavior was also found in strongly coupled holographic chiral
anomalous systems [29, 34{36]. In [29] we found that when there is no axial charge dissipation, the
longitudinal magnetoconductivity indeed has a pole at ! = 0 which leads to a function
in the real part of the longitudinal magnetoconductivity. The real part of the conductivity
at zero frequency excluding the function is a monotonically decreasing function of B and
decreases from
T at B = 0 to 0 at B = 1. The coe cient in front of i=! of the imaginary
part is a monotonically increasing function of B and increases from B2 at B = 0 to linear
system coincides with the weakly coupled result, and was also found in experiments [20].
There are at least two ways to introduce axial charge dissipations into the holographic
system [36]. The rst is by explicitly breaking the U(
1
)A symmetry with a U(
1
)A charged
scalar in the bulk which has a nonzero source at the boundary. The second way is to make
the U(
1
)A gauge eld massive [35, 37, 38] so that there is no U(
1
)A gauge symmetry in the
bulk anymore. The two ways are in fact equivalent in the following sense: the equations
for the perturbations are the same for the two mechanisms at zero charge and axial charge
density after choosing suitable gauges and substituting the mass of the U(
1
)A gauge eld
by the background scalar eld. In fact the massive U(
1
)A case is one special limit of the
explicit breaking case where the mass of the scalar eld is chosen to be zero. With explicit
breaking of the axial charge conservation symmetry, we found that the DC conductivity
is composed of two terms and the nonconstant term has an exact B2 dependence on
the magnetic eld B. This qualitatively agrees with the experimental result of [21]. In
both of the holographic zero density systems with and without axial charge dissipation,
the hydrodynamic results agree with the holographic results as long as 5 is large enough
to stay in the hydrodynamic regime while B can be very large which is outside of the
hydrodynamic regime.
Previous study of negative magnetoresistivity in holographic chiral anomalous systems
focused on the probe limit, where the magnetic eld cannot be very large so that
backreactions of the magnetic eld to gravity are not important. To study the large B behavior
more accurately, we need to take into account the backreaction e ects of the magnetic
eld. In this paper, we study the holographic zero charge and axial charge density systems
with the backreactions of the magnetic
eld. This is also a rst step towards the study
of magnetotransport behavior in holographic
nite charge and axial charge density chiral
anomalous systems, where backreactions of guage elds should always be considered. In
this paper we consider both the cases without and with axial charge dissipation for the
zero density system. We nd that in the case without axial charge dissipation, the small
frequency longitudinal magnetoconductivity deviates from the probe limit at larger B=T 2
region. At B=T 2 !
1, the imaginary part of the longitudinal magnetoconductivity
coincides with the probe limit result while the real part diverges for backreaction strength
larger than a critical value, in contrast to being zero in the probe limit. In the case with
axial charge dissipation, at large B=T 2 the DC longitudinal magnetoconductivity becomes
linear in B, which deviates from the exact B2 behavior for the probe limit.
{ 2 {
The rest of the paper is organized as follows. In section 2, we will calculate the
longitudinal magnetoconductivity with backreactions to the gravity without axial charge
dissipation at both
nite and zero temperature. In section 3, we add axial charge
dissipations and calculate the longitudinal magnetoconductivity at nite temperature. Section 4
is devoted to conclusion and discussions.
U(
1
)A holographic system with magnetic
without axial charge dissipation
In this section, we calculate the magnetoconductivity in the presence of backreactions of
the magnetic eld to the gravity at both zero charge density and zero axial charge density
without introducing any dissipations and compare the result with the probe case, especially
at the large magnetic eld limit. We will consider the following action1
1
4e2 (F 2 +F 2)+
3
A
F F
+3F
F
S =
Z
d x
5 p
g
1
where the gauge elds V and A
correspond to the vector and axial U(
1
) currents
respectively.
is the Newton constant, e is the Maxwell coupling constant and
is the
ChernSimons coupling constant. Here we did not introduce any dissipation terms and
according to the hydrodynamic formula in [29] we will get a function at zero frequency
which leads to an in nite DC magnetoconductivity.
The equations of motion are 1 2
g
R
R
12
2
1We have set the curvature scale L = 1.
ds2 =
{ 3 {
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
The background equations of motion become
f
f +
f
+
2n0
n
= 2e22 is a dimensionless constant, which represents the strength of backreactions.
For this system, there are three second order equations and one rst order equation coming
from the Einstein's equations of motion and only three of them are independent. For
convenience in numerics, we choose the three equations above to eliminate the second
derivative of h in the equations: (2.8) is a linear combination of the tt and xx components
of the Einstein's equations of motion, (2.9) is the zz component and (2.10) is the rr
component, which is a rst order equation. Note that in the regime of classical gravity,
1, while can be arbitrarily large or small depending on the ratio of =e. At zero charge and axial charge density, these equations of motion coincide with the equations for the case with only one U(1) gauge eld [39, 40].
One exact solution to this system is the BTZ
R2 solution
with the AdS3 radius being 1=p3. However, we cannot nd irrelevant deformations to
ow this solution to asymptotic AdS5 geometry and only marginal deformations can be
found, which render the near horizon geometry no longer BTZ
R2 any more. Thus at
nite temperature, it is more convenient to directly expand the solutions at the horizon as
follows
The asymptotic AdS5 boundary behaviors of the metric elds are
f ' r2 1 +
n ' r2 1 +
h ' r2 1 +
f 2
2f0 + r02
r
f 2
2f0 + r02 +
r
f 2
2f0 + r02
r
B2 ln r
6r4
B2 ln r
12r4
B2 ln r
6r4
M
r4 +
+ nr42 +
2rn42 +
;
where M and n2 are parameters which are determined by the horizon data and f0 can be
eliminated by performing a coordinate translation r ! r
f0. This coordinate
transformation changes the position of the horizon but does not change the temperature of the
{ 4 {
geometry. In this system there is a conserved quantity along the radial direction f0hnp h0fn
associated with a scaling symmetry in the background equations of motion. We can also
h
derive this radially conserved quantity from a linear combination of the tt and zz
components of the Einstein's equations of motion. From this conserved quantity we nd that at
zero temperature n2 = M=2, and at nonzero temperature 2n2 = M
In numerics, n0 and h0 can be xed to arbitrary values and
nally need to be rescaled
according to the boundary coe cients in front of r2 in n and h. In this case, the physical
value of the magnetic eld B will also be rescaled and become di erent from the input value
of B. In our numerics we x n0 and h0 to numerically convenient values for simplicity,
and we can read out the physical value of the magnetic eld from the boundary values
horizon, we can integrate the equations to the boundary and produce background solutions
characterized by the temperature T and the physical magnetic eld B.
2.2
Longitudinal magnetoconductivity
To calculate the longitudinal magnetoconductivity in the backreacted geometry above, we
consider perturbations
Vz = vze i!t;
At = ate i!t on the background solutions. As we
are studying the system at zero charge and axial charge density, these perturbations do
not couple to the metric perturbations. The equations for vz and at are
and we can simplify them into one single equation for vz
At the boundary, the asymptotic behavior for vz is
vz = 0:
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
and the de nition of the conductivity is [41]
current, whose de nition can be found in [29, 42].
Without an exact background solution, it is not possible to solve this equation
analytically. We instead solve it numerically by integrating the equation from the horizon to the
boundary with infalling boundary condition at the horizon. As shown in [29], in general for
a chiral anomalous system we need to impose three kinds of dissipations in order to make
{ 5 {
sipations. Here as a special case of zero charge and axial charge density, only the axial
charge dissipation is needed for a nite DC magnetoconductivity, which we will consider in
the next section. In this section, with no axial charge dissipation mechanism, the imaginary
part of the longitudinal magnetoconductivity behaves as 1=! at ! ! 0 and the real part
consequently gets a function at ! = 0, which means that the DC magnetoconductivity
is divergent. The longitudinal magnetoconductivity takes the form of
zz =
E +
i
w c0
(2.21)
at low frequency, where E is the quantum critical conductivity.
Di erent from the probe limit, the backreacted background solutions depend on the
value of
B but not
while the perturbations only depend on
B so the
nal result of
the magnetoconductivity will depend on all three parameters of ,
B=T 2 increases, the e ects of backreactions will become more apparent.
In gure 1, we show the real part and the imaginary part of the AC
magnetoconductivity as a function of !=T for di erent values of B=T 2 separately at xed
see from the coe cient in front of 1=! in the imaginary part, the height of the function
grows as a function of B=T 2. As B increases, the gap region in the real part becomes wider
and wider as can be seen from the gure, which is consistent with the fact that weight is
transferred to the ! ! 0 region as B increases. At larger values of !=T quasinormal modes
start to show up which lead to peaks in the real part of zz and as B increases, more and
higher peaks will arise. This behavior was also found in the axial charge dissipation system
and B=T 2. As
or
in the probe limit [35].
In
gure 2, we show the real part of the DC magnetoconductivity (excluding the
function), i.e. the quantum critical conductivity
E , as a function of B=T 2 for various
and . In the left gure, we x
= 1 and choose
= 1; 50; 100 respectively.
In the right gure, we x = 1 and choose = 0:5; 1; 5 respectively. When B = 0,
Re zz(0) =
T , which is universal regardless of the value of
or . From the left gure in
{ 6 {
with increasing B, which deviates from the probe case, but the qualitative behavior is the
same and as we will see later Re zz(0) nally vanishes at B ! 1 as long as
is not too
large. From the right gure we can see that at
= 1, Re zz(0) decreases monotonically as
B increases at values of not too small.
However, when
is very large while
very small, Re zz(0) would start to grow
monotonically as B increases. Figure 3 shows that this would be the case for
This behavior is related to the divergence of Re zz(! ! 0)=T for small
limit B=T 2 ! 1 as we will explain in the next subsection for the zero temperature limit.
=16 at the
< p
Note that in this gure, the horizontal axis is
zz(0) depends on both
and B=T 2 separately.
B=T 2 instead of B=T 2, and this shows that
In gure 4, we show the imaginary part of the DC magnetoconductivity Im zz(! ! 0)
as a function of B for various values of
and . Because there is no dissipation and
Im zz(0) behaves as 1=! near ! ! 0, we plot the coe cient in front of 1=! in Im zz(0) in
the gure. When
increases the deviation from the probe limit becomes more apparent,
but similar to the real part, the qualitative behavior is still the same as the probe limit.
{ 7 {
= 0:5; 1; 5.
longitudinal magnetoconductivity for
In gure 5, we zoom in at the small B=T 2 region. At small B=T 2, as shown in gure 5,
!Im zz(! ! 0) is proportional to B2 and the coe cient does not depend on
as we
will see from the hydrodynamic formula below, which means that it is the same as in the
probe case at leading order. At large B=T 2, !Im zz(! ! 0) is linear in B. This result
is qualitatively the same as in the probe case, however, the quantitative di erence due to
backreaction becomes apparent for large B=T 2 and large . To investigate the large B
region more carefully we will study the system at zero temperature which corresponds to
the B=T 2 ! 1 limit in the next subsection. Surprisingly we will see that at B=T 2 ! 1
the behavior of zz(! ! 0) goes back to the probe result. From
gure 3 we can see that at
when the real part starts to diverge at B=T 2 ! 1, the leading order
in ! behavior of the imaginary part remains qualitatively the same as those with other
{ 8 {
As shown in [29] the hydrodynamic formula for the small ! longitudinal
magnetoconductivity at both zero charge and axial charge density, which the holographic probe system
obeys is
where
susceptibility. To calculate
5, we start from the following equation for at which can be
obtained from equations (2.16) for ! ! 0
zz =
E +
i (8 B)2
!
5
;
p
Here an integration constant from the equation of motion for vz has been chosen to be
zero from the boundary conditions at the horizon. At nite temperature and small B=T 2,
we can solve this equation order by order in B=T 2 and the leading order contribution to
5 only depends on B from the background small
B=T 2 corrections, which means that
at small B=T 2 and small
B=T 2, the leading order contribution to !Im zz(0) is (28 2BT)22 ,
which is the same as the probe limit and is subject to
B=T 2 and B2=T 4 order corrections.
This is also consistent with our numerical ndings. This hydrodynamic formula should be
valid in the hydrodynamic regime:B=T 2
1, which is indeed the case as in this regime
the leading order result is the same as the probe limit result. In the following subsection
we will see that even at zero temperature, this hydrodynamic formula still agrees with the
holographic result.
2.3
Zero temperature
The zero temperature limit of this system is equivalent to the large B=T 2 limit at nite
temperature. Due to numerical di culties at large B=T 2 in the
nite temperature
calculation, in this subsection we study the system at exact zero temperature to approach
the B=T 2
R2 [39, 43, 44]. We need irrelevant perturbations at the horizon to
ow this solution to
!
1 limit. At zero temperature, an exact solution to this system is AdS3
asymptotic AdS5 solutions. The near horizon solution with irrelevant perturbations is
ds2 =
3r2(1 + f1r +
)dt2 +
B
p
p
" are higher order corrections and f1 has to be negative in order to ow to
asymptotic AdS5 solutions. The value of
can be solved from the equations of motion
for the perburbations to be
3). As B can always be absorbed into rescaling
of x and y, and f1 can also be rescaled to
1 by rescaling r, it seems that we can only
get one e ective value of B at zero temperature. However, when we scale r to rescale f1
to
1, the boundary behavior of gxx will change accordingly with a di erent coe cient
in front of r2 and leads to a di erent physical value of B: the physical magnetic eld for
f1 =
1 is ( f1)2= times the physical magnetic eld for other values of f1. Thus tuning
{ 9 {
the near horizon parameter f1 will give solutions with di erent values of B, though these
solutions are in fact equivalent physically as B is the only scale in the background solutions
at zero temperature.
From the background equations it looks like that the value of
only a ects the details
of the one to one correspondence between the horizon initial value of f1 and the nal value
of the physical magnetic eld, however, with di erent values of
the background geometry
is di erent even for the same physical magnetic eld. Thus , which does not appear in the
equations of perturbations, would still a ect the conductivity at zero temperature. The
only dimensionful quantity at zero temperature is B, thus from dimensional analysis, we
know that Re zz(0)
geometry depends on
p
B while !Im zz(0)
B. Due to the fact that the background
while the equations of perturbations only depend on , the behavior
=
p : for =
p
Re zz(0)=p
as !32 =
p
1
.
of zz(! ! 0) is expected to depend on both
and . As we will explain below, there
are three di erent kinds of qualitative behavior of zz(! ! 0) depending on the value of
= 1=32, Re zz(0)=p
B = 0 at leading order in !, and for =
B is a constant at zero frequency, for =
> 1=32,
p
< 1=32, Re zz(0)=pB diverges
p
This characterization of the three di erent kinds of qualitative behaviors can be derived
from the IR equations using the near far matching method [45] as follows. The equation
of motion for vz(r) reduces to the following in the IR region r
p
B
vz00 + z +
v0
r
!2
9r4
The infalling solution to this equation is the Bessel Kfunction K 1p6
3ir! . Expanding
16
coe cient in front of the two linearly independent solutions vz(
1
)j! r pB = r p +
this function at the boundary of the IR region !
r
B we can get the relative
and
p
16
vz(2)j! r pB = r p +
solutions scales as ! p
32
16
r p
and r p
16
of this region. It turns out that the relative coe cient of the two
with a complex coe cient. The two linearly independent solutions
are both real so the boundary coe cients vz(10;2) and vz1
(1;2) associated with
these two solutions are all real. Substituting these into the formula for zz in (2.20) it is
easy to see that the leading order in the imaginary part Im zz(! ! 0)
leading order contribution to the real part scales as Re zz(! ! 0)
explicitly the scaling behavior of the small frequency longitudinal magnetoconductivity at
!32 =p
2vz(11)
i!vz(10) while the
1. Thus more
zero temperature is the following.
For 3p2 < 1, Re zz(! ! 0)
For 3p2 = 1, Re zz(! ! 0)
c1( ; ) B p
! 32 1 and Im zz(! ! 0)
p
B
B! d( ; );
c2( ; ) B and Im zz(! ! 0)
B! d( ; );
For 3p2 > 1, Re zz(! ! 0)
0 and Im zz(! ! 0)
B! d( ; ),
where c1;2( ; ) and d( ; ) are constants which might depend on
depend on ! or B. The condition that this behavior only exists for p
and
while do not
> 32 shows that
B for large values of compared to .
p
this is a backreaction e ect which cannot be seen in the probe limit. This also explains the
strange monotonically increasing behavior for the nite temperature Re zz(! ! 0) with
Note that to compare this result with the nite temperature case of last subsection,
we should focus on the B scaling instead of the ! scaling behavior because in numerics we
always have a small while nonvanishing value of !. The B scaling behavior for the real
part of the longitudinal magnetoconductivity is
Re zz(! ! 0)
< 1, the real part of the nite temperature DC longitudinal
magnetoconductivity would diverge at B=T 2 ! 1, which is consistent with the numeric
result of last subsection.
We con rm this analytic nding with numerics. Numerically we obtain the zero
temperature background solutions with di erent values of magnetic eld by choosing di erent
initial values of f1 at the horizon. Then we solve the equation of motion for vz with infalling
boundary condition at the horizon and read the boundary coe cients of vz0 and vz1 with
the solutions for vz.
For 3p2
> 1, we numerically checked that for a continuous range of parameters
and ,
region p
Re zz(! ! 0)
0 and Im zz(! ! 0)
8 B! , which coincides with the large B=T 2 probe
limit result at leading order in !. This is also consistent with the large B=T 2 behavior
of the backreacted
nite temperature results in this parameter region. This numerical
nding shows that in the small
< 32
the result for the DC longitudinal
magnetoconductivity still agrees with the probe limit result quantitatively at leading order
in !. However, at subleading orders of ! in both the real and imaginary parts of zz,
Then we choose
Re zz(! ! 0) indeed scales as c1pB p!
B
= 5=32. The imaginary part Im zz(! ! 0)
scales as c2pB where c2 is around 1:21 at
At
B! d( ; ) where d is 8 for this set of values of
and . We expect
and
the leading order in ! behavior of Im zz(! ! 0) is always
. The zero temperature divergence of the quantum critical conductivity was also found
in Einsteindilaton systems at zero density when there is no chiral anomaly [46].
We can now check if the hydrodynamic formula is still valid at zero temperature, which
is already out of the hydrodynamic regime. At zero temperature the equation for at is still
the same as the nite temperature one of (2.23) and we can solve it numerically on the zero
temperature background. From dimensional analysis 5
B, and numerically we nd that
for any value of , which is larger or smaller or equal to (32 )2, we always have 5 = 8 B.
By substituting
5 into the hydrodynamic formula we
nd that this formula still gives
the exact holographic result at leading order in ! even at T = 0 which is outside the
hydrodynamic regime. The fact that 5 = 8 B for all values of
and
is also consistent
with that the imaginary part of zz is always 8 B=! at leading order in !. At the same
time, the explicit value of the quantum critical conductivity cannot be obtained from the
hydrodynamic formula.
The results of this subsection show that in holography we can nd a parameter region
the quantum critical conductivity always vanishes at B=T 2
in which the real part of the longitudinal DC magnetoconductivity, i.e. the quantum critical
conductivity diverges at B=T 2 ! 1, in contrast to the previous probe limit result where
Adding axial charge dissipation
In this section, we add axial charge dissipation to the backreacted zero density system of
last section to get a
nite DC longitudinal magnetoconductivity. As shown in [36], there
are two simple mechanisms to encode axial charge dissipation: one is to introduce a mass
for the UA(
1
) gauge eld and the other is to source the system by an axially charged scalar
eld. However, for the massive UA(
1
) gauge
eld case, there exists a problem that the
scaling dimension of the axial current has changed, so in this section we use the second
way to introduce the axial charge dissipation. The massive scalar corresponds to a massive
operator which can be interpreted as the mass of the dual fermions. We will consider the
HJEP09(216)
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
following action2
S =
Z
d x
5 p
g
with
1
iqA
where the gauge elds V and A
correspond to the vector and axial U(
1
) currents
respectively and
is a complex scalar eld with mass m. As in [36], we choose m2 =
3
throughout this paper to match the dimension of the dual massive operator with the
dimension of the weak coupling limit.
The equations of motion are
R
R
12
1
2
g
2
e2 F
F
r F
+
F
+ F
F
2
e2
F
2
F
2We have set the curvature scale L = 1.
and
The equations become
We solve this system at nite temperature with a nite magnetic eld B at zero charge
and axial charge density. The assumption for the background solutions is
f
+
f
n0
n
+
perature solutions to this system, so we numerically integrate the equations to produce
background solutions. The leading order near horizon expansion of the elds are
and
" denotes higher order corrections which can be solved order by order given
the leading order parameters. The horizon radius can always be rescaled to r0 = 1. The
free parameters are the temperature T , the e ective physical magnetic eld related to n0
or the input value B~ and the initial value 0 which is related to the boundary value of .
At the asymptotic AdS5 boundary the leading order expansions of the elds are
M
' r
1
f ' r
n ' r
h ' r
B2
6
B2
12
B2
6
+
+
+
ln r
r3
3
2
6
6
6
M 4 ln r
M 4 ln r
M 4 ln r
r4 +
r4 +
r4 +
+
r+3 +
;
;
;
;
where M corresponds to the source of the axial charged scalar eld
and
+ gives the
response to the source. The parameter f0 can be set to zero by a coordinate transformation
r ! r
f0, which does not change the temperature. With the horizon parameters T , n0
and 0, we can integrate the system to the boundary to get solutions at temperature T ,
with magnetic eld B and scalar source M .
3.2
To calculate the longitudinal magnetoconductivity we consider perturbations
vz(r)e i!t, At = at(r)e i!t, Ar = ar(r)e i!t,
=
1(r; t) + i 2(r; t) =
Vz =
1(r)e i!t +
i 2(r)e i!t on the background above, where 1 decouples from other modes. As discussed
extensively in [36], there are two kinds of gauge choices we can choose, ar = 0 or 2 = 0,
based on the fact that the equations in the bulk are invariant under the transformation
A
!
2 !
2 + q
+
n0
h0
2h
n0
n
+
h0
2h
n h
+
!2vz +
f
8 Bpha0t = 0;
f n
2iq! 2
= 0;
and the equation for ar is
8i B!vz
f 2nph
i!a0t
+
2q 0
f
At the horizon, we have the ingoing boundary conditions
vz ' (r
r0) 4 T v(0) +
v(0) 5B2 + 2n20( 12 + ms2 (
20
)) ( 2 iT + !)!
192 2T 2n20(2 T
i!)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
(3.25)
(3.26)
(3.27)
(3.29)
i!
p
i!
i!
where v(0) and a(
1
) are two arbitrary constants.
At the boundary we have the following expansions
With the gauge ar = 0, the equations for these perturbations are
zz =
i!vz0
i!
2
:
Ba(
1
) h0(4 T
T n0(2 T
i!)
i!) !
(r
r0) + : : :
!
!
at ' (r
r0) 4 T a(
1
)(r
r0) + : : :
2 ' (r
r0) 4 T
p
32v(0) B T + a(
1
) h0n0(4 T
i!)
p
8 h0 T n0 (0)q
!
+ : : : ;
1
2
M 2 20 + !
iat0M q + 20!
ln r
r3 +
We can solve these equations numerically by integrating from the horizon to the
boundary with the boundary condition that the source of at and 2 is 0. The conductivity can
be calculated from
B=0.1
B=0.5
B=1
20
30
ω
Using the fact that the system is invariant under the residual symmetry at ! at + i! ,
where
is a constant independent of r, we will be able to generate
2 !
2
solutions with at = 0 for each independent numerical solution. Then we can use the two
free parameters at the horizon to generate solutions which has no source of 2 at the
boundary. In
gure 6 we show the AC longitudinal magnetoconductivity for M=T =
and B=T 2 = 0:1 2; 0:5 2;
2 respectively. We can see from the gures that after adding
this axial charge dissipation, the zero frequency pole in the imaginary part indeed vanishes
and instead a drude peak develops at small frequency even for M=T
O(1), i.e. when the
axial charge conservation symmetry is completely broken. As B increases, the height of
the drude peak also increases which means that the axial relaxation time increases with B.
At larger B quasinormal modes start to develop at large values of !.
As can be seen from the numerics above, with the axial charge dissipation we
have a
nite DC longitudinal magnetoconductivity. In this case, we can calculate the
DC conductivity using the radially conserved quantity [47] following [36, 48]. Consider
V = (0; 0; 0; 0; Et + vz(r)) and A = (at(r); 0; 0; 0; 0), the equations are now
and
(3.30)
(3.31)
(3.32)
(3.33)
(3.34)
The radially conserved current is
and we have Jz(
1
) = Jz(r0). At the horizon, we have the ingoing boundary condition
n
h0
2h
+
n h
+
8 Bpha0t = 0
f n
f
= 0
Jz(r) =
pfn v0
h z
8 Bat;
vz(r0) '
Et
ln(r
r0)
at ' n0ph0q2 2(r0)
:
E
T
1.4 × 1014
6 × 1029
5 × 1029
above. The red lines are slope 1 functions at the large B region, which indicates that at large
B=T 2, zz(0) is indeed a linear function of B=T 2.
Thus we have the DC longitudinal magnetoconductivity
zz = Jz(
1
)=E = p
+
n0
h0
This formula contains two parts of contributions. The rst part is n0= h0 which reduces
to T in the probe limit. This part now also has a dependence on the background magnetic
eld. The rest is the second part, which reduces to exact B2 behavior in the probe limit.
Due to backreactions, the B2 scaling behavior of this part might also become di erent. We
numerically checked that the analytic result agrees with our numerical results. With this
analytic formula for the DC longitudinal magnetoconductivity, we can reach for arbitrary
large B region. Thus we do not need to go to the zero temperature limit to work on the
large B behavior and we give the zero temperature background solutions to this system in
the appendix. In the following we focus on the small M region where 5 is large enough to
p
stay in the hydrodynamic regime.
Di erent from the universal B2 behavior of the probe limit, after taking into account
the backreactions to the geometry, n0, h0 and 0 now all depend on both B and M . At
small B=T 2, the leading order dependence on B in all these parameters should be the
same as the probe limit and deviations from the probe limit only arise at larger B=T 2 and
. In the following we mainly focus on the large B=T 2 behavior of the DC longitudinal
magnetoconductivity at xed small values of M=T and large . In gure 7 we plot the DC
n
T h0
5 × 107
3 × 1018
longitudinal magnetoconductivity
(left) and M=T = 0:00005
(right) at
zz(0) as a function of B=T 2 at xed M=T = 0:005
= 200. At large B=T 2, zz(0) grows linearly in
B=T 2. To analyze the scaling of zz(0) on B more explicitly, it is better to study the two
parts in the analytic formula (3.35) separately.
B
T2
p
1
γ1
100
104
106
HJEP09(216)
B
T2
B
T2
104
p
p
p
We denote the scaling exponent of B=T 2 in the rst part n0= h0 in the formula (3.35)
as f , i.e. n0= h0 ' c(M=T )(B=T 2) f at large B=T 2. In numerics we can get the value
of the scaling exponent f using 1 = B(n0= h0)0=(n0= h0) and by de nition this value
p
p
that we obtained only has the meaning of the scaling exponent when it remains a constant
in a nite region of B=T 2. In gure 8 we show the dependence of n0= h0 and the value of
p
1 at xed M=T = 0:005 (top
gure) and M=T = 0:00005 (bottom) separately. Due to
numerical constraints, we can reach for much larger values of B=T 2 for the M=T = 0:00005
case. As we can see from the
gure, the value of 1 reaches a constant 1 at large B,
indicating a scaling behavior at large B with scaling exponent being f = 1, in contrast
to the behavior of n0= h0 '
p
T + csB2 at small B=T 2 and p
B=T 2, where cs denotes a
constant independent of B. However, at large B=T 2 this term is not the main contribution
to zz(0) in the formula (3.35) as this term is much smaller than the second part.
In the second part of formula (3.35), the numerator in n0p
32 2B2
h0q2 2(r0) is exactly B2
and the full dependence of this term on B is determined by the dependence on B in the
denominator n0ph0 2(r0). We denote the scaling exponent of B in n0ph0 2(r0) as s for
large B, i.e. n0ph0 2(r0) ' c(M; T )B s . In numerics we can get the value of the scaling
T3
8 × 108
6 × 108
4 × 108
2 × 108
1
of the analytic formula for the DC longitudinal magnetoconductivity (3.35) as well as
exponent s using 2 = B(n0ph0 2(r0))0=(n0ph0 2(r0)) and by de nition this value that
we obtained only has the meaning of the scaling exponent when it remains a constant in a
nite region of B=T 2. In gure 9 we plot the dependence on B of n0ph0 2(r0) as well as 2
at xed M=T = 0:005 (top gure) and M=T = 0:00005 (bottom) separately. For the case
of M=T = 0:005 , due to numerical constraints, we cannot reach very large B=T 2 region,
but we can already see that 2 is approaching 1 slowly as B becomes larger, indicating a
scaling behavior with s = 1. In the gure of M=T = 0:00005
we can already see that 2
almost goes to 1 at large B=T 2.3 Substituting this scaling behavior into the second part
of the analytic formula for zz(0) (3.35), we can see that the second part in the formula
also goes linearly in B at large B=T 2, compared to the B2 behavior of the small B=T 2
limit. Note that the second part is much larger than the rst part in the analytic formula
of the DC conductivity. Thus we can see that after considering backreaction e ects, the
DC longitudinal magnetoconductivity is linear in B at large B=T 2, which is di erent from
the exact B2 behavior of the probe limit. This scaling behavior coincides with the weakly
coupled kinetic result qualitatively [3, 31]. In one of the experiments [20], the same scaling
behavior was also found.
3However we cannot at present rule out a power law which deviates slightly from 1 due to the numerical
5.0 × 10171.0 × 10181.5 × 10182.0 × 10182.5 × 10183.0 × 1018 T 2
1016
1017
1018
From the hydrodynamic formula, at small B=T 2 and large axial charge relaxation time,
the DC longitudinal magnetoconductivity obeys the following formula
DC =
E + (8 B)2 5
;
5
B
T 2
(3.36)
(3.37)
τ5T
350000
1.5 × 1010
1.0 × 1010
5.0 × 109
B
B
0.6
0.4
where 5 is the axial charge relaxation time. We can calculate 5 and 5 numerically using
this same setup but with di erent boundary conditions at the asymptotic AdS5 boundary.
For
5, we choose the boundary condition that vz and
2 are sourceless at the boundary.
We can also simplify the three equations for perturbations into one equation of motion for
at at zero frequency
a0t0 + a0t
n0
n
+
h0
2h
64 2B2at
n2f
f
2q2 2at = 0:
5 can be determined from the zero momentum quasinormal mode under the boundary
condition that all three elds vz, at and 2 are sourceless at the boundary. When we nd a
pure imaginary quasinormal mode at frequency
I!I we can get 5 = 1=!I . The detailed
procedure of this calculation can be found in [36]. Here we show the numerical results for
these two quantities. In gure 10, we show the dependence of 5 and its scaling exponent
at large B ( 5 ' c 5 B 5 ) on B=T 2 for two xed values of M=T = 0:005 (top); 0:00005
5
(bottom),
= 1. Note that in the gure, we de ned
5 = B 50 = 5, which only
has the meaning of the scaling exponent when it reaches a constant in a certain region of B.
120
100
80
60
40
20
0
We can see that 5 increases as B increases and reaches a nite and constant value at large
B=T 2 ! 1, i.e. at B=T 2 ! 1, the scaling exponent
5 ! 0.4 This means that at xed
M , there will be an upper limit in 5 no matter how large the magnetic eld is and this is
very di erent from the probe limit result, where at large B=T 2,
5 ! 1 and 5 diverges
at in nite B=T 2. We will see later that this caused the deviation in the dependence of the
DC longitudinal magnetoconductivity on B at B=T 2
! 1 from the probe limit result.
At small B it is expected that 5
M 2 at small M=T , which is the result from the
probe limit. Here we show in
gure 11 that at two large and
xed values of B=T 2, we
still have 5
M 2 at small M=T . The holographic axial charge relaxation time and its
property was also studied recently in a top down model in [49] in AdS/QCD.
In gure 12, we plot the dependence of 5 and its scaling exponent
5 at large B
( 5 ' c 5 B 5 ) on B=T 2 for two xed values of M=T = 0:005 (top); 0:00005 (bottom),
= 1. Note that in the gure, we de ned
5 = B 05= 5, which only has
the meaning of the scaling exponent when it reaches a constant in a certain region of
B. We can see that 5 is a monotonically increasing function of B and at B=T 2
5 grows linearly in B, which is the same as the probe limit result.
With the scaling
! 1
behaviors of 5 and
5 we can see that the hydrodynamic formula also predicts a linear in
B behavior for the longitudinal DC magnetoconductivity at B=T 2
! 1. We also checked
numerically that the leading order contribution in the hydrodynamic formula (3.36), i.e. the
second term (8 B)2 5 agrees with the leading order contribution in the analytic formula
32 2B2=n0p
h0q2 2(r50) as can be seen from
gure 13. This shows that in this backreacted
holographic system with axial charge dissipation, the hydrodynamic formula is still valid as
long as 5 is large enough to stay in the hydrodynamic regime, while B=T 2 can be in nitely
large, which is outside the hydrodynamic regime.
4We cannot rule out the possibility that the scaling exponent is slightly above 0 due to the numerical
5.0 × 106
values of M=T = 0:005 (top); 0:00005 (bottom),
= 1.
1.00
0.98
1.5 × 108
32 2B2=n0ph0q2 2(r0) for two xed values of M=T = 0:005 (left); 0:00005 (right),
longitudinal magnetoconductivity (3.36) over the leading order contribution in the analytic formula
= 200
and
= 1.
4
Conclusion and discussion
In this paper, we considered the backreaction e ects of the magnetic eld to the holographic
longitudinal magnetoconductivity for zero charge and axial charge density chiral anomalous
systems. Backreaction e ects are important at large B=T 2 and large backreaction strength
. In the case without axial charge dissipation, the longitudinal magnetoconductivity has
a pole in the imaginary part at ! = 0. The small frequency result deviates from the probe
limit at larger B=T 2 region. At B=T 2
! 1, we instead work in the zero temperature limit
and
nd that the imaginary part of the small frequency longitudinal magnetoconductivity
coincides with the probe limit result while the real part of the DC longitudinal
magnetoconductivity diverges for backreaction strength
larger than a critical value c = (32 )2,
in contrast to being zero in the probe limit. In the case with axial charge dissipation,
the negative magnetoresistivity behavior still exists after including backreactions. At large
B=T 2 the DC longitudinal magnetoconductivity becomes linear in B, which deviates from
the exact B2 behavior for the probe limit. Surprisingly we also found that for both cases
the hydrodynamic formula for the small frequency longitudinal magnetoconductivity
obtained in [29] still gives the holographic result at zero temperature, which is already out of
the hydrodynamic regime.
The calculations in this paper are a rst step to the study of holographic negative
magnetoresistivity for nite charge and axial charge density systems, where the
backreactions of the gauge elds are important to the gravity background. At nite charge density,
momentum relaxation is needed in order to have a nite DC longitudinal
magnetoconductivity, and at nite axial charge density, energy dissipation will be needed. The next step in
this direction would be to add momentum dissipations in the holographic system [50{58] at
nite charge density and compare the holographic result with the hydrodynamic formula.
At nite chemical potential and a
nite magnetic eld background, there exists an
instability to spatially modulated phases as shown in [59], which possibly leads to much richer
magnetotransport behavior. We will report the study of magnetoresistivity in holographic
nite density chiral anomalous systems in the future work.
It is still an open question how to add energy dissipations in holography. At nite
axial charge density, it would be interesting to check if there is indeed still a pole at ! = 0
after including momentum and axial charge dissipations. Another interesting question is to
study the axial charge relaxation and momentum relaxation time from the memory matrix
formalism [50, 60] in the hydrodynamic regime for chiral anomalous systems with a
background magnetic eld and also check it in strongly coupled holographic systems. Finally,
as was found in [26], chiral anomaly also induces strong suppression of the thermopower
in a chiral anomalous system. It would be interesting to study this e ect from both the
hydrodynamic and holographic point of view.
Acknowledgments
We would like to thank RongGen Cai, Sean Hartnoll, Karl Landsteiner, Yan Liu, Koenraad
Schalm and Jan Zaanen for useful discussions. The work of Y.W.S. was supported by the
European Union through a Marie Curie Individual Fellowship MSCAIF2014659135. The
work of Q.Y. was supported by National Natural Science Foundation of China (No.11375247
and No.11435006).
This work was also supported in part by the Spanish MINECO's
\Centro de Excelencia Severo Ochoa" Programme under grant SEV20120249. Q.Y. would
like to thank the hospitality of IFT during the completion of this work.
The equations of motion are
R
1
2
g
2e2 (F 2 +F 2) (D
R
12
2
e2 F
2
F
r F
+
The assumption for the background solutions is
F
2
e2
F
F
F
+ F
F
dr2
f (r)
where we have introduced an j j4=2 term for the convenience of analytic calculation at
zero temperature, which does not a ect the qualitative properties of transport coe cients.
HJEP09(216)
ds2 =
Zero temperature background solutions with axial charge dissipation
In this appendix, we present the zero temperature background solutions in the case with
axial charge dissipations in the presence of a background magnetic eld. We consider the
following action
S =
Z
d x
R + 12
m2
1
4e2 F
2
2
1
4e2
F 2 +
3
)2 ;
A
F F
+ 3F
F
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
and
The equations become
f 0h0
2f h
f 0n0
f n
+
h0n0
hn
+
n02
2n2
f 00
2f
n00
n
n0 f 0
n
f
n0
4n
6
f
where
= 2 2=e2 and we have rescaled e
exact solution to the equations above is AdS3
and =e2 !
R2 with a constant scalar
. At zero temperature, an
ds2 =
r2dt2 +
+
+
+
4
2f
B2
0
2
2
3
f
= 0;
= 0
= 0;
n =
B
3
8
1
14
);
(1 + f2r +
r
1 +
2
1+ 38
r2(1 + f1r +
);
1
;
where
can be tuned to get di erent values of physical B and M .
At zero temperature, the equations for the perturbations vz, 2 and at are the same
as equations (3.20) of the
nite temperature case. The
4 term appears in the equation
of motion for 2 but does not change the equation of motion for ar. When we derive the
equation of motion for 2 from the three equations (3.20), the 4 term will arise
automatically from the equation of motion of the background scalar eld. At zero temperature,
in the near horizon region it is di cult to solve for the near horizon behavior of the three
elds vz, at and 2 at r
1 while w can be smaller or bigger than r. However, we can get
the near horizon behavior at r
w
1. f1 and f2 are two free parameters which
To ow this solution from the horizon to asymptotic AdS5 we need to nd appropriate
irrelevant perturbations. Thus up to the rst order in perturbations the near horizon
solution becomes
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)
(A.18)
(A.19)
(A.20)
+
where
20 =
8 vz0 r 3
q
9
8
+
i!at0
6(1 + 38 )q
! r 3
=
;
and
represent subleading order corrections at order r , r, r and so on.
With these boundary conditions in principle we can solve the zero temperature case
numerically and the result would only depend on B=M 2,
and . This corresponds to the
B=T 2
B=T 2 limit, which we already obtained in the nite temperature section, we will not study
! 1 and M=T ! 1 limit. As we are more interested in the small M=T while large
the zero temperature longitudinal magnetoconductivity here.
Open Access.
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Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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