Negative magnetoresistivity in holography

Journal of High Energy Physics, Sep 2016

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 field. We calculate the longitudinal magnetoconductivity in the presence of back-reactions of the magnetic field 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 find that the quantum critical conductivity grows with increasing magnetic field 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 find the negative magnetoresistivity behavior. The DC longitudinal magnetoconductivity scales as B in the large magnetic field limit, which deviates from the exact B 2 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 field limit.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP09%282016%29122.pdf

Negative magnetoresistivity in holography

Revised: June magnetoresistivity in holography Ya-Wen 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 AdS-CFT Correspondence; Gauge-gravity 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 semi-metal 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 non-constant 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 Chern-Simons 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 K-function 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 Einstein-dilaton 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 Rong-Gen 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 MSCA-IF-2014-659135. 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 SEV-2012-0249. 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. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [arXiv:1511.04050] [INSPIRE]. 78 (2008) 074033 [arXiv:0808.3382] [INSPIRE]. HJEP09(216) (2009) 191601 [arXiv:0906.5044] [INSPIRE]. [arXiv:0908.4189] [INSPIRE]. 03 (2011) 023 [arXiv:1011.5107] [INSPIRE]. [5] J. Erdmenger, M. Haack, M. Kaminski and A. Yarom, Fluid dynamics of R-charged black holes, JHEP 01 (2009) 055 [arXiv:0809.2488] [INSPIRE]. [6] N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Dutta, R. Loganayagam and P. Surowka, Hydrodynamics from charged black branes, JHEP 01 (2011) 094 [arXiv:0809.2596] [7] D.T. Son and P. Surowka, Hydrodynamics with Triangle Anomalies, Phys. Rev. Lett. 103 [8] H.-U. Yee, Holographic Chiral Magnetic Conductivity, JHEP 11 (2009) 085 [9] Y. Neiman and Y. Oz, Relativistic Hydrodynamics with General Anomalous Charges, JHEP [10] K. Landsteiner, E. Megias and F. Pena-Benitez, Gravitational Anomaly and Transport, Phys. Rev. Lett. 107 (2011) 021601 [arXiv:1103.5006] [INSPIRE]. [11] K. Landsteiner, E. Megias, L. Melgar and F. Pena-Benitez, Holographic Gravitational Anomaly and Chiral Vortical E ect, JHEP 09 (2011) 121 [arXiv:1107.0368] [INSPIRE]. [12] K. Jensen, R. Loganayagam and A. Yarom, Thermodynamics, gravitational anomalies and cones, JHEP 02 (2013) 088 [arXiv:1207.5824] [INSPIRE]. [13] D.-F. Hou, H. Liu and H.-c. Ren, A Possible Higher Order Correction to the Vortical Conductivity in a Gauge Field Plasma, Phys. Rev. D 86 (2012) 121703 [arXiv:1210.0969] [14] M. Ammon and J. Erdmenger, Gauge/gravity duality: Foundations and applications, Cambridge University Press (2015). [15] H. Nastase, Introduction to the ADS/CFT Correspondence, Cambridge University Press [16] J. Zaanen, Y. Liu, Y.W. Sun and K. Schalm, Holographic Duality in Condensed Matter Physics, Cambridge University Press (2015). [arXiv:1505.04772] [INSPIRE]. [17] K. Landsteiner and Y. Liu, The holographic Weyl semi-metal, Phys. Lett. B 753 (2016) 453 [18] K. Landsteiner, Y. Liu and Y.-W. Sun, Quantum phase transition between a topological and a trivial semimetal from holography, Phys. Rev. Lett. 116 (2016) 081602 [arXiv:1511.05505] HJEP09(216) arXiv:1503.02630 [INSPIRE]. steered by a magnetic eld, arXiv:1503.08179 [INSPIRE]. [24] J. Xiong et al., Signature of the chiral anomaly in a Dirac semimetal: a current plume [25] F. Arnold et al., Negative magnetoresistance without well-de ned chirality in the Weyl semimetal TaP, Nature Commun. 7 (2016) 1615 [arXiv:1506.06577] [INSPIRE]. [26] M. Hirschberger et al., The chiral anomaly and thermopower of Weyl fermions in the half-Heusler GdPtBi, arXiv:1602.07219. Phys. 24 (1963) 419. [27] L.P. Kadano and P.C. Martin, Hydrodynamic equations and correlation functions, Annals [28] S.A. Hartnoll, P.K. Kovtun, M. Muller and S. Sachdev, Theory of the Nernst e ect near quantum phase transitions in condensed matter and in dyonic black holes, Phys. Rev. B 76 (2007) 144502 [arXiv:0706.3215] [INSPIRE]. [29] K. Landsteiner, Y. Liu and Y.-W. Sun, Negative magnetoresistivity in chiral uids and holography, JHEP 03 (2015) 127 [arXiv:1410.6399] [INSPIRE]. [30] D. Roychowdhury, Magnetoconductivity in chiral Lifshitz hydrodynamics, JHEP 09 (2015) 145 [arXiv:1508.02002] [INSPIRE]. [31] D.T. Son and B.Z. Spivak, Chiral Anomaly and Classical Negative Magnetoresistance of Weyl Metals, Phys. Rev. B 88 (2013) 104412 [arXiv:1206.1627] [INSPIRE]. [32] E.V. Gorbar, V.A. Miransky and I.A. Shovkovy, Chiral anomaly, dimensional reduction and magnetoresistivity of Weyl and Dirac semimetals, Phys. Rev. B 89 (2014) 085126 [arXiv:1312.0027] [INSPIRE]. [33] P. Goswami, J.H. Pixley and S. Das Sarma, Axial anomaly and longitudinal magnetoresistance of a generic three dimensional metal, Phys. Rev. B 92 (2015) 075205 [arXiv:1503.02069] [INSPIRE]. [34] G. Lifschytz and M. Lippert, Anomalous conductivity in holographic QCD, Phys. Rev. D 80 (2009) 066005 [arXiv:0904.4772] [INSPIRE]. [35] A. Jimenez-Alba, K. Landsteiner and L. Melgar, Anomalous magnetoresponse and the Stuckelberg axion in holography, Phys. Rev. D 90 (2014) 126004 [arXiv:1407.8162] [INSPIRE]. [36] A. Jimenez-Alba, K. Landsteiner, Y. Liu and Y.-W. Sun, Anomalous magnetoconductivity and relaxation times in holography, JHEP 07 (2015) 117 [arXiv:1504.06566] [INSPIRE]. [37] U. Gursoy and A. Jansen, (Non)renormalization of Anomalous Conductivities and Holography, JHEP 10 (2014) 092 [arXiv:1407.3282] [INSPIRE]. [38] I. Iatrakis, S. Lin and Y. Yin, The anomalous transport of axial charge: topological vs non-topological uctuations, JHEP 09 (2015) 030 [arXiv:1506.01384] [INSPIRE]. [arXiv:0908.3875] [INSPIRE]. [40] S. Janiszewski and M. Kaminski, Quasinormal modes of magnetic and electric black branes versus far from equilibrium anisotropic uids, Phys. Rev. D 93 (2016) 025006 [arXiv:1508.06993] [INSPIRE]. [41] G.T. Horowitz and M.M. Roberts, Holographic Superconductors with Various Condensates, Phys. Rev. D 78 (2008) 126008 [arXiv:0810.1077] [INSPIRE]. Lect. Notes Phys. 871 (2013) 433 [arXiv:1207.5808] [INSPIRE]. [42] K. Landsteiner, E. Megias and F. Pena-Benitez, Anomalous Transport from Kubo Formulae, [43] E. D'Hoker, P. Kraus and A. Shah, RG Flow of Magnetic Brane Correlators, JHEP 04 (2011) 039 [arXiv:1012.5072] [INSPIRE]. [44] K.A. Mamo, Enhanced thermal photon and dilepton production in strongly coupled N = 4 SYM plasma in strong magnetic eld, JHEP 08 (2013) 083 [arXiv:1210.7428] [INSPIRE]. [45] T. Faulkner, H. Liu, J. McGreevy and D. Vegh, Emergent quantum criticality, Fermi surfaces and AdS2, Phys. Rev. D 83 (2011) 125002 [arXiv:0907.2694] [INSPIRE]. [46] E. Kiritsis and J. Ren, On Holographic Insulators and Supersolids, JHEP 09 (2015) 168 [arXiv:1503.03481] [INSPIRE]. [47] N. Iqbal and H. Liu, Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm, Phys. Rev. D 79 (2009) 025023 [arXiv:0809.3808] [INSPIRE]. [48] A. Donos and J.P. Gauntlett, Thermoelectric DC conductivities from black hole horizons, JHEP 11 (2014) 081 [arXiv:1406.4742] [INSPIRE]. 105001 [arXiv:1602.03952] [INSPIRE]. [49] E.-d. Guo and S. Lin, Quark mass e ect on axial charge dynamics, Phys. Rev. D 93 (2016) HJEP09(216) (2013) 649 [arXiv:1212.2998] [INSPIRE]. [arXiv:1311.3292] [INSPIRE]. (2014) 101 [arXiv:1311.5157] [INSPIRE]. [50] S.A. Hartnoll and D.M. Hofman, Locally Critical Resistivities from Umklapp Scattering, Phys. Rev. Lett. 108 (2012) 241601 [arXiv:1201.3917] [INSPIRE]. [51] G.T. Horowitz, J.E. Santos and D. Tong, Optical Conductivity with Holographic Lattices, JHEP 07 (2012) 168 [arXiv:1204.0519] [INSPIRE]. [52] Y. Liu, K. Schalm, Y.-W. Sun and J. Zaanen, Lattice Potentials and Fermions in Holographic non Fermi-Liquids: Hybridizing Local Quantum Criticality, JHEP 10 (2012) 036 [arXiv:1205.5227] [INSPIRE]. [53] A. Donos and S.A. Hartnoll, Interaction-driven localization in holography, Nature Phys. 9 [54] A. Donos and J.P. Gauntlett, Holographic Q-lattices, JHEP 04 (2014) 040 [55] T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05 [56] D. Vegh, Holography without translational symmetry, arXiv:1301.0537 [INSPIRE]. [57] R.A. Davison, K. Schalm and J. Zaanen, Holographic duality and the resistivity of strange metals, Phys. Rev. B 89 (2014) 245116 [arXiv:1311.2451] [INSPIRE]. eld theories with chiral anomaly from holography, JHEP 03 (2016) 164 [arXiv:1601.02125] [INSPIRE]. [1] D.E. Kharzeev , J. Liao , S.A. Voloshin and G. Wang , Chiral magnetic and vortical e ects in high-energy nuclear collisions | A status report , Prog. Part. Nucl. Phys . 88 ( 2016 ) 1 [2] P. Hosur and X. Qi , Recent developments in transport phenomena in Weyl semimetals , Comptes Rendus Physique 14 ( 2013 ) 857 [arXiv: 1309 .4464] [INSPIRE]. [3] H.B. Nielsen and M. Ninomiya , Adler-Bell-Jackiw Anomaly And Weyl Fermions In Crystal, Phys. Lett. B 130 ( 1983 ) 389 [INSPIRE]. [4] K. Fukushima , D.E. Kharzeev and H.J. Warringa , The Chiral Magnetic E ect, Phys. Rev . D [19] G.H. Wannier , Theorem on the Magnetoconductivity of Metals, Phys. Rev. B 5 ( 1972 ) 3836 . [20] H.-J. Kim et al., Dirac versus Weyl Fermions in Topological Insulators: Adler-Bell-Jackiw Anomaly in Transport Phenomena , Phys. Rev. Lett . 111 ( 2013 ) 246603 [arXiv: 1307 .6990] [21] Q. Li et al., Observation of the chiral magnetic e ect in ZrTe5 , Nature Phys . 12 ( 2016 ) 550 [22] X. Huang et al., Observation of the chiral anomaly induced negative magnetoresistance in 3D Weyl semi-metal TaAs , Phys. Rev. X 5 ( 2015 ) 031023 [arXiv: 1503 .01304]. [23] C. Zhang et al., Observation of the Adler-Bell-Jackiw chiral anomaly in a Weyl semimetal , [39] E. D'Hoker and P. Kraus , Magnetic Brane Solutions in AdS, JHEP 10 ( 2009 ) 088 [58] A. Donos and J.P. Gauntlett , Novel metals and insulators from holography , JHEP 06 ( 2014 ) [59] M. Ammon , J. Leiber and R.P. Macedo , Phase diagram of 4D [60] A. Lucas and S. Sachdev , Memory matrix theory of magnetotransport in strange metals , Phys. Rev. B 91 ( 2015 ) 195122 [arXiv: 1502 .04704] [INSPIRE].


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP09%282016%29122.pdf

Ya-Wen Sun, Qing Yang. Negative magnetoresistivity in holography, Journal of High Energy Physics, 2016, 122, DOI: 10.1007/JHEP09(2016)122