Magnetoresistance in relativistic hydrodynamics without anomalies

Journal of High Energy Physics, Jun 2017

We present expressions for the magnetoconductivity and the magnetoresistance of a strongly interacting metal in 3 + 1 dimensions, derivable from relativistic hydrodynamics. Such an approach is suitable for ultraclean metals with emergent Lorentz invariance. When this relativistic fluid contains chiral anomalies, it is known to exhibit longitudinal negative magnetoresistance. We show that similar effects can arise in non-anomalous relativistic fluids due to the distinctive gradient expansion. In contrast with a Galilean-invariant fluid, the resistivity tensor of a dirty relativistic fluid exhibits similar angular dependence to negative magnetoresistance, even when the constitutive relations and momentum relaxation rate are isotropic. We further account for the effect of magnetic field-dependent corrections to the gradient expansion and the effects of long-wavelength impurities on magnetoresistance. We note that the holographic D3/D7 system exhibits negative magnetoresistance.

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Magnetoresistance in relativistic hydrodynamics without anomalies

Accepted: May Magnetoresistance in relativistic hydrodynamics without anomalies Andrew Baumgartner 0 1 3 Andreas Karch 0 1 3 Andrew Lucas 0 1 2 0 Stanford , CA, 94305-4060 , U.S.A 1 Seattle, WA , 98195-1560 , U.S.A 2 Department of Physics, Stanford University 3 Department of Physics, University of Washington , USA We present expressions for the magnetoconductivity and the magnetoresistance of a strongly interacting metal in 3 + 1 dimensions, derivable from relativistic hydrodynamics. Such an approach is suitable for ultraclean metals with emergent Lorentz invariance. When this relativistic uid contains chiral anomalies, it is known to exhibit longitudinal negative magnetoresistance. We show that similar e ects can arise in nonanomalous relativistic uids due to the distinctive gradient expansion. In contrast with a Galilean-invariant uid, the resistivity tensor of a dirty relativistic uid exhibits similar angular dependence to negative magnetoresistance, even when the constitutive relations and momentum relaxation rate are isotropic. We further account for the e ect of magnetic eld-dependent corrections to the gradient expansion and the e ects of long-wavelength impurities on magnetoresistance. We note that the holographic D3/D7 system exhibits negative magnetoresistance. 1 Introduction 2 Relativistic hydrodynamics in a magnetic eld 2.1 2.2 2.3 2.4 3.1 3.2 Weak magnetic elds Strong magnetic elds Linear response Thermal transport 3 \Microscopic" examples decreasing function of B, or the resistance is a monotonically increasing function of B [1]. However, the recent discovery of Weyl semimetals | materials in which conduction bands intersect at distinct points in the Brillioun zone | shows that this is not always the case [2{9]. In Weyl semi-metals, quasi-particles with momentum values near the intersection points (Weyl points) are described by the massless Weyl Hamiltonian H = v~ ir eA~ (1.1) where v is the velocity, ~ is the pseudospin operator, and correspond to the chirality of the quasi-particle. It is a well known result in quantum eld theory that in chiral elds are applied to the sample: e 2 F (?F ) = e 2 is the scattering time for quasiparticles to scatter from the neighborhood of one Weyl point to another, and A A = A is the deviation in the axial charge density from equilibrium. Applying an in nitesimal electric eld Ei, and using Ohm's law to extract the electrical conductivity, one nds1 e 2 description at weak coupling [12], as well as a hydrodynamic [13, 14] or holographic [15] description at strong coupling. Evidence for such an angular structure was found in the recent experiments [2{9]. Already at weak coupling it has been demonstrated that NMR is possible in nonanomalous systems [16]. In the present paper, we describe the hydrodynamic gradient expansion in background magnetic elds and derive a hydrodynamic equation for ij , analogous to (1.5), without any assumption of chirality. We are inspired in part by the holographic D3/D7 system, whose eld theory dual is that of N = 2 super Yang-Mills (SYM) hypermultiplets progagating through an N = 4 SYM plasma [17]: we will show that this system exhibits NMR. We will also see that it is possible to obtain positive magnetoresistance within hydrodynamics, and will present two di erent `microscopic' mechanisms for this. 1The anomaly itself drives a charge separation which yields an axial charge growing linearly with time crystal lattice, we anticipate the following expression for ij : In section 2, we will show how (1.7) generically appears in relativistic hydrodynamics, The inverse of ij , the resistivity tensor ij , will also have similar structure: ij ij + ijkBk + BiBj B2 : with all ; ; 6 = 0, and relate these coe cients to the hydrodynamic dissipative coe cients. An important di erence between Galilean-invariant uids and more general uids (including relativistic uids) is the fact that the charge current is not proportional to the momentum density in the latter case. Hence, we will nd that 6= 0, in contrast to the Galilean invariant case. In fact, up to a brief discussion of thermal transport, our discussion of hydrodynamic charge transport is also valid for any non-Galilean invariant system. Nowhere do we assume the existence of any (emergent) axial anomaly. Unlike in (1.5), there is no reason (a priori) to expect 2 to be parametrically large. Still, we note that typical anomalous NMR observed in experiment is not an order-of-magnitude enhancement. Hydrodynamic transport in background elds has been applied successfully to describe strongly correlated materials starting with the work of [18] on 2+1 dimensional physics. More recently, this (relativistic) hydrodynamic approach to transport has also been applied successfully to understand experimental transport data from clean samples of graphene [19, 20]. These studies take as input the hydrodynamic transport coe cients, and then use the structure of the hydrodynamic equations to give expressions for the dependence of the transport properties on external elds and particle number density. Starting from relativistic hydrodynamics instead of Galilean hydrodynamics, one nds a number of distinct predictions in 2+1 dimensions such as B-dependent lifetimes for cyclotron modes (a violation of Kohn's theorem) [18]. One important aspect of this procedure is that one has to be careful to work in a consistent expansion scheme. Hydrodynamics itself is a gradient expansion, where @ is treated as a small parameter. In the usual gradient expansion, since B~ = r A~, the results of [18] are, strictly speaking, only valid to linear orders in B. In this limit, one must treat multiple small parameters in the theory as \equally small" and only then perform the perturbative expansion. If one wants to, for example, see the motion of hydrodynamic poles (such as the cyclotron resonance) in the conductivity as a function of B, one needs to include all higher transport coe cients involving arbitrary powers of B. Since in our work all the interesting physics appears at quadratic order in B, we will develop an expansion scheme in which B is considered to be zeroth order in derivatives [21]. The article is organized as followed: in section 2 we present our hydrodynamic calculation, including the full conductivity and resistivity tensors. We then discuss some interesting limits and their physical interpretations. In section 3 we compare our results to that of the D3/D7 system and a simple toy model made of \electron-hole plasma" in 3 + 1 { 3 { (1.6) (1.7) Relativistic hydrodynamics in a magnetic eld Weak magnetic elds dimensions. The former is shown to exhibit negative magnetoresistance, while the latter has positive magnetoresistance. We discuss a hydrodynamic model for the momentum relaxation time in section 4, and show how the magnetic eld generically leads to anisotropic momentum relaxation. We conclude in section 5. As this paper was being nalized, [22] appeared, which contains some overlap with section 2. These terms can be found to rst order in the derivative expansion by requiring positivity of the divergence of the entropy current [24]. Such an analysis gives = = q(g (g + u u ) + F u + + u u ) : (2.4) Hydrodynamics is the low energy e ective description of any interacting quantum eld theory, valid for uctuations whose wavelength is much larger than a `thermalization scale': when quasiparticles are well-de ned, this scale is simply the mean free path of the quasiparticles. When we look on length scales long compared to the mean free path, the system appears to be in local thermal equilibrium, thereby allowing us to describe the global dynamics in terms of conserved quantities. In this paper, these conserved quantities will be charge, energy and momentum. The dynamical equations in the presence of external elds are r J r T where the last term allows for the dissipation of momentum due to impurities [18]; it can be derived rigorously when the disorder strength is small from multiple approaches [23]. The form of the 1= term in (2.1) is only sensible to linear order in the spatial components of u . Following the conventions of [18], we can write the current and energy-momentum tensor as J T = u + + Jmag = (" + p)u u + pg + + Tmag where and are dissipative contributions which arise at rst order in derivatives and a uid frame can be chosen in which they satisfy the following orthogonality relations u = u = u = 0: (2.1a) (2.1b) (2.2a) (2.2b) (2.3) HJEP06(217)54 The parameter q is the \quantum critical conductivity", the conductivity in the absence of charges and external elds. D denotes the number of spatial dimensions, which in this paper will generically be 3. The last term in eqs. (2.2a) and (2.2b) are contributions due to polarization of the material. They are present already in thermal equilibrium. Their contribution to currents in the bulk will be compensated by surface currents, rendering them unmeasurable in any experimental set up [25]. Alternatively, one can derive these terms using variational techniques such as in [21]. Reference [18] presented formulas for the conductivity and resistivity tensors within relativistic hydrodynamics in 2+1 dimensions. Here, we comment on the extension of this will see that in 3 + 1 dimensions, anisotropic (E~ B~ )Bi terms generically arise in Ji. 2.2 Strong magnetic elds As we mentioned in the introduction, one important caveat in the work of [18] and the work that followed is whether the hydrodynamic expansion has been systematically applied. In the constitutive relation (2.4) we only kept terms to rst order in the gradient expansion, treating F as being rst order in the gradient expansion. The nal answers hence only are valid up to linear order in E and B. Even assuming we may treat 1= as rst order in derivatives, all of the novel relativistic phenomenology of [18] requires inclusion of terms proportional to qB2 in the conductivity! As we emphasize in this paper, there are additional hydrodynamic contributions to the conductivity at order B2. For Abelian background elds, which includes electromagnetism, one can de ne an alternate expansion scheme in which the magnetic eld strength is treated as order 0 in the gradient expansion, and only derivatives of B~ count as gradients. Namely, we should really keep all orders in F at the start of the calculation. This is especially important for us since we are interested in NMR, which only occurs at order B2. Since our goal is to obtain the linear response relation (1.4), we can consider \weak" electric elds and \strong" magnetic elds, i.e. E O( 1 ). Thus, we will develop a more general constitutive relation that keeps all orders in B~ , but is linear in E~ . This is easiest to do using non-relativistic notation where E~ and B~ are treated separately. Note that we require B . T 2 as a point of principle; if this inequality is violated, then the hydrodynamic description should be replaced by an alternative description. For example, in the limit of extremely large B- elds, a hydrodynamic framework for low-lying Landau levels becomes appropriate [26]. We consider the thermodynamic quantities in (2.2a) and (2.2b) to be functions of , T and B2, and treat ; T and u as the degrees of freedom that respond to external perturbations to the system. The uctuations of "; p and will then be determined by the equation of state, as is standard. As in [18], we assume that in equilibrium the uid velocity is u of the uid u = (1; ~0), and so perturbations from equilibrium allows us to treat the velocity i = vi as O(@). Thus, any quadratic terms in ui can be ignored in the gradient expansion. We also assume that the applied electromagnetic elds are static { 5 { t ! t; ! ; Bi ! Bi; xi ! xi + vit; ( + P ) ! ( + P ); Ei ! Ei + ijkvj Bk: sources; strictly speaking, electromagnetism is a gauge theory and dynamical gauge elds complicate the hydrodynamic description [22, 27], although the physics is only qualitatively di erent under extreme magnetic elds. Finally, although it is not rigorous, we will for now assume the \mean eld" approximation that disorder modi es the momentum conservation equation simply through the factor 1= , as in (2.1), is correct. These approximations hone in on the changes to the gradient expansion that occur in a background magnetic eld. We will relax this assumption in section 4. Since without gradients and background electric eld the hydrodynamic velocity is zero, keeping only terms linear in E also means keeping only leading orders in v. Rotation invariance demands that the only tensor structures allowed in J i are vi, Ei, ijkBj vk, ijkBj Ek and BiBj Ej . The exact combination of these terms that is allowed to appear is further constrained by boost invariance. Since we are working to linear order in E and hence v, we can restrict ourselves to Galilean boosts which act as: Ji ! Ji + vi ; ! T ti + ( + P )vi; (2.5a) current reads Galilean boost: uid frame to x 2 As we are not considering inhomogeneous ows, we can exploit boost invariance by boosting into the rest frame of the uid with vi = 0. In this frame the most general form of the J i = c(B2)Ei + d(B2)Ej Bj Bi + ~(B2) ijkBj Ek: The full constitutive relation can be recovered by acting on this rest frame current with a J i = [ (B2) + ~(B2)B2]vi ~(B2)BiBj vj + c(B2)(Ei + "ijkvj Bk) +d(B2)Ej Bj Bi + ~(B2) ijkBj Ek: The coe cients can, in general, depend on B2. Finally, we use our freedom to rede ne the Another way to interpret what we have found is the simple statement that The rst order correction to the current J i, which was before simply proportional to q, is now proportional to a matrix ij which inherits the rotational symmetry breaking pattern of the external magnetic eld. Since entropy production, which occurs at quadratic order 2In principle one could add transport coe cients analogous to ~, c and d that appear in (2.7) also in the momentum current. These can however be absorbed by changing the hydrodynamic frame, that is by rede ning the velocity as vi ! vi + A1Ei + A2 ijkBjEk + A3EjBjBi. The 3 coe cients A1;2;3 contain enough freedom to eliminate the 3 analogues of ~, c and d in the momentum current. { 6 { in E, should be proportional to EiJi, and the rst term in J i does not contribute to entropy production as it arises at zeroth order in hydrodynamics, we conclude that the matrix should be positive de nite. This constrains c 0; c + dB2 0: We also observe that the coe cient ~ is dissipationless, and does not appear to be constrained. The form of the constitutive relation can be further constrained if we impose charge conjugation symmetry. Assuming that the underlying critical theory has charge conjugation symmetry, and latter is only broken by the explicit presence of the charge carries via demands symmetry under ji ! ji; This symmetry requires ~ = 0. Generically, it is possible for ~ to be an odd function of . But for simplicity, we will often set ~ = 0 in what follows to simplify the equations. We con rm in appendix A that starting with the most general relativistic constitutive relation including terms up to order F 3 indeed yields eq. (2.7) up to O(B2), when restricting to terms linear in E. The transport coe cients c, ~ and d appear as linear combinations of the various terms appearing in the relativistic analysis. 2.3 Linear response We are now in a position to determine ij in linear response. The energy and charge conservation equations can be ignored so long as the uid is homogeneous [18]; the momentum conservation equation reads: ( + P ) i! + 1 v i vi = Ei + "ijkJ j Bk: (2.12) Plugging (2.12) into the constitutive relation (2.7) gives the following matrix expression ij + c(B2) = B2 ij 2 BiBj "ijkBk J j + c(B2) ij + c(B2) "ijkBk + dBiBj Ej : (2.13) ij (2.10) + P (2.11) (2.14a) (2.14b) (2.14c) Without loss of generality, we let B~ = Bz^. With this, we obtain the following expressions for the conductivity: zz = 2 with the corresponding resistivity given by the inverse matrix: The constants in c(B2) and d will depend on the microscopic details of the theory, and their sign will determine if NMR is present. Indeed, (2.14a) is reminiscent of (1.5), even though this uid is not chiral: letting c c0 + c1B2, and similarly for d, we see that so long as c1 + d0 > 0 zz is an increasing function of B2, and zz is a decreasing function of B2, as is (1.5). However, the main experimental test for anomaly-induced NMR is the dramatic angular dependence of the resistivity. Using the de nitions in (1.7), it is straightforward to show that = = = 2 + c(B2) B2 c(B2) + 2 4 + ( 2 + B2 2) c(B2)2 + 2 2 c(B2)2 B2 c(B2)2 + 2 c(B2) + 3 B 4 + ( 2 + B2 2) c(B2)2 + 2 2 c(B2)2 ; 2 4 66 1 + 1 + 1 + c2B2 2 + c c2B2 2 + c 2 2 + c 1 + d B2 2 + c ; 3 77 : 5 c2B2 2 + c 1 + c 2 + c c2B2 2 + c = < 0: At B = 0, one can easily check that = = 0, as must happen, since the theory becomes isotropic. Allowing B 6= 0 but assuming d = 0, we nd that d > c 3 0 c0 + 2 { 8 { We hence conclude that the hydrodynamics of [18] can exhibit similar angular dependence in the resistivity to (1.5), despite the fact that it the hydrodynamic equations are manifestly isotropic. This e ect is dependent on the breaking of Galilean invariance, which allows for the coe cient c 6= 0. Adding the d-dependence back in, we conclude that only for d su ciently negative is it possible for > 0. At small B, one can show that is necessary in order for the angular dependence to appear as \positive" magnetoresistance. (2.15a) (2.15b) (2.15c) (2.16a) (2.16b) (2.16c) (2.17) (2.18) If we include non-vanishing ~, then the conductivity matrix generalizes to: zz = 2 xy = current and the heat current in response to applied electric elds and thermal gradients: HJEP06(217)54 (2.19a) (2.19b) (2.19c) (2.20) (2.21) (2.22) (2.23) The heat current is de ned as [18] i ! J Qi = ij T ij ! T ij T ij Ej ! Q i T ti J i with the chemical potential of the uid. A priori, such a computation can be quite subtle, since it appears as though we need to account for more terms in the derivative expansion to x the constitutive relations for @j T . However, consider the following arguments. Firstly, we may use the standard Landau frame in which (2.8) is exact. Using the thermodynamic relation + P = T s, with s the entropy density, we conclude that in an electric eld Ei (but keeping @iT = 0): Qi = T svi ij (Ej + jklvkBl): We have already solved for the velocity eld vi in an applied electric eld in our computation of ij , so hence we obtain straightforwardly the matrix xx = xy = zz = sT c B2c + Secondly, we use Onsager reciprocity which states that ij (B) = ji( B); since all o diagonal elements of the transport matrices are antisymmetric, we conclude that ij = ij . { 9 { Next, we can imagine turning o the eletric eld and only applying an external temperature gradient. The momentum conservation equation then reads (2.24) (2.25) (2.26a) (2.26b) (2.26c) (2.27) we may combine (2.21), (2.24) and (2.25) to obtain ij : vi = ijkJj Bk T s J i = T : c xx = xy = zz = B2c c 2 + ( + sT )2 + 2 + s2T 2 From these results we can conclude that the constitutive relation for the current must include a linear temperature gradient as As in an ordinary relativistic uid, we conclude that there are no new dissipative coe cients associated with thermo-magnetic response. We note that all thermoelectric coe cients in the xy plane are identical to those presented in [18]. This is to be expected, since only the longitudinal components of the transport matrix are sensitive to the anisotropies introduced by the Ej Bj Bi term in (2.7). 3 3.1 \Microscopic" examples N = 4 SYM plasma We now wish to compare our formalism with the conductivity of Nf massive N = 2 supersymmetric hypermultiplets owing through an N = 4 SYM plasma with gauge group SU(Nc) at temperature T . This model was studied extensively starting with [28] and the conductivities in the background of constant electromagnetic eld with generic orientations was worked out in [17]. We take the limits Nc ! 1 with large but nite 't Hooft coupling = gY2MNc, allowing the use of holographic techniques. The avor hypermultiplets are dual to Nf D7 branes [29] embedded in a xed AdS-Schwarzschild background. Furthermore, we will work in the probe limit Nf Nc so that we may neglect the back reaction of the probe branes on the supergravity elds. This allows us to treat the plasma as stationary, and focus on the dynamics of the avor elds alone. Speci cally, this limit allows for an apparent dissipation of momentum. The avor elds lose energy to the plasma at a rate of order Nc so only at times of order Nc will the back reaction on the Nc2 plasma degrees of freedom be non-negligible. This momentum relaxation is rather `peculiar', and so the theory of transport in probe brane models di ers in important ways from other models of transport [23]. In particular, it is unclear whether or not a `weak disorder' limit exists. As such a limit was required in order to rigorously include in our hydrodynamic model of transport, there is a priori no reason to expect exact quantitative agreement between our hydrodynamic model and this holographic model. It is known that generic holographic models disagree with the hydrodynamics of [30{32] at next-to-leading order in : this can crudely be thought of as arising due to -dependent corrections to the hydrodynamic constitutive relations. The conductivity of the propagating hypermultiplets in generic constant background elds was found in [17]. They considered an E eld that is xed along the x-axis, while z plane. This can be mapped onto our formalism by rewriting their results in terms of the basic constitutive relation (1.7). For small electric elds, they found xx = ~ 1 + b2x s 1 + b2 1 + Nf2Nc2T 2 ~216 2 cos6 ?(1 + b2); xy = ~bz ; 1 + b2 xz = bxbz 1 + b2x xz (3.1) where bi = qBTi2 and ~ = qT 2 . This can be brought into the form (1.7) with coe cients (3.2a) (3.2b) (3.2c) (3.3) limit [23, 30{32]. itly compute In the limits consistent with where 0 = Nf2Nc2p 64 cos6 ? and q = 2 p . This results look quite di erent from the hydrodynamic form. One could ask whether the D3/D7 answer can be t into the hydrodynamic framework by a particular choice of transport coe cients. At large , the coe cients c, d and ~ are generically -dependent. T 3 and B qT 2 limits, [17] have shown that probe brane models appear = qT 2 at leading order in . At this order, one trivially nds a Drude-like conductivity: ij 2 ij= . Since c, d and ~ arise at next-to-leading order in this limit, their unique determination requires specifying at order 0. It is unclear whether this question is even `well-posed', in light of the subtleties that arise in transport beyond the weak disorder However, given the exact magnetic eld dependence of the resistivity, we can nonperturbatively compute the magnetoresistance in both B and . Firstly, one can explic= = = q T 2q 2 0T 2(B2+q2T 4) + 1 zz = q 2 + 2q T 2 0 (B2 + q2T 4) ; which is clearly a decreasing function of B2. Secondly, the ratio of the longitutinal to transverse resistivities is 2 0T 2 xx = 1 + zz q( 2 + 2q 0T 6) B2 > 1 (3.4) implying the resistivity along the direction of the magnetic eld is supressed relative to the transverse directions. As expected, in the B ! 0 limit, the ratio goes to one since the theory becomes isotropic. In this section we present a simple classical cartoon of a uid where we can compute the coe cients c and d 6= 0. More precisely, let us consider a toy model of two charged one of charge density ^ (the + uid) and the other of charge density ^ (the uids, uid), analogous to electron and hole uids in graphene. Unlike in graphene [33], we will suppose that the momentum of these two charged uids is also an almost conserved quantity. For simplicity, we assume that all other properties, such as enthalpy M^ , of these two are identical, and we also only consider the hydrodynamic gradient expansion to rst order uids in derivatives. The net current is given by the sum of currents in the uids: J = J+ + J . The spatial components of these currents are given by J +i = ^v+i + ^q Ei + "ijkv+jBk ; J i = ^vi + ^q Ei + "ijkvj Bk : ^q is the `quantum critical conductivity' for each microscopic uid, and will be important to include. The presence of ^q has broken Galilean invariance microscopically, and will result in a non-Galilean magnetoresistance. The momentum quasi-conservation equations of the two uids are i!M^ v+i = ^Ei + "ijkJ +jBk i!M^ vi = ^Ei + "ijkJ j Bk (v+i (vi vi ); v+i): =M^ governs the rate at which the electron/hole uids exchange momentum. (3.5) and (3.6) form a set of equations which can be solved straightforwardly: upon doing so, we nd that 2i!M^ ^2 + ^q 2 + B2 ^q 2i M^ ! + B2 ( ^2 + 2^q ) iM^ !)^q i!M^ 2^q + ^ 2 : ; 2i!M^ 2i M^ ! ^2 + ^q 2 + B2 ^q i!M^ (!M^ )2 + B2 ^2 + 2^q i!M^ (3.7a) (3.7b) governs the rate of relaxation between the two uids. Assuming that is small enough that it can be treated within the gradient expansion of hydrodynamics, one can show using that the second law of thermodynamics implies Perhaps more intuitively, (3.8) can also be understood from the requirement that thermal equilibrium v i = 0 is stable. The approximations we have made in the last step of (3.7) are valid in the limit !M^ . In this limit, we expect single uid hydrodynamics with net charge density zero. ? = c and k = c + dB2. Upon making the identi cations M = 2M^ ; k = 2^q + ? = k + ^ 2 B2 ^q ; d = ^q : (x) T (x) T we see that (2.14) and (3.7) agree. Furthermore, we nd an expression for From (3.8), we conclude that d < 0. Using (2.18), we see that this model will exhibit positive magnetoresistance if momentum relaxation is strong enough. 4 Momentum relaxation rate So far, we have used a `mean eld' description of momentum relaxation. It is also possible that upon adding a magnetic eld, momentum can relax more or less e ciently in the direction of the magnetic eld. In this section, we will perturbatively compute the rate of momentum relaxation in a uid, disordered by very long wavelength inhomogeneity in an externally imposed chemical potential [20, 34]. In such a limit, the transport coe cients may be computed by solving the hydrodynamic equations in an inhomogeneous medium, which can be shown to be: jklvkBl jklvkBl 2 = 0; = 0; = ijkJ jBk: (4.1c) (4.1a) (4.1b) Suppose that (x) = +u^(x), with u perturbatively small. One can compute to leading order in u by either solving (4.1) in an inhomogeneous background [20, 34, 35], or by using the memory function formalism [23, 36{38]. For our purposes, it will be easier to do the latter. What one nds is that vi in (2.12) should be replaced by ij vj , with ij Z d3k ( 2 )3 kikj j (k)j2 lim !!0 with the retarded Green's function evaluated in the translation invariant theory, and (k) the Fourier transform of (x). The only non-zero contributions to ij will be at least O(u2). For simplicity in what follows, we will neglect the anisotropic corrections to the `quantum critical' conductivity q which can arise in a magnetic eld. The hydrodynamic Green's functions may be found by the following prescription [39]. Suppose the hydrodynamic equations of motion take the schematic form Let AB be the susceptibility matrix: AB = @'A=@ B, with B the thermodynamic conjugate variables to 'A. For us, 'A = ( ; ; T ti) and A = ( T =T; T =T; vi), and [39] The hydrodynamic retarded Green's function is From the equations of motion in a magnetic eld, we nd (neglecting viscous e ects, for simplicity, as these are subleading in the limit where ^(x) is extremely slowly varying [20, ikib1 + i 1 ijkkj Bk ikib2 + i 2 ijkkj Bk imkBk iki ikj ij+c ijkBk +P c +P jm + +P mjnBn AB = B ( + P ) ij 1 C : A GRAB(k; !) = M (k)((M (k) i!) 1 AB : 2 = b2 = T (4.2) (4.3) (4.4) (4.5) 1 C ; A (4.6) (4.7a) (4.7b) (4.8) 34, 35]) MAB = B with the parameters 0 1 = b1 = 0 1k2 T Recall the de nition of c in (2.7). nd is that the spectral weight then diverges as k 2: Using these equations, we now compute the (k; !) ! 0 limit of GR(k; !). What we lim !! ! 2 + c2B2 = A c [k2 ( 2 + 2c2B2) c2(kiBi)2] + ; where the thermodynamic prefactor A = ) It is straightforward to see that spectral weight is enhanced by a magnetic eld parallel to the wave vector. Hence, assuming (k) is random, upon performing the angular integral in (4.2), we nd that zz > xx = yy. Since zz= xx > 1, this type of inhomogeneity tends towards `positive magnetoresistance' by causing zz= xx to increase, relative to the scenario with homogeneous. At higher orders in k, the spectral weight of the charge density picks up additional HJEP06(217)54 contributions. Some of these contributions are proportional to viscosity, which has been neglected so far in our discussion. In the presence of a nite magnetic eld, the viscosity becomes a fourth-rank tensor with multiple non-vanishing coe cients: see e.g. [22, 27]. Other contributions are proportional to third-order corrections (in spatial gradients) to the charge current J ; these can be thought of as k2 corrections to the coe cient c in (4.8). Nevertheless, it is instructive to compare (4.8) with a calculation of the spectral weight only accounting for viscous dissipation. Hence, let us set ij = 0, and assume that the only dissipative coe cient is the isotropic shear viscosity . In this case, one nds that (4.9) (4.10) (4.11) with lim !! A0 = ! = A0 B2 2(3k2 + kz2) + k6 2 B2 2kz2 + k6 2 Due to the divergence in the denominator of (4.10) as kz ! 0, it is clear from (4.2) that zz will be smaller than xx; the factor of kz2 in the formula for zz will cancel this small factor in the denominator, while no such cancellation will arise for xx. If viscous e ects are large, inhomogeneous chemical potentials lead to negative magnetoresistance. The angular dependence is thus qualitatively distinct from the case where ij 6= 0. 5 Conclusion In this article we have presented a relativistic hydrodynamic theory of magnetotransport in 3 + 1 dimensions. Depending on the `microscopic' models of interest, it is possible to obtain both positive and negative magnetoresistance within our framework. NMR is, in some ways, a more generic e ect for a relativistic or non-Galilean invariant uid: arising not only from the spatial anisotropies caused by the presence of background magnetic elds, but also from the particular structure of relativistic hydrodynamics. In particular, we found that the holographic D3/D7 system exhibited negative magnetoresistance. A two- uid cartoon of electron-hole plasma which exhibits positive magnetoresistance can also be found. Smooth disorder potentials also imply a slight positive magnetoresistance. In Galilean invariant uids, it has been shown [40{42] that magnetoresistance is sensitive only to viscosity. However, we have shown in section 4 that for other gradient expansions, this no longer remains the case. It is not clear whether magnetoresistance is a good viscometer for electron uids arising from general band structures. This may not be the case, even if the temperature is low enough that thermal e ects are negligible. One di culty, as we have noted, is that the viscosity itself is a fourth-rank tensor which can become anisotropic in the presence of a magnetic eld. Further work to resolve this question is warranted. Acknowledgments We thank Anton Andreev for comments on a draft of this paper. The work of AB and AK is supported, in part, by the US Department of Energy under grant number DE-SC0011637. AL was supported by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant GBMF4302. A Constitutive relations to third order in F Here we show that the non-relativistic constitutive relations can be derived from the most general covariant expression up to third order in the eld strength. The expression is J = u + q1F u + q2F F u + q3F 2F u + q4F + q5F F F u u u + q6F F (?F ) u + q7" F F u F ~F u ~ + q8F (?F ) (?F ) u + q9(?F ) F F u + q10(?F ) F (?F ) u : (A.1) We will only keep track of terms to \ rst order" in E, recalling that vi is rst order in E while B is zeroth order. The rst term simply gives which is expected by Galilean invariance. 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Andrew Baumgartner, Andreas Karch, Andrew Lucas. Magnetoresistance in relativistic hydrodynamics without anomalies, Journal of High Energy Physics, 2017, 54, DOI: 10.1007/JHEP06(2017)054