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Magnetic oscillations in a holographic liquid V. Giangreco M. Pulettia, S. Nowlingb, L. Thorlaciusa,c, T. Zinggd a University of Iceland, Science Institute, Dunhaga 3, 107 Reykjavik, Iceland b Jamestown Community College, 525 Falconer St, Jamestown, NY 14701, USA c The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova University Centre, 106 91 Stockholm, Sweden d Helsinki Institute of Physics P.O. Box 64, FI-00014 University of Helsinki, Finland We present a holographic perspective on magnetic oscillations in strongly correlated electron systems via a fluid of charged spin 1/2 particles outside a black brane in an asymptotically anti- 5 de-Sitter spacetime. The resulting back-reaction on the spacetime geometry and bulk gauge field 1 0 gives rise to magnetic oscillations in the dual field theory, which can be directly studied without 2 introducing probe fermions, and which differ from those predicted by Fermi liquid theory. y PACSnumbers: 11.25.Tq,04.40.Nr,71.10.Hf,75.45.+j a M Introduction: The de Haas – van Alphen effect [1] andthusextendstheso-calledelectronstarmodel[12,13] 1 refers to quantum oscillations in the magnetization as to include a magnetic field and finite temperature. We 2 a function of 1/B, present in metals at low temperature obtain oscillations in the magnetization directly from ] T andstrongmagneticfieldB. Thisphenomenonisgen- the bulk gravitational physics by incorporating Landau h erallyassociatedwiththeFermisurfaceandtheobserved quantization into the charged fluid description. The t - oscillations are usually interpreted in terms of Fermi liq- method differs conceptually from earlier probe fermion p uid theory and quasi-particles [2]. The measurement of computations of magnetic oscillations in an electron star e h quantum oscillations is a standard tool for investigat- background [14, 15] and it predicts a dependence on the [ ing the electronic structure of metallic systems and this magnetic field and temperature of the oscillation ampli- extends to strange metallic phases, where fermion exci- tude that departs from Fermi liquid theory. 2 v tations are strongly correlated and the quasi-particle de- The Model: Our approach is based on an extension 9 scriptionbreaksdown. Theobservationofquantumoscil- of the electron star geometry, developed in [12, 13, 16, 5 lations, in combination with ARPES experiments, leads 17]. For related work see also [18, 19]. This geometry 4 to the conclusion that low temperature physics in such is obtained by coupling Einstein – Maxwell theory to a 6 systemsisstillgovernedbyaFermisurface,althoughthe chargedperfectfluid,ofnon-interactingfermionsofmass 0 . underlyingphysicsisnotfullyunderstood, seee.g.[3,4], m and charge normalized to one in units of the Maxwell 1 and may not conform to the standard Fermi liquid pic- coupling constant e, 0 ture. In this paper, we provide a novel view of quantum 5 1 (cid:90) √ 1 oscillationsthatisnottiedtoaquasi-particledescription S = d4x −g(R−2Λ) : but is instead based on a holographic representation of 2κ2 v thestronglycorrelatedelectronsystemintermsofadual 1 (cid:90) √ (cid:90) √ i − d4x −gF Fµν − d4x −gL , (1) X gravitational model. 4e2 µν fl r a In recent years, gauge/gravity duality [5–7], has been where κ2 =8πGN is the gravitational coupling, the AdS applied to model strongly coupled dynamics in vari- lengthscaleLisgivenintermsofthenegativecosmolog- ous condensed matter systems including strange metals ical constant Λ = −3/L2, and κ/L (cid:28) 1 corresponds to (for reviews see e.g. [8, 9]). Holographic systems at fi- theclassicalgravity(largeN)regime. Thebulkfermions nite charge density exhibit interesting non-Fermi liquid aretreatedinaThomas–Fermiapproximation,validfor behavior, revealed for instance in spectral functions of model parameters satisfying probe fermions [10, 11]. Here we develop a simple holo- κ graphic model for strongly correlated fermions in a mag- mL(cid:29)1, e2 ∼ (cid:28)1. (2) L netic field and use it to study magnetic oscillations in an unconventional setting. The model involves a fluid of In [20, 21] more refined computations involving holo- chargedspin1/2particlesoutsideadyonicblackbranein graphic Fermi systems confirmed that the electron star 3+1dimensionalanti-deSitter(AdS)spacetime. Ittakes qualitatively reproduces essential features of these mod- into account the back-reaction due to charged bulk mat- els,evenbeyonditsa prioriregimeofvaliditydefinedby ter on the spacetime geometry and the bulk gauge field equation (2). 2 Previous work on magnetic effects in holographic met- Thisconvenientlyleadstosimpleexpressionsforthelocal als [14, 22–27] has not taken into account the full back- chemical potential and magnetic field, reaction on the geometry due to the presence of charged e e matter in a non-vanishing magnetic field at finite tem- µ (r)= aˆ(r), H (r)= Bˆr2. (8) loc κ loc κL perature. The semi-classical approximation used in the electron star construction, suitably generalized to finite Inserting this ansatz into the Einstein-Maxwell field B and T, also allows the back-reaction to be included equations coupled to a charged fluid leads to a system in the B (cid:29) T regime, which is of primary interest to of ODE’s for {aˆ(r),cˆ(r),gˆ(r)}, which can be solved by the study of magnetic oscillations. Including the back- numerical methods along the lines of [16]. The numeri- reaction provides direct access to the underlying strong cal algorithm is implemented on dimensionless field vari- coupling dynamics without having to introduce probe ables, denoted by a hat, obtained from their dimension- fermions. fulcounterpartsbyabsorbingappropriatepowersofκ,e, We work in a 3+1 dimensional spacetime with local and L. For more detail, we refer to the Appendix. coordinates (t,x,y,r), which asymptotes to AdS and The local chemical potential µ vanishes both at the 4 loc which is static, stationary and has translational symme- horizon, r = 1, and at the AdS boundary, r = 0. It fol- tryorthogonaltotheradialdirection. Atanygivenpoint lows that (cid:96) can only be positive inside a finite range filled in the spacetime the fluid is at rest in a local Lorentz r > r > r , defining the radial region where the fluid is i e frameeA,andthefluidvelocityisgivenbyuµ =eµ. The supported. The region between the horizon and the in- µ 0 state of the charged fluid is completely determined by a neredge ofthe fluid1>r >r isdescribed bya vacuum i local chemical potential and magnetic field, dyonicblackbranesolution. Wealsohaveavacuumsolu- tion in the region outside the fluid r > r > 0, but with µ =A uµ , H = e[µeν]F , (3) e loc µ loc 1 2 µν different black brane parameters due to the additional where A is a U(1) gauge potential, and F the cor- mass and charge of the intervening fluid. The magnetic µ µν responding field strength tensor. This means that the field is the same in all three regions as the fluid particles charge density σ, the pressure p, and the magnetization do not carry magnetic charge, but the bulk magnetiza- density η of the fluid depend only on µ and H . An tion varies due to the presence of the fluid. loc loc equation of state for the fluid is obtained as in [13], ex- Atypicalprofileforthefluidchargedensityσˆ isshown cept now the constituent dispersion relation is that of in fig. 1. The fluid does not extend all the way to the Dirac fermions in a magnetic field, (cid:96) E2 =m2+k2+(cid:0)2(cid:96)+1(cid:1)γH ±γH , (4) Σ (cid:96) loc loc where γ is a constant proportional to the gyromagnetic 1. ratio of the fermions. The index (cid:96) ≥ 0 labels Landau levels and the the last term on the right hand side is due to Zeeman splitting. Assuming 1/N effects due to 0.5 bulk thermalization can be neglected [16, 17], the local pressure, charge and magnetization of the fluid are then obtained by filling states up to the Fermi level given by µ . The number of occupied Landau levels is thus a loc 0. r local quantity, 0. 0.5 1. (cid:96) =(cid:22)µ2loc−m2(cid:23) , (5) FIG.1. Profileofthefluidchargedensityσˆ,withparameters filled 2γHloc chosen so that max(cid:96)filled =5. which is determined by the equations of motion and varies with radial position in the bulk geometries of in- horizon or the boundary and the bump-like shape of the terest. In particular, there is no fluid in regions where profile reveals the presence of jumps in the local number (cid:96) <0. filled of filled levels. The field variables only depend on the radial coordi- The AdS/CFT dictionary relates thermodynamic nate r and we choose the following parameterization for quantities of the dual field theory to properties of the the metric, bulkmetricandgaugefield. TemperatureTˆ andentropy L2 (cid:18) cˆ(r)2 (cid:19) SînthedualfieldtheoryaregivenbytheHawkingtem- ds2 = − dt2+dx2+dy2+gˆ(r)2dr2 , (6) r2 gˆ(r)2 peratureandentropyoftheblackbrane,whiletheenergy Eˆ,chemicalpotentialµˆ,chargeQˆ,magneticfieldstrength and non-zero components of the gauge potential, BândmagnetizationMˆ oftheboundarydualarereadoff eLcˆ(r)aˆ(r) eL fromtheasymptoticbehaviorofthebulkfields. Thefree A = , A = Bˆx. (7) t κ rgˆ(r) y κ energyfollowsfromevaluatingtheon-shellaction,andis 3 found to satisfy the standard thermodynamic relation, included. In the limit of vanishing magnetic field, the different Landau levels collapse to a continuum and one Fˆ =Eˆ−SˆTˆ −µˆQˆ . (9) obtains a softer dependence, pˆ∝ (µˆ−mˆ)5/2, that leads to a third order phase transition as was found in [17]. The field equations of the bulk theory give rise to the following equation of state for the dual theory (see Ap- Atnon-vanishingmagneticfieldonecanalsoapproach pendix for details), the phase transition by varying Bˆ at fixed temperature. In this case we find ∆Fˆ ∝(Bˆ −Bˆ)2 and the phase tran- c 3 Eˆ=SˆTˆ +µˆQˆ−MˆBˆ, (10) sition is again of second order. We anticipate that going 2 beyond the Thomas – Fermi approximation will change which agrees with the corresponding relation for dyonic the nature of the phase transition. Indeed, a first or- black branes in AdS . der phase transition was found at Bˆ = 0 using WBK 4 Results: The electron fluid solution is only found for wave functions for the fermions in [21] and we expect restrictedvaluesofBândTˆ. Intheabsenceofamagnetic this would be the case at finite Bâs well. field it was already observed in [16, 17], that there is a critical temperature Tˆc above which there is no electron (cid:84)(cid:96) fluidandaphasetransitiontoavacuumblackbranecon- 1 (cid:96) figuration occurs. A non-zero magnetic field brings two (cid:84)c newaspects. Firstofall,thetransitiontemperaturegoes 10(cid:45)2 downmonotonicallyasBîsincreaseduntilitreacheszero at a critical magnetic field Bˆ , above which no electron c fluid is supported at any temperature. This is evident 10(cid:45)4 from our numerical solutions, but can also be inferred from the analytic dyonic black brane solution (provided 10(cid:45)6 (cid:96) intheAppendix). RaisingeitherBôrTˆ lowersthemax- 50 100 150 200 1(cid:66) imum value reached by the local chemical potential µ loc as a function of r in the dyonic black brane background. (cid:144) FIG. 2. Phase diagram of dyonic electron fluid solutions for For sufficiently high Bând/or Tˆ the number of occupied mˆ =0.5,withaxesnormalizedsuchthattheboundarychem- levels(cid:96)filledin(5)willbenowherepositivesothatnofluid ical potential is µˆ = 1. The different color shadings mark can be supported and the vacuum dyonic black brane is parameter regions with different maximum numbers of occu- the only available solution. pied Landau levels, increasing from left to right. In the limit Second, the order of the phase transition changes. Bˆ→0theedgesbetweentheregionsallasymptotetoTˆc,the maximaltemperatureatwhichtheelectronfluidissupported Whereas in [16, 17] it was found to be of third order, at Bˆ=0. it becomes second order in the presence of a magnetic field. This can be shown by a similar analytic argument as was used in [17] for the Bˆ= 0 case. Consider a tem- The Bˆ−Tˆ phase diagram reveals a periodic feature peraturejustbelowthetransitiontemperatureatBˆ(cid:54)=0, in 1/Bˆ. A representative plot for mˆ = 0.5 is displayed keepingthemagneticfieldfixed. Theconditionµˆ >mˆ in fig. 2. Changing the value of mˆ in the numerical cal- loc isthensatisfiedinanarrowbandintheradialdirectionin culationsdoesnotsignificantlyaffectthephasediagram, the dyonic black brane solution and inside this band the apart from changing the critical values on the axes. Dif- fermions can occupy the lowest Landau level only. Fur- ferentcolorsmarkdifferentvaluesofthemaximumvalue thermore, the back-reaction on the geometry due to the of filled Landau levels (cid:96) , which increases from left to filled fermionfluidcanbeneglectedattemperaturesveryclose right. At low temperatures the edges between regions tothetransition. Inthepresenceofthefermionfluidthe with a different number of occupied levels occur at equal free energy of the system is lowered by an amount given intervals in 1/Bˆ. This periodic feature is even more ap- by the on-shell action of the fluid, i.e. the integral of parent in the plots showing the magnetization Mˆ as a the fluid pressure. Near the phase transition the pres- function of Bˆ in fig. 3. For temperatures close to the sure, given by equation (20) in the Appendix, scales as criticaltransitiontemperatureTˆ,themagnetizationdif- c pˆ∝(µˆ−mˆ)3/2. Ashortcalculationalongthelinesof[17] fersonlyslightlyfromthatofadyonicblackbraneatthe then results in a free energy difference, ∆Fˆ ∝(Tˆ −Tˆ)2, same temperature and magnetic field strength but when c between the solution with a fluid and a vacuum dyonic the temperature is lowered the magnetization oscillates. black brane, indicating a second order phase transition. The oscillations are clearly visible when Bˆ (cid:29) Tˆ, which The same behavior is also seen in our numerical solu- is the regime where the de Haas – van Alphen effect is tions of the field equations with the full back-reaction observed experimentally. 4 (cid:45)(cid:77)(cid:96) (cid:66)(cid:96) m(cid:96) (cid:61)9.(cid:215)10(cid:45)1, (cid:84)(cid:96)c (cid:61)2.84(cid:215)10(cid:45)3 (cid:45)(cid:77)(cid:96) (cid:66)(cid:96) m(cid:96) (cid:61)5.(cid:215)10(cid:45)1, (cid:84)(cid:96)c (cid:61)7.45(cid:215)10(cid:45)2 (cid:45)(cid:77)(cid:96) (cid:66)(cid:96) m(cid:96) (cid:61)1.(cid:215)10(cid:45)1, (cid:84)(cid:96)c (cid:61)6.36(cid:215)10(cid:45)1 c 2.5 (cid:144) (cid:144) b (cid:144) c c 2. 2. a (cid:72)a(cid:76) b (cid:72)b(cid:76) (cid:72)a(cid:76) 2.3 (cid:72)c(cid:76) 0 b (cid:72) (cid:76) 1. (cid:72)c(cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:45)2. (cid:72) (cid:76) 2.1 (cid:72) (cid:76) c (cid:96) (cid:96) (cid:96) 400 800 1200 1 (cid:66) 75 150 225 1 (cid:66) 20 40 60 1 (cid:66) (cid:144) (cid:144) (cid:72) (cid:76) (cid:144) FIG. 3. Magnetization vs. magnetic field strength for various values of mˆ. Solid lines correspond to the electron fluid geometry, dashed ones to a dyonic black brane at same temperature and chemical potential. The labels denote temperatures Tˆ/Tˆ =0.9(a), 0.3(b), 3·10−3 (c). Intheleftmostplot(b)wasomittedduetotoomuchvisualoverlapwiththeothercurves. c In the rightmost plot the relative sign of Mˆ and Bˆ changes as a function of magnetic field, indicating a crossover from a diamagnetic to a paramagnetic state. ` Another feature is that the magnetization of the elec- D1 B tron fluid configuration is lower than that of a dyonic black brane with the same parameters. This is more H ê L 60 pronounced as the value of mˆ is lowered and when mˆ becomes small enough, the state crosses over from diamagnetic to paramagnetic. The local magnetization is 40 the sum of two contributions, a diamagnetic one which originates from the black brane and a paramagnetic one 20 due to the fluid which is a gas of free electrons at zero ` temperature. Varying the parameter mˆ tunes the gravi- 10-6 10-4 10-2 T tationalattractionbetweentheblackbraneinthecenter and the electron fluid surrounding it. The weaker the FIG.4. PeriodofthedeHaas–vanAlphenoscillationsasa interaction, the larger the fluid region can grow and the function of temperature for mˆ =0.5, the dotted vertical line marks the critical temperature where the solution makes the more dominant the paramagnetism becomes. transition to a dyonic black brane. For small values of the magnetic field, the overall amplitude of the magnetization is linear in Bˆ, as can be seen in fig. 3. This differs from the behavior predicted by the Landau theory of Fermi liquids via the Kose- ture for the transition to the dyonic black brane, above vich – Lifshitz formula [2, 28]. It also differs from earlier which the electron fluid is no longer supported. holographic results obtained in a probe limit in [24, 25] Discussion: We have presented a holographic model and appears to be due to the gravitational back-reaction fora2+1dimensionalsystemofstronglycorrelatedelec- which is included in our model. trons in a magnetic field, involving 3+1 dimensional The plots in figure 3 do not show any overlap of os- fermions treated in a Thomas – Fermi approximation in cillations with different periods, suggesting that a single an asymptotically AdS dyonic black brane background, Fermi surface is responsible for the phenomenon. This is taking into account both the gravitational and electro- inagreementwith[15,24],whereitwasarguedthatmag- magnetic back-reaction due to the charged matter. The netic oscillations are dominated by a single (extremal) system exhibits de Haas – van Alphen oscillations that Fermi momentum, despite the large number of holo- appear to be dominated by a single sharp Fermi surface, graphicallysmearedFermisurfacesinthissystem,which while the oscillation amplitude has a non-Fermi liquid turn into a continuum in the Thomas – Fermi limit. character that departs from earlier probe fermion com- Figure 4 shows the period in 1/Bˆ of the quantum os- putations. cillations vs. temperature. It reveals a similar trend as Whilethesemi-classicalmodelstudiedhereprovidesa is observed in experiments, where the period is constant relatively simple framework for numerical computations, at low temperature but increases with rising T until the it is rather crude. A Thomas – Fermi treatment of bulk oscillations get washed out at higher temperatures. Our fermions is known to wash out some quantum features numerical results suggest that the oscillation period di- that are present in more realistic models [15, 29] and verges in the holographic model at the critical tempera- the sharp edges of the fluid profile for individual Lan- 5 dau levels can introduce fictitious non-analyticities into whichcompletelydeterminethestateofthechargedfluid observables that involve derivatives acting on the bulk atagivenpointinthebulkspacetime. Inparticular, the fields [17]. The latter problem can presumably be reme- electric current and magnetic polarization, which can be died by introducing thermal effects in the bulk fluid or expressedintermsoflocalchargeandmagnetizationden- by replacing the anisotropic electron star by a quantum sities, Jµ =σuµ, Mµν =2ηe[µeν], and the on-shell fluid 1 2 many-bodymodelbasedonWKBwavefunctionsforbulk Lagrangian density, given by the pressure L = −p for fl Landau levels, along the lines of [21], where the tails the static solutions we are considering, are all functions of the fermion wave functions naturally smooth out the of µ and H . loc loc edgesfoundinthefluiddescription, butweleavethisfor Theformalismweareusingderivesfromsocalledspin future work. fluid models, which have been studied in general relativ- Acknowledgements: We would like to thank N. Buc- ity since the 1970’s [30–35]. We have only presented the ciantini,E.Kiritsis,V.Jacobs,K.Schalm,andJ.Zaanen minimalingredientsneededtodescribethestaticgeome- for helpful discussions. This work was supported in part tries that are of interest here, but the full formalism can by the Icelandic Research Fund and by the University also handle more general dynamical backgrounds. of Iceland Research Fund. V.G.M.P. and L.T. acknowl- Fluid variables: The bulk variables that describe the edge the Swedish Research Council for funding under fluid are the charge density σ, the magnetization density contracts623-2011-1186and621-2014-5838,respectively. η and the pressure p. The fluid components are locally T.Z. was partially supported by the Nederlandse Organ- free fermions in an external magnetic field along the ra- isatie voor Wetenschappelijk Onderzoek (NWO) under dial direction, with dispersion relation theresearchprogramoftheStichtingvoorFundamenteel (cid:16) (cid:17) Onderzoek der Materie (FOM). E2 =m2+k2+ 2(cid:96)+1 γH ±γH , (15) (cid:96) loc loc where the index (cid:96) ≥ 0 labels the Landau levels, γ is a Appendix constant proportional to the gyromagnetic ratio of the constituent fermions, and the ± in the rightmost term is Fieldequations: Weadapttheelectronstarconstruc- due to Zeeman splitting. There is a degeneracy between tion developed in [12, 13, 16, 17] to include the effects of different sign Zeeman states in adjacent Landau levels. a background magnetic field. Our starting point is the The sum over levels can therefore be rearranged into a action (1) in the main text, which describes a charged sum(cid:80)(cid:48) ,wheretheprimeindicatesinsertingarelative fluid coupled to Einstein – Maxwell theory. We will be (cid:96)≥0 factor of 1/2 in the (cid:96)=0 term. The density of states is consideringstaticsolutionsdescribingachargedfluidsus- pendedabovethehorizonofaplanardyonicblackbrane (cid:88)(cid:48) E n(E)=βγH θ(E2−(cid:15)2) , (16) in 3+1 dimensional asymptotically AdS spacetime with loc (cid:96) (cid:112)E2−(cid:15)2 radial electric and magnetic fields and translation sym- (cid:96)≥0 (cid:96) metry in the two transverse directions. The field equa- (cid:112) where β is a constant and (cid:15) = m2+2(cid:96)γH is the tions are given by (cid:96) loc energy in the Landau level labelled by (cid:96). In the limit R − 1Rg +Λg =κ2(cid:0)Tem+Tfl +TJ +TM(cid:1) , of weak magnetic field the sum over Landau levels can µν 2 µν µν µν µν µν µν be replaced by an integral, which is easily performed to ∇νFµν =e2(Jµ+∇νMµν) , (11) reproducethedensityofstatesusedtoconstructanelec- where Jµ and Mµν are the fluid current and magnetiza- tron star in zero magnetic field in [13]. tion tensor, Thelocalchargedensityσisobtainedfromthedensity of states via δL δL Jµ =− fl , Mµν =−2 fl , (12) δAµ δFµν (cid:90) µloc σ = n(E)dE, (17) and the stress energy tensors are given by 0 1 (cid:16) 1 (cid:17) Tem = F F λ− g F Fλσ , andthepressureandmagnetizationdensityareobtained µν e2 µλ ν 4 µν λσ fromthechargedensitybythethermodynamicrelations, Tfl =−g L , µν µν fl ∂p ∂p TµJν =−J(µAν)+uλAλu(µJν)−uλJλu(µAν), =σ, =η, (18) ∂µ ∂H TM =M F λ+uλF ρu M −uλM u F ρ.(13) loc loc µν λ(µ ν) λ (µ ν)ρ λρ (µ ν) analogous to the electron star [13]. The constant of in- Let eA denote a local Lorentz frame where the fluid is µ tegration in p is fixed such that p vanishes for σ = 0. at rest. The fluid four velocity is then given by uµ = Using the density of states in (16) leads to the following eµ and the fluid components experience a local chemical 0 explicitexpressionsforthefluidvariablesintermsofthe potential and a local magnetic field, local chemical potential and local magnetic field, µ =A uµ, H = e[µeν]F , (14) loc µ loc 1 2 µν 6 σ =βγH (cid:88)(cid:48)θ(cid:0)µ2 −(cid:15)2(cid:1)(cid:113)µ2 −(cid:15)2, (19) loc loc (cid:96) loc (cid:96) (cid:96)≥0 (cid:34) (cid:32)(cid:112) (cid:33)(cid:35) p= γβH (cid:88)(cid:48)θ(cid:0)µ2 −(cid:15)2(cid:1) µ (cid:113)µ2 −(cid:15)2−(cid:15)2log µ2loc−(cid:15)2(cid:96) +µloc , (20) 2 loc loc (cid:96) loc loc (cid:96) (cid:96) (cid:15) (cid:96) (cid:96)≥0 (cid:34) (cid:32)(cid:112) (cid:33)(cid:35) η = γβ(cid:88)(cid:48)θ(cid:0)µ2 −(cid:15)2(cid:1) µ (cid:113)µ2 −(cid:15)2−(cid:0)2(cid:15)2−m2(cid:1)log µ2loc−(cid:15)2(cid:96) +µloc . (21) 2 loc (cid:96) loc loc (cid:96) (cid:96) (cid:15) (cid:96) (cid:96)≥0 The matter stress-energy tensors in (13) reduce to Wehaveintroducedtwoauxiliaryfunctions. Oneisqˆ(r), which is related to the value of the local electric field Tfl =pg , TJ =µ σu u , µν µν µν loc µ ν by eterF = e r2qˆ. The other is Mˆ, which originates 1 0 3 tr κL TM =− H η(e1e1 +e2e2). (22) fromthefunctionalderivative δS oftheon-shellaction, µν 2 loc µ ν µ ν δFxy and whose value at r = 0 is the magnetization in the We proceed to solve the combined Einstein and Maxwell boundary theory, Mˆ = lim Mˆ(r). Finally, energy- r→0 equations for this system with the above expressions for momentum conservation can be expressed as the fluid variables. Metric and gauge field ansatz: Parameterizing non- dpˆ =σˆdaˆ +2rBˆηˆ. (32) vanishing components of the tetrad as dr dr Lcˆ(r) L L The following quantity e0 = , e1 =e2 = , e3 = gˆ(r), (23) t r gˆ(r) x y r r r (cid:34) (cid:35) 3+pˆ 3 2aˆqˆ r(Bˆ2+qˆ2) and the gauge potential as Yˆ =cˆ − − +2BˆMˆ − (33) r3 r3gˆ2 rgˆ 2 eLcˆ(r)aˆ(r) eL At = κ rgˆ(r) , Ay = κ Bˆx, (24) is constant along the radial direction r when the field equations are satisfied, and later on we use this to de- yieldsthefollowinglocalchemicalpotentialandmagnetic termine (40), the equation of state in the dual boundary field, field theory. e e µ (r)= aˆ(r), H (r)= Bˆr2. (25) Solutions: In the presence of a charged fluid we have loc κ loc κL tosolvethefieldequations(27)-(31)numerically. Asdis- Hatsdenotedimensionlessquantitiesandwefindituseful cussedinthemaintext,thefluidisonlysupportedwhere to convert all parameters and field variables into dimen- the local chemical potential is larger than the minimum sionless form [13], energystateinthelowestLandaulevel. Insolutionswith κ e4L2 κ a non-vanishing fluid profile this condition is met inside mˆ = m, βˆ= β, γˆ = γ, a radial range 1 > r > r > r > 0, where r = 1 is the e κ2 eL i e radial location of the brane horizon, and r = 0 marks κ κL µˆ = µ , Hˆ = H , spatial infinity of the spacetime. loc e loc loc e loc In the region 1 > r > r , there is no fluid and the σˆ =eκL2σ, pˆ=κ2L2p, ηˆ=eκLη. (26) i solution of the field equations is a dyonic black brane, Final form of the field equations: The Einstein and aˆ(r) Maxwell equations (11), with stress-energy tensors given cˆ(r)=1, =Qˆ(1−r), rgˆ(r) by (22) and the ansatz (23)-(24) for the metric and the Maxwell gauge field, reduce to a system of first order 1 2+Qˆ2+Bˆ2 Qˆ2+Bˆ2 =1− r3+ r4. (34) ordinary differential equations, gˆ(r)2 2 2 dcˆ 1 r =− cˆgˆ2aˆσˆ, (27) The local chemical potential grows as we move out- dr 2 wards from the horizon and by equation (5) in the main rdgˆ =−3gˆ− 1gˆ3(Bˆ2r4+qˆ2r4−6−2pˆ+2aˆσˆ), (28) textthelowestLandaulevelcanbeoccupiedifµˆ2loc ≥mˆ2. dr 2 4 The radial position r = r outside the black brane hori- i rdaˆ =−aˆ −r2gˆqˆ− aˆgˆ2(Bˆ2r4+qˆ2r4−6−2pˆ), (29) zonwherethisconditionisfirstsatisfieddefinestheinner dr 2 4 edge of the fluid. The dyonic black brane solution then dqˆ 1 provides initial data at r = r for the subsequent nu- r =− gˆσˆ, (30) i dr r2 merical evaluation of the system of equations (27)-(31) dMˆ ηˆ 1 inthefluidregion. Athightemperature,theconditionis r =Bˆr− + aˆgˆ2σˆMˆ . (31) notsatisfiedanywhere outsidethebranehorizon. Inthis dr r 2 7 casetherewillbenofluidandtheonlysolutionisthedy- onic black brane itself. 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