]>NUPHB14558S0550-3213(19)30026-410.1016/j.nuclphysb.2019.01.018The AuthorsHigh Energy Physics – TheoryFig. 1Different modes when dˆ=600. The dashed lines are real part of ωˆ and the full lines are the imaginary part. Common colors indicate same modes. From top-left corner clockwise, φ=π2, 15π32, 12π59, and, 0. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)Fig. 1Fig. 2The logarithm of ratio of the analytical zero temperature value and the thermal numeric value of Im ω. The scaling of logdˆ has been chosen such that −13logdˆ=−13logd+logT+O(1). The temperature increases to the right. The circles represent the results from our computations and the line segments have been added to help analysis. For the left-hand figure, where 0.02dˆ1/3=kˆx≠0=kz, the slopes are from the left, 0, 73, −73 and −1 and for the right-hand figure, where 0.02dˆ2/9=kˆz≠0=kx, 0, 53, −53, and −13. The slopes conform with our analytic results.Fig. 2Fig. 3The real part of ωˆ for different dˆ scanned at various kˆx and kˆz. The deep blue indicates the value zero (hydrodynamic regime) while lighter shades are non-zero and positive (collisionless regime). Completely white regions we have not scanned over.Fig. 3Fig. 4The DC conductivity for various dˆ. The circles correspond to numerical values while the curve is our prediction for σˆxx(ω=0)=σˆzz(0)≡σˆL.Fig. 4Fig. 5The AC conductivity for various dˆ as a function of ωˆ. Dashed curves represent imaginary parts, solid curves represent real parts of the conductivities. Purple is for the zz component while blue is for the xx component. The figures on the same row have equal dˆ. From the top row to bottom, dˆ=6, 60, 600, and 6000. Left panel figure is the zoomed-in version of the right panel, i.e., focusing on smaller frequency window.Fig. 5Fig. 6The diffusion coefficient for a range of dˆ. The circles mark the numerical values while the solid curve is our analytical prediction.Fig. 6Low-energy modes in anisotropic holographic fluidsGeorgiosItsiosagitsios@gmail.comNikoJokelabcniko.jokela@helsinki.fiJarkkoJärveläbcjarkko.jarvela@helsinki.fiAlfonso V.Ramallodealfonso@fpaxp1.usc.esaInstituto de Físíca Teórica, UNESP-Universidade Estadual Paulista, R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo 01140-070, SP, BrazilInstituto de Físíca TeóricaUNESP-Universidade Estadual PaulistaR. Dr. Bento T. Ferraz 271, Bl. IISao PauloSP01140-070BrazilbDepartment of Physics, P.O. Box 64, FIN-00014 University of Helsinki, FinlandDepartment of PhysicsP.O. Box 64University of HelsinkiFIN-00014FinlandcHelsinki Institute of Physics, P.O. Box 64, FIN-00014 University of Helsinki, FinlandHelsinki Institute of PhysicsP.O. Box 64University of HelsinkiFIN-00014FinlanddDepartamento de Física de Partículas, Universidade de Santiago de Compostela, SpainDepartamento de Física de PartículasUniversidade de Santiago de CompostelaSpaineInstituto Galego de Física de Altas Enerxías (IGFAE), E-15782 Santiago de Compostela, SpainInstituto Galego de Física de Altas Enerxías (IGFAE)Santiago de CompostelaE-15782SpainAbstractIn this work we will study the low-energy collective behavior of spatially anisotropic dense fluids in four spacetime dimensions. We will embed a massless flavor D7-brane probe in a generic geometry which has a metric possessing anisotropy in the spatial components. We work out generic formulas of the low-energy excitation spectra and two-point functions for charged excitations at finite baryon chemical potential. In addition, we specialize to a certain Lifshitz geometry and discuss in great detail the scaling behavior of several different quantities.1IntroductionHolography has shown to be a useful tool to study various gauge field theories particularly at finite density [1–4]. The field has matured to the point where the low hanging fruit has been picked, while the more complicated problems have been sitting aside until recent years. One physically very relevant situation, where a complicated problem occurs, is a system where some or all of the spacetime symmetries are broken. Even in one of the simplest cases, where the translational symmetry is spontaneously broken in only one of the field theory directions leads to tedious calculations and numerical work. However, despite involved numerics, there is already a vast literature addressing holographic striped phases [5–25] and in particular the conductivities of charged excitations [26–33].Another way of breaking the symmetries is to maintain homogeneity but singling out some of the spatial directions and study anisotropic situations. The dual gravity backgrounds for many field theories possessing anisotropy have been constructed [34–43]. Given the fact that anisotropic backgrounds are in many ways computationally much tamer than inhomogeneous ones, surprisingly, not much has been said about the transport of excitation spectra or conductivities, for a recent example see, however, [44].In this paper, we set out to study a family of anisotropic backgrounds with generic metric components. We introduce a probe D7-brane thus adding fundamental degrees of freedom in the otherwise dual pure-glue field theories. We are particularly interested in low-energy excitations and transport properties of fundamental degrees of freedom. The backgrounds we have in mind have only been constructed numerically in the literature, and our aim in this paper is not to reconstruct them. Instead, upon generalizing the methods developed in [45–47], we leave our results in forms that are directly applicable if numerical backgrounds are plugged in. Here we are content with discussing general behavior with varying amounts of density. However, we do cross-check our analytic formulas numerically in certain Lifshitz geometries and find excellent agreement.We continue with the introduction by a technical review of the dual gravity setups that are relevant in the present case and we will put them in a broader physical context. Here we are going to be a bit cavalier about relating the conventions in different contexts and write down the formulas as in the original papers. However, in the rest of the paper we pay careful attention to all the numerical factors and keep consistent conventions and will in particular relate the two metrics discussed below.Most of the backgrounds with anisotropy found so far are inspired by the one obtained in [41], which corresponds to a system of D3- and D7-branes. The latter are completely dissolved in the geometry and induce a RR axion χ which is linear in one of the spatial Minkowski coordinates. In section 3 of [41] the authors found a running solution which interpolates between the scaling solution in the deep IR and the AdS5×S5 geometry in the UV. This solution has zero temperature and was found numerically. They also found, in section 7, a scaling D4–D6 solution. In this case there are two anisotropic directions w1 and w2 and the anisotropy is induced by a RR two-form F2∝dw1∧dw2. In appendix B they work out a ten-dimensional Lifshitz solution corresponding to a D3–D5 system with F-string sources. This solution is written in eq. (8.1) of [41] and corresponds to a dynamical exponent z=7.A more general class of anisotropic gravity solutions was found in [42,43]. This solution has non-zero temperature in general and generalizes the running solution of [41]. The solution found in [42,43] is numeric, although there are some analytic expressions for the functions in some limits. For example, there are expressions for small anisotropic parameter (actually for a≪T) (see appendix D in [43]). There are also expressions near the boundary (section 3) and for small T≪a (appendix E).The anisotropic background of Mateos–Trancanelli (MT) has been used, for example, in [48,49] to study the thermal photon production in a plasma. They embedded flavor D7-brane probes in the MT background and analyzed the fluctuations of the worldvolume gauge field (at zero charge density). The goal was to obtain the current-current correlators for photons with k0=|k→| and to get the photon production rate for different angles and energies. In [48] the quarks are massless, while in [49] the embedding of the flavor branes corresponds to massive quarks. In [50] the authors considered the effect of a constant magnetic field on the photon spectrum.In [51] the author studied the running of the shear viscosity η in the MT solution. The anisotropy induces a dependence of the shear viscosity with the scale, which manifests itself in a temperature dependence of η which violates the lower bound on the shear viscosity to the entropy ratio bound [52]. In [53] the author analyzed the Chern–Simons diffusion rates for the MT solution. To obtain this quantity one has to analyze the fluctuations of the axion field χ.The papers [54] and [55] deal with a generalization of the MT solution to the case in which a U(1) gauge field is added. This corresponds to an R-charge chemical potential. The black holes constructed are of the Reissner–Nordström type. In this solution the internal S5 is deformed, which corresponds to new internal components of the RR five-form F5. These two papers mimic the MT one, but the solution depends on an additional charge parameter Q, in addition to the axionic parameter a. They also address the thermodynamics in the presence of the chemical potential. The conductivities of this background were derived in [56].The MT approach is of course not the only one to generate anisotropy in holography. Indeed, already some time before MT, the authors of [57] found a solution of Einstein's equation corresponding to an anisotropic energy momentum tensor (with two pressures) and they analyzed the quasinormal modes for R-charge diffusion. The recent paper [58] obtained a new solution which, apart from the dilaton and axion, has an extra scalar field X. They argued that this new solution is thermodynamically preferred over the MT one at low temperatures.In an interesting and detailed paper [59] a new anisotropic solution was found. In this case the anisotropy is induced by a dilaton profile of the type ϕ=ρz (ρ being a constant similar to a in MT). Now the metric at the IR is of the type AdS4×R (and not Lifshitz). The section 3 includes the analysis of the anistropic thermodynamics in this setting.Another way of getting supergravity solutions with anisotropy is by considering backgrounds dual to non-commutative gauge theories. As an example of these, in [60], the charge diffusion in the D1–D3 solution of Maldacena–Russo is studied. The author solves Maxwell's equation in this background and derives the longitudinal and Hall conductivities.After this rather lengthy review of the existing literature, let us summarize our aim. We will address the problem of studying the charge transport properties of an anisotropic plasma using top-down holographic methods. Our main motivation to follow top-down approach, instead of a bottom-up approach, is that the field theory dual is well-established and the anisotropy has a well-defined origin in the gauge theory. We will concentrate on a particular setup in which the anisotropy is introduced by a space-dependent axion, which corresponds to N=4, (3+1)-dimensional super Yang–Mills deformed by a theta angle linearly dependent on one of the coordinates. The corresponding supergravity backgrounds have been obtained in references [41] and [42,43].We will start discussing the background geometry which has a general metric possessing anisotropy as in [42,43]. We then embed a probe D7-brane in this geometry and discuss the associated thermodynamics in Sec. 2 to the extent possible without specifying a particular solution. We then switch to discussing the fluctuation spectra of the flavor degrees of freedom in Sec. 3. We also work out the two-point functions and make a non-trivial check of the formulas. In Sec. 4 we specialize to the gravity solution found in [41] and evaluate the formulas laid out in the preceding section. In Sec. 5 we further perform a numerical analysis and show that our analytical formulas agree with the numerics very accurately. Sec. 5 contains a brief summary and an outlook of possible outgrowths of our work. Some computational details are relegated in App. A.2Gravity background and flavor thermodynamicsWe consider the low-energy physics on a probe brane in a spatially anisotropic background. The background setup was originally studied in [42,43]. The action is that of a type IIB supergravity where we only have a dilaton, an axion, and a RR five-form,(2.1)S=12κ102∫d10x−g[e−2ϕ(R+4∂Mϕ∂Mϕ)−12F12−14⋅5!F52], where M=0,…,9 and F1=dχ is the axion field strength. The metric Ansatz in the string frame is(2.2)ds2=Ls2u2(−FBdt2+dx2+dy2+Hdz2+du2F)+Ls2ZdΩS52, where all the components depend only on the radial coordinate u, which is 0 at the UV boundary. The RR five-form is set to be self-dual and chosen to be F5=α(ΩS5+⋆ΩS5), where ΩS5 is the volume element of a five-sphere and α is a constant determined by flux quantization. The axion is linear in this Ansatz: χ=az.The authors of [42,43] studied mostly solutions that are asymptotically AdS. This enables us to set some regularity conditions. Due to freedom in parametrization, we can set H(0)=B(0)=1 at the boundary along with ϕ(0)=0. The function F is the blackening factor and vanishes at the horizon, uH. The equations of motion for these Ansätze require α=4e−ϕ(0)Ls2=4Ls2 and F(0)=1/Z(0). Finally, the Ansatz was further simplified by setting(2.3)H=e−ϕ,Z=eϕ2, which restricts F(0)=Z(0)=1. With these, we only need to find solutions for F, B, and ϕ.The solutions emerging with these Ansätze exhibit scaling behavior with pure AdS5 metric at the UV boundary and it becomes more and more anisotropic closer to the horizon. The anisotropicity is controlled by the axion strength, a2. These solutions can be found analytically near the UV boundary at the low- and high-temperature limits. For intermediate temperatures, only numerical results are available. Due to the difficulty in obtaining analytical solutions, most of the following calculations will be presented with the metric in (2.2) although in a more condensed notation. Later on, we will consider a special case of fixed-point Lifshitz metric in Sec. 4.We now wish to embed a probe D7-brane into the geometry. We first find a classical solution and study its thermodynamics and then move on to the fluctuations in Sec. 3. For the metric of the five-sphere, we will be using the fibration(2.4)dΩS52=dθ2+sin2θdψ2+cos2θdΩS32. The probe D7-brane will span the directions (t,x,y,z,u,Ω3). We furthermore turn on a gauge field on the brane, F=At′(u)du∧dt. We have chosen the gauge Au=0. The brane will obey the dynamics following from the DBI action:(2.5)SDBI=−T7∫d8xe−ϕ−det(g+F), where the metric g is the induced metric on the brane, i.e.,(2.6)ds82=Ls2u2(−FBdt2+dx2+dy2+Hdz2+du2F)+Ls2Zcos2θdΩS32. Most of the time, we will set Ls=1 to avoid cluttering the equations. Moreover, notice that we choose to absorb the factors of 2πα′ in the definitions of the gauge fields. We only focus on massless fundamentals, so it is consistent to integrate over the internal directions as the embedding does not vary inside Ω3. In other words, ψ=const. and θ=0 are consistent solutions to the equations of motion.We choose the following conventions for the use of indices. Lower case Latin letters i,j,… denote spatial directions. Greek letters μ,ν,… correspond to Poincaré coordinates or the coordinates along the boundary. Capital Latin letters A,B,… correspond to all the coordinates of the metric. In addition, prime indicates differentiation with respect to u.The action evaluates to(2.7)SDBI=−T7VΩ3Ls3∫due−ϕgxxgzzZ3(|gtt|guu−At′2). The Ω3 is the volume of the three-sphere while V is the 4-volume of the space spanned by t, x, y, and z.We see that At is a cyclic variable so it can be easily solved from the Euler–Lagrange equations to give(2.8)At′=d|gtt|guuW+d2,W≡e−2ϕLs6Z3gxx2gzz. Here, d is a constant and we will show that it is proportional to the particle density.The chemical potential is(2.9)μ=∫0uHduAt′, while the particle density is(2.10)ρ=−1V(∂Ωgrand∂μ)V,T=−1V(∂d∂μ)V,T(∂Ωgrand∂d)V,T=T7Ω3d, where Ωgrand=−Son−shell is the unregularized grand potential,(2.11)Ωgrand=T7VΩ3Ls14∫0uHdu|gtt|guuWd2+W=T7VΩ3∫0uHdue−3ϕ4u−5BLs12+d2u6e32ϕ.For further thermodynamic computations, the on-shell action needs to be regularized. The simplest way for this is to subtract the 0-density action,(2.12)Sreg.(on−shell)=−T7VΩ3Ls3∫0uHdu|gtt|guu(Wd2+W−W). If d was small, we would also need to consider the contributions from the d=0 solution. The proper regularization of the d=0 action has been done in [61]. We only care about energy differences, so this suffices to our needs.The temperature is given by the formula(2.13)TH=B(uH)|F′(uH)|4π.The energy density can be found with a Legendre transformation ϵ=ΩgrandV+μρ. In order to determine the pressures along the different directions, let us put the system in a box of sides Lx, Ly and Lz. Then, the pressure along direction q can be found with(2.14)pq=−LqV∂Ωgrand∂Lq,q=x,y,z. The computation of pressures seems like a trivial task, after all, all the unknown quantities depend only on u. However, the z direction is a special case and the action depends non-trivially on Lz as seen in the explicit Lifshitz scaling solution of section 4. With these, we can also compute the speed of sound with(2.15)cq2=(∂pq∂ϵ)s. Later in Sec. 4 we will explicitly evaluate all the formulas in this section in a particular geometry.3Low-energy modesWe now move on to considering the spectrum of fluctuations of the gauge fields. We modify the action g+F→(g+F+f) and expand to second order in f where fμν=∂μaν−∂νaμ. To expand the action, we need to expand the determinant(3.1)−det(g+F+f)=−det(g+F)det(1+(g+F)−1f). We can use the following expansion(3.2)det(1+X)=1+TrX2+(TrX)28−TrX24+O(X3). The first order terms vanish as we are fluctuating around a saddle point. The inverse of (g+F) is(3.3)(g+F)−1=[−guuguu|gtt|−At′2At′|gtt|guu−At′200⋯−At′|gtt|guu−At′2|gtt|guu|gtt|−At′200⋯00gzz−10⋯000gxx−1⋯⋮⋮⋮⋮⋱]=G+J, where G is the diagonal part while J is the antisymmetric part of the matrix. In the above matrix, t is the first coordinate, u the second, z the third etc.The second order term of the determinant is(3.4)−det(g+F+f)2ndord.=|gtt|guu−At′2(JABJCD8−GADGBC+JDAJBC4)×fABfCD. We get the equations of motion from the Euler–Lagrange equations. We make the assumption that aμ do not depend on the spherical coordinates. The equations are(3.5)∂NW|gtt|guuW+d2(JNMJAB2−GMAGBN−JBNJMA)fAB=0 for all M. It turns out that in our case when only the tu components of J are non-zero, the J matrices do not contribute to either the action nor the equations of motion. They will not appear in the rest of the calculations.Our gauge condition is au=0. We get an important constraint equation from this by setting m=u(3.6)Gtt∂tat′+∑iGii∂iai′=0⇔∂tat′−∑ivi2∂iai′=0,vi2=−GiiGtt. For spatial or temporal coordinates, we have(3.7)aμ″+∂ulog(|gtt|guuWW+d2GμμGuu)aμ′−∑λGλλGuu∂λfμλ=0.We make a further assumption. Due to rotational invariance in the xy-plane, we assume that aμ's are independent of y. Then we Fourier transform along directions t, x, and z with(3.8)aμ(u,t,x,z)=∫R3dωdkxdkzaμ(u,ω,k→)ei(k→⋅(x,z)−ωt). The equations of motion as functions of ω and k→=(kx,kz) are(3.9)ωat′+(kxvx2ax′+kzvz2az′)=0at″+∂ulog(guu|gtt|WW+d2GuuGtt)at′+GttGuu(vx2kxEx+vz2kzEz)=0ax″+∂ulog(guu|gtt|WW+d2GuuGxx)ax′−GttGuu(ωEx+kzvz2ω(kxEz−kzEx))=0,az″+∂ulog(guu|gtt|WW+d2GuuGzz)az′−GttGuu(ωEz−kxvx2ω(kxEz−kzEx))=0, where we have already used gauge-invariant quantities(3.10)Ex=ωax+kxat,Ez=ωaz+kzat. Using the gauge constraint and definitions of Ex and Ez, we can solve the derivatives of aμ in terms of Ex and Ez(3.11)at′=1−ω2+(vx2kx2+vz2kz2)(vx2kxEx′+vz2kzEz′)ax′=−1−ω2+(vx2kx2+vz2kz2)((ω−kz2ωvz2)Ex′+kxkzωvz2Ez′)az′=−1−ω2+(vx2kx2+vz2kz2)((ω−kx2ωvx2)Ez′+kxkzωvx2Ex′).Plugging these into the above equations of motion, we can express the equations of motion in terms of the gauge-invariant fields. Using suitable linear combinations, we can find equations that have the 2nd derivative on only one field. For Ex it is(3.12)Ex″+(∂ulog(guu|gtt|WW+d2GuuGxx)+∂ulog(vx2)kx2vx2ω2−(vx2kx2+vz2kz2))Ex′+kxkzvz2∂ulog(vx2)Ez′ω2−(vx2kx2+vz2kz2)−GttGuu(ω2−(vx2kx2+vz2kz2))Ex=0, and for Ez it is(3.13)Ez″+(∂ulog(guu|gtt|WW+d2GuuGzz)+∂ulog(vz2)kz2vz2ω2−(kz2vz2+kx2vx2))Ez′+kxkzvx2∂ulog(vz2)ω2−(kz2vz2+kx2vx2)Ex′−GttGuu(ω2−(kz2vz2+kx2vx2))Ez=0.We can now start solving these equations. There are two important cases that can be solved analytically. For T=0, we will find a zero sound like dispersion relation and for T≠0 we will find a diffusion dispersion relation. The strategy for both cases involves solving the equations when ω,k→0 with the near-horizon limit and then demanding that these two limits commute. We will begin with the T=0 case.3.1T=0: zero sound modeFirst, we take the limit u→∞. We need the asymptotic behavior of the metric components,(3.14)F=F0uαf(1+O(1u)),B=B0u−αb(1+O(1u)),ϕ=ϕ˜0−αϕlog(uLs+O(1)). To preserve Lorentz invariance for the (t,x,y) components, we set αb=αf. We relax the conditions on (2.3) by setting H=H0e−ϕ, Z=Z0eϕ/2. The series expansions below are valid if αϕ<4, αf>−2, and αf−2αϕ>−2, which are the cases of interest to us, i.e., for the pure AdS metric and for the metric appearin later in (4.1).The equations of motion for E decouple in the asymptotic limit(3.15)Ex″+4+αf2uEx′+ω2B0F02uαfEx=0Ez″+4+αf−2αϕ2uEz′+ω2B0F02uαfEz=0. We require ingoing boundary conditions so the solution to these equations is(3.16)Eq=Fqurq/2Hrq2−αf(1)(2u1−αf2ωB0F0(2−αf)),rx=2+αf2,rz=2+αf−2αϕ2, where Fq is an integration constant and Hrq(1) is the Hankel function of the first kind. The ω→0 limit of these are(3.17)Eq≈Fq(1+icot(πrq2−αf)Γ(1+rq2−αf)ωrq2−αf(B0F0(2−αf))rq2−αf−iΓ(rq2−αf)urqπ(B0F0(2−αf))rq2−αfωrq2−αf).Second, we take the low-frequency limit of equations (3.9). The equations decouple and ax′ and az′ are easily solved. From these, we obtain Ex′ and Ez′ and then integrate them(3.18)aq″+∂ulog(guu|gtt|WW+d2GuuGqq)aq′=0,q=x,z. These are easily integrated and the gauge-invariant fields are(3.19)Ex=Ex,0+∫0udu˜[Cxωu˜2d2+e−ϕ/2HZ03Ls12u˜−6BF−kxωe−ϕ2HZ03Ls12B(Cxkx+Czkz)u˜8(d2+e−ϕ/2HZ03Ls12u˜−6)32]≡Ex,0+ωCxJx(u)−kx(Cxkx+Czkz)ωI(u)Ez=Ez,0+∫0uu˜[CzωHu˜2d2+e−ϕ/2HZ03Ls12u˜−6BF−kzωe−ϕ2HZ03Ls12B(Cxkx+Czkz)u˜8(d2+e−ϕ/2HZ03Ls12u˜−6)32]≡Ez,0+ωCzJz(u)−kz(Cxkx+Czkz)ωI(u), where, in the second step, we have defined the integrals Jx(u), Jz(u), and I(u). In the above expressions, Cx, Cz, Ex,0, and Ez,0 are integration constants. The next step is to approximate these expressions at the near-horizon limit(3.20)Ex=Ex,0+ωCxJˆx−kx(Cxkx+Czkz)ωIˆ−ωCx2B0dF0(2+αf)u−αf2−1Ez=Ez,0+ωCzJˆz−kz(Cxkx+Czkz)ωIˆ−ωCz2e−ϕ˜0H0B0dF0(2−2αϕ+αf)uαϕ−αf2−1Lsαϕ, where(3.21)Jˆx≡Jx(u=∞),Jˆz≡Jz(u=∞),Iˆ≡I(u=∞). We now match these expansions with the ones in (3.17). We first solve the Fx and Fz coefficients and then solve for Ex,0 and Ez,0, which will give us two linear equations in terms of Cx and Cz. Imposing the Dirichlet boundary conditions (Eq,0=0), the only way to obtain a non-trivial solution is to require singularity of the linear equation, which will give us the dispersion relation. After a few straightforward steps, we get(3.22)(Ex,0Ez,0)=−1ω××(−Iˆkx2+Jˆxω2−μxω6−αf2−αf−Iˆkxkz−Iˆkxkz−Iˆkz2+Jˆzω2−μzω6−αf−2αϕ2−αf)(CxCz), where the coefficients are(3.23)μx=−π(tan(παf2−αf)+i)(1B0F0(2−αf))42−αfdΓ(6−αf4−2αf)2μz=πH0e−ϕ˜0(tan(π(αϕ−αf)2−αf)−i)(1B0F0(2−αf))4−2αϕ2−αfdLsαϕΓ(6−αf−2αϕ4−2αf)2. The condition for the determinant to be zero, i.e., the dispersion relation, is given by the equation(3.24)(JˆzIˆω2−kz2−ω6−αf−2αϕ2−αfμzIˆ)(JˆxIˆω2−kx2−μxIˆω6−αf2−αf)=kx2kz2.The 1st order solution to this equation gives us the zero sound mode(3.25)ω2=kx2IˆJˆx+kz2IˆJˆz=k2(IˆJˆxcos2φ+IˆJˆzsin2φ)≡(cx2cosφ2+cz2sinφ2)k2≡k2cq2. The next order terms give us the damping of the mode. By defining ω=cqk+δω, we can extract(3.26)δω=k42−αfcq2αf2−αfIˆ2(μxJˆx2cos2φ+(kcq)−2αϕ2−αfμzJˆz2sin2φ).3.2T≠0: diffusion modeThis time we set T≠0 and take the near-horizon limit u→uH and the low-frequency limit ω∼k2→0. Note that we are implicitly expecting to find a diffusive solution, i.e., ω=−iDk2.We first take the near horizon limit of the equations of motion(3.27)Ex″+(1u−uH+bx)Ex′+(cx(u−uH)2+dxu−uH)Ex+fxEz′=0Ez″+(1u−uH+bz)Ez′+(cz(u−uH)2+dzu−uH)Ez+fzEx′=0, where the coefficients are(3.28)bx=B′(uH)2B(uH)+2uH−H0Z03Ls12(34ϕ′(uH)+3uH−1−kx2ω2B(uH)F′(uH))(d2uH6e3ϕ(uH)2+H0Z03Ls12)+F″(uH)2F′(uH)bz=B′(uH)2B(uH)+2uH+ϕ′(uH)−H0Z03Ls12(3uHϕ′(uH)+12)4uH(d2uH6e3ϕ(uH)2+H0Z03Ls12)+F″(uH)2F′(uH)+Z03Ls12kz2B(uH)eϕ(uH)F′(uH)ω2(d2uH6e3ϕ(uH)2+H0Z03Ls12)fx=Z03Ls12kxkzB(uH)eϕ(uH)2F′(uH)ω2(d2uH6eϕ(uH)+H0Z03Ls12e−12ϕ(uH))=eϕ(uH)H0fzdx=dz=−ω2(B′(uH)F′(uH)+B(uH)F″(uH))B(uH)2F′(uH)3−Ls12Z03(kz2eϕ(uH)+H0kx2)F′(uH)(d2uH6e3ϕ(uH)2+H0Z03Ls12)cx=cz=ω2B(uH)F′(uH)2.We solve these equations using the Frobenius series, i.e., Eq=Fq(u−uH)αq(1+βq(u−uH)+…), where αq, Fq and βq are all coefficients that might depend on ω and k. With the expansion, we can solve for αq's and βq's(3.29)αq=−icq=−iωB(uH)|F′(uH)|≡αβx,z=−dx,zFx,z+α(bx,zFx,z+fx,zFz,x)(1+2α)Fx,z, where we have chosen an explicit sign for αq in order to have an infalling solution. In the expression for βq, one either chooses the first indices or the second indices for all the terms. Taking the low-frequency limit with ω∼k2∼ϵ2, β's take the value(3.30)Fqβqkq=−iZ03Ls12eϕ(uH)B(uH)(FxH0e−ϕ(uH)kx+Fzkz)ω(d2uH6e3ϕ(uH)2+H0Z03Ls12)+O(ε2).For the other order, we once again first solve for ax′ and az′ and then write down the Eq′. The solution reads(3.31)aq′=Cq(guu|gtt|WW+d2GuuGqq)−1,q=x,z, which we then plug into the expression for E′'s(3.32)Ex′=ω2ax′−(kx2vx2ax′+kzkxaz′vz2)ωEz′=ω2az′−(kz2vz2az′+kxkzax′vx2)ω. To respect our low-frequency expansion, we can neglect the ω2 terms as ax′ and az′ should be of the same order.The integrated expressions for E's then read(3.33)Eq=Eq,0−∫0udu˜kqωe−ϕ2HZ03Ls12B(u˜)(Cxkx+Czkz)u˜8(d2+H0Z03Ls12e−3ϕ(u˜)/2u˜−6)3/2=Eq,0−kq(Cxkx+Czkz)ωI(u)≈Eq,0−kq(Cxkx+Czkz)ω×(I(uH)+e−ϕ(uH)2H(uH)Z03Ls12B(uH)uH8(d2+H0Z03Ls12e−3ϕ(uH)/2uH−6)3/2(u−uH)), where we have also done a near-horizon expansion.We must now match our two solutions. First, we set (u−uH)α=Enh and Cq∝Fq, then match the two terms in the expansions. We get the relation(3.34)(Ex,0Ez,0)=−iωuH2d2+e−3ϕ(uH)/2H0Z03Ls12uH−6×(ω+ikx2DxikxkzDxikxkzDxH(uH)(ω+ikz2DxH(uH)))(CxCz), where(3.35)Dx=I(uH)uH2d2+e−3ϕ(uH)/2Z03H0Ls12uH−6. A non-trivial solution to Dirichlet boundary conditions is provided only when the matrix is singular, i.e., when(3.36)ω=−i(kx2+kz2/H(uH))Dx, which corresponds to a diffusion mode.3.3Two-point functionsWe now move on to compute two-point functions of this system both in zero temperature and finite temperature. From these two-point functions we can extract the conductivity with which we can do a non-trivial consistency check through the Einstein relation.Including all prefactors in (3.4), we get the 2nd order Lagrangian(3.37)L=−T7Ls3|gtt|guuWW+d2(∑A<BGAAGBB(fAB)22). First of all, we drop all the fields in directions other than t, z, or x. To use gauge invariant quantities, we write the bracketed sum first as(3.38)∑A<BGAAGBB(fAB)2=Guu(Gtt(at′)2+Gxx(ax′)2+Gzz(az′)2)+GttGxxEx2+GttGzzEz2+1ω2GxxGzz(Exkz−Ezkx)2, where we already performed a Fourier transform. All the multiples of fields are to be interpreted as(3.39)EqEq′=Eq⁎(k,ω)Eq′(k,ω). The term with derivatives with respect to u can then be written as(3.40)Gxx(Gzzkz2+Gttω2)(Ex′)2+Gzz(Gxxkx2+Gttω2)(Ez′)2−(GxxGzzkxkz)(Ez′Ex′+Ex′Ez′)ω2(ω2Gtt+kz2Gzz+kx2Gxx)/Guu.We now do a partial integration with respect to u for the terms which include Eq′ terms. The strategy is to integrate the complex conjugated field and then differentiate the rest of the expression. Making use of equations of motion, it turns out that all the remaining bulk integrals cancel so we are left with only a surface integral.(3.41)Son−shell(2)=T7Ω3Ls32∫d4k|gtt|guuWGuuW+d2×Gxx(Gzzkz2+Gttω2)ExEx′+Gzz(Gxxkx2+Gttω2)EzEz′−(GxxGzzkxkz)(EzEx′+ExEz′)ω2(ω2Gtt+kz2Gzz+kx2Gxx)|u→0. As a next step, we consider a low-energy limit, yielding(3.42)Son−shell(2)=T7Ω3Ls32∫d4k1ω(Ex,0(k)⁎Cx(k)+Ez,0(k)⁎Cz(k)). The final steps involve expressing Cq's in terms of the boundary values and taking functional derivatives with respect to the boundary values.3.3.1T=0We invert equation (3.22) and get(3.43)(CxCz)=ω(Iˆkz2−Jˆzω2+μzω6−αf−2αϕ2−αf)(Iˆkx2−Jˆxω2+μxω6−αf2−αf)−Iˆkx2kz2×(Iˆkz2−Jˆzω2+μzω6−αf−2αϕ2−αf−Iˆkxkz−IˆkxkzIˆkx2−Jˆxω2+μxω6−αf2−αf)(Ex,0Ez,0).The on-shell action, in terms of the boundary values, now takes the form(3.44)Son−shell(2)=T7Ω3Ls32×∫d4k1(Iˆkz2−Jˆzω2+μzω6−αf−2αϕ2−αf)(Iˆkx2−Jˆxω2+μxω6−αf2−αf)−Iˆ2kx2kz2×((μzω6−αf−2αϕ2−αf−Jˆzω2+Iˆkz2)Ex,02+(μxω6−αf2−αf−Jˆxω2+Iˆkx2)Ez,02−Iˆkxkz(Ex,0Ez,0+Ez,0Ex,0)). Now we will simply take functional derivatives of the on-shell action to obtain the current-current correlators. Bear in mind that ∂∂aμ(k)=∂Ei(k)∂aμ(k)∂∂Ei(k). First, the tt correlator is(3.45)〈Jt(−k)Jt(k)〉=δδat⁎(k)δδat(k)S(2)=T7Ω3Ls3(μxω6−αf2−αfkz2+μzω6−αf−2αϕ2−αfkx2−ω2(kx2Jˆz+kz2Jˆx))(Iˆkz2−Jˆzω2+μzω6−αf−2αϕ2−αf)(Iˆkx2−Jˆxω2+μxω6−αf2−αf)−Iˆ2kx2kz2.Second, we consider direction q=cosχxˆ+sinχzˆ, i.e. the current Jq=Jxcosχ+Jzsinχ. In addition, for a more condensed notation, we use the notation kx=kcosφ and kz=ksinφ. The current two-point function becomes(3.46)〈Jq(−k)Jq(k)〉=〈Jx(−k)Jx(k)〉cos2χ+〈Jz(−k)Jz(k)〉sin2χ+〈Jx(−k)Jz(k)+Jz(−k)Jx(k)〉2sin2χ=T7ω2Ω3Ls3×μxω6−αf2−αfsin2χ+μzω6−αf−2αϕ2−αfcos2χ−ω2(Jˆzcos2χ+Jˆxsin2χ)+Iˆk2sin2(φ−χ)(Iˆkz2−Jˆzω2+μzω6−αf−2αϕ2−αf)(Iˆkx2−Jˆxω2+μxω6−αf2−αf)−Iˆ2kx2kz2. On the other hand, two perpendicular directions with q⊥=sinχxˆ−cosχzˆ have the two-point function(3.47)〈Jq(−k)Jq⊥(k)〉=T7ω2Ω3Ls3(sinχcosχ[μzω6−αf−2αϕ2−αf−μxω6−αf2−αf−ω2(Jˆz−Jˆx)]+Iˆk2sin(2(φ−χ))2)(Iˆkz2−Jˆzω2+μzω6−αf−2αϕ2−αf)(Iˆkx2−Jˆxω2+μxω6−αf2−αf)−Iˆ2kx2kz2.The conductivity tensor can be computed with the relation(3.48)σij(ω)=1iω〈Ji(−ω,k→=0)Jj(ω,k→=0)〉. In the low-frequency limit, the longitudinal and Hall conductivities are(3.49)σqq(ω)=iT7Ω3Ls3(Jˆzcos2χ+Jˆxsin2χ)JˆzJˆx1ω, and(3.50)σqq⊥(ω)=iT7Ω3Ls3sinχcosχ(Jˆz−Jˆx)JˆzJˆx1ω. The iω singularity resembles Drude conductivity and implies a delta peak for the real part of the conductivity at ω=0.3.3.2T≠0Inverting the matrix in (3.34) gives(3.51)(CxCz)=iuH2d2+e−3ϕ(uH)/2H0Z03Ls12uH−6H(uH)(ω+i(kx2+kz2H(uH))Dx)(H(uH)(ω+ikz2DxH(uH))−ikxkzDx−ikxkzDxω+ikx2Dx)×(Ex,0Ez,0), which we use to express the on-shell action in terms of field boundary values(3.52)Son−shell(2)=iT7Ω3Ls3d2+e−3ϕ(uH)/2H0Z03Ls12uH−6uH22×∫d4kH(uH)(ω+iDxH(uH)kz2)Ex,02+(ω+iDxkx2)Ez,02−ikxkzDx(Ez,0Ex,0+Ex,0Ez,0)ωH(uH)(ω+i(kx2+kz2H(uH))Dx).The two-point functions can then be easily computed. We list the same components as above(3.53)〈Jt(−k)Jt(k)〉=iT7Ω3Ls3d2+e−3ϕ(uH)/2H0Z03Ls12uH−6uH2(H(uH)kx2+kz2)H(uH)(ω+i(kx2+kz2H(uH))Dx)〈Jq(−k)Jq(k)〉=iT7Ω3Ls3ωd2+e−3ϕ(uH)/2H0Z03Ls12uH−6uH2(ω(H(uH)cos2χ+sin2χ)+ik2Dxsin2(φ−χ))H(uH)(ω+i(kx2+kz2H(uH))Dx)〈Jq(−k)Jq⊥(k)〉=iT7Ω3Ls3ωd2+e−3ϕ(uH)/2H0Z03Ls12uH−6uH2(ωcosχsinχ(H(uH)−1)+ik2Dxsin2(φ−χ)2)H(uH)(ω+i(kx2+kz2H(uH))Dx). The corresponding DC conductivities are(3.54)σqq=T7Ω3Ls3d2+e−2ϕ(uH)H(uH)Z(uH)3Ls12uH−6uH2(cos2χ+sin2χH(uH))σqq⊥=T7Ω3Ls3d2+e−2ϕ(uH)H(uH)Z(uH)3Ls12uH−6uH2(1−1H(uH))cosχsinχ. We see that these only depend on the near-horizon physics.3.3.3Einstein relationEinstein relation relates conductivity to susceptibility and diffusion in a non-trivial manner,(3.55)σ=Dχc. The validity of this relation has previously been checked and verified in many other holographic settings, starting with [63]. Susceptibility is computed with(3.56)χc−1=(∂μ∂ρ)T=Ls9T7Ω3∫0uHdue−2ϕBHZ3u8(d2+e−2ϕHZ3Ls12u−6)3/2=I(uH)T7Ω3Ls3. A simple algebraic exercise shows that the Einstein relation is indeed satisfied.4Fixed-point Lifshitz metricNo analytical non-trivial solutions for the system discussed in Sec. 2 are known. We wish to compare our analytical low-energy expressions to numerical ones. If we relax the regularity conditions near the boundary, we can find a closed form fixed-point Lifshitz-like solution, originally discovered in [41]. The solution is(4.1)α=6aZ05/2H0Ls5,eϕ=eϕ0(Lsu)4/7=83a2Ls2H0Z0(Lsu)4/7B=33B0Z049(Lsu)2/7,F=4933Z0(uLs)2/7(1−u22/7uH22/7)H=H0(uLs)4/7,Z=Z0(Lsu)2/7, where a is the axion parameter that determines the strength of the anisotropy and B0, H0, Z0 are free dimensionless parameters. Without losing generality, we can set them to unity. To obtain the original form of the fixed point metric in [41] from our solution above, we need to make a few modifications. By setting and scaling(4.2)u=Ls8Rs7r−7/6,{t,x,y}→1112Ls7Rs6{t˜,x˜,y˜},z→1112Ls5Rs4w˜,a=1211Rs4Ls5β, the constant Ls drops from the expressions and we obtain the string frame metric(4.3)ds2=R˜s2(r7/3(−f(r)dt˜2+dx˜2+dy˜2)+r5/3dw˜2+dr2r5/3f(r))+Rs2r1/3dΩS52f(r)=1−(rHr)11/3,eϕ=223βr2/3,R˜s2=1112Rs2,α=7211Rs4β. In the Einstein frame dsE2=e−ϕ/2ds2 and with rH=0, we see that the metric(4.4)dsE2=R˜E2(r2(−dt˜2+dx˜2+dy˜2)+r4/3dw˜2+dr2r2)+RE2dΩS52, exhibits a Lifshitz-like scaling with {t˜,x˜,y˜}→k{t˜,x˜,y˜}, w˜→k2/3w˜, and r→k−1r. In the metric, RE2=3β22Rs2. Notice that one cannot consider the isotropic limit in this solution due to double scaling limit.This solution was originally obtained by considering the one-form and the RR five-form to be sourced by N D3- and k D7-branes s.t. α=(2π)4NVol(S5), β=kLw, where Lw is the period of the w˜ coordinate. From low-energy flat space perspective, the branes are extended along the following directions(4.5)M4×S1×X5t˜x˜y˜rw˜s1s2s3s4s5ND3××××kD7××××××××.We will specialize our above solutions and computations to this special background metric. We will scale out thermal factors by redefining the radial coordinate and setting the horizon to reside at 1. The rescaled quantities we will decorate with hats as follows:(4.6)ω=ωˆuH−6/7Ls−1/7,kx=kˆxuH−6/7Ls−1/7,kz=kˆzuH−4/7Ls−3/7Ex=EˆxuH−2Ls2,Ez=EˆzuH−12/7Ls12/7. The only dimensionless parameter we thus have is the rescaled charge density dˆ,(4.7)dˆ=duH18/7Ls39/7eϕ0∝(μ0Ls)9/4Ls3T3, where high dˆ corresponds to high particle density or equivalently to low temperature and vice versa for low dˆ. Similar interpretations apply to kˆ and ωˆ. We will begin with the thermodynamic expressions and then move on to the low-energy excitations.4.1ThermodynamicsWe start by direct evaluation of the thermodynamic quantities as calculated in the general framework in Sec. 2. First, the temperature is given by(4.8)T=B(uH)|F′(uH)|4π=11121πLs1/7uH6/7. The regularized on-shell action is(4.9)−Ωgrand=Sreg.on−shell=−T7VΩ3∫0uHdu33e−ϕ0Ls54/77u33/7(11+dˆ2−1)=T7VΩ333e−ϕ0Ls54/726uH26/7(F12(−1318,12,518,−dˆ2)−1)=uH→∞T7VΩ333d13/9e4ϕ0/9Γ(518)Γ(119)26πLs1/3, where the last line is the zero-temperature result. The chemical potential is(4.10)μ=∫0uHdud337d2+e−2ϕ0Ls78/7u36/7(Lsu)15/7=μ0−Ls33F12(29,12;119;−dˆ−2)8(uH/Ls)8/7, where(4.11)μ0=11d4/9e4ϕ0/9Γ(29)Γ(518)123πLs1/3 is the zero-temperature chemical potential. The pressure and energy density expressions at low temperatures are(4.12)ϵ=T7Ω3[33d13/9e4ϕ0/952πLs1/3Γ(518)Γ(29)−33116π4/3d222/3Ls7/3T4/3]+O((LsT)13/3)py=px=T=0−ΩgrandVpz=T=0139px, from which we can compute the speed of sound at zero temperature(4.13)cs,y2=cs,x2=49,cs,z2=5281.4.2Low-energy excitations and conductivityAs for the low energy excitations, we can express the solution to the zero sound equation (3.24) in terms of the fixed point metric. The 1st order solution and the imaginary correction are(4.14)ωˆ2=49kˆx2+2dˆ2/9Γ(518)Γ(119)Γ(19)Γ(718)kˆz2Imδωˆ|φ=0=−9π3/2Γ(209)4223dˆ4/9Γ(29)2Γ(518)Γ(53)2kˆx7/3Imδωˆ|φ=π/2=−9(311)2/3π3/2Γ(518)5/6Γ(209)22dˆ27(Γ(19)Γ(718))11/6Γ(119)6Γ(43)2kˆz5/3, where we only report the x and z direction for the imaginary part to avoid cluttering the notation. Notice that the speed of first sound and zero sound in the x direction are equal. On the other hand, the speed of zero sound in the z direction depends on the particle density, making the zero sound fundamentally spatially anisotropic.As for the diffusion mode, we have (3.36) in terms of the fixed-point metric(4.15)ωˆ=−i(kˆx2+kˆz2)331+dˆ2F12(518,32;2318;−dˆ2)10.We can study the behavior of the diffusion in different limits of dˆ.(4.16)Dx∼1T,Dz∼Ls2/3T1/3,dˆ→0Dx∼μ0(LsT)7/3,Dz∼μ0(LsT)5/3,dˆ→∞.Finally, we express the DC conductivities in terms of this metric. The longitudinal and transverse conductivities in direction q=xˆcosχ+zˆsinχ and q⊥=xˆsinχ−zˆcosχ are(4.17)σqq=T7Ω3Ls60/7uH−4/7e−ϕ01+dˆ2(cos2χ+(LsuH)4/7sin2χ),σqq⊥=T7Ω3Ls60/7uH−4/71+dˆ2(1−(LsuH)4/7)cosχsinχ. Likewise, the behavior of DC conductivities in two different limits are(4.18)σxx∼T7Ls26/3T2/3,σzz∼T7Ls28/3T4/3,dˆ→0σxx∼T7Ls41/12μ09/4T7/3,σzz∼T7Ls49/12μ09/4T5/3,dˆ→∞.In the next section, we compare our analytical expressions to numerical ones.5Numerical verificationThe numerical methods we use are standard, see, e.g., [62]. All of our solutions are obtained at finite temperature.5.1Transition from hydrodynamic to collisionless regimeIn our analytical computations, we saw diffusion and zero sound modes at finite temperatures and zero temperatures, respectively. It is to be expected that, in numerical computations, both modes can appear, and, for different momenta and particle densities, the other is the preferred low-energy mode. The zero sound mode will naturally have to receive some thermal corrections [8,64,65].It turns out that the leading contributions at low momenta come from the diffusion mode. When we scan higher momenta, we see the emergence of a non-zero real part. The higher the particle density or the lower the temperature, the lower the transition momenta. In the numerical analysis, we can also see other purely imaginary modes. See Fig. 1 for examples. In addition, x and z directions behave differently as the system is fundamentally spatially anisotropic and the speed of zero sound is radically different for the two directions.The real part of ωˆ receives little thermal corrections when it comes to ∇kˆωˆ. The case is very different for the imaginary part. We can study this more quantitatively by fixing k and d and scaling kˆ and dˆ appropriately to probe the effect of the temperature. For kz=0, we fix kˆx=0.02dˆ1/3 and, for kx=0, we pick kˆz=0.02dˆ2/9. We compare this to the imaginary part of the zero sound at various dˆ to extract the thermal correction to Im ω. The Fig. 2 contains the logarithmic comparisons of the numerical and analytic values at T=0. At low temperatures and high temperatures, the numerical and analytic values agree. At intermediate temperatures, a thermal power law correction can be extracted. For kx≠0, the thermal correction is Imω(T)=Imω(T=0)(1+γxT7/3) and for kz≠0, it is Imω(T)=Imω(T=0)(1+γzT5/3), where γ's are unknown coefficients.In addition, the point of transition between the hydrodynamic regime and collisionless regime has a simple scaling law for d→∞. The scaling of critical momentum and angular frequency in the two cardinal directions, x and z, are(5.1)kx,c∼Ls7/3d−4/9uH−2∼T7/3μ0Ls7/3,ωx,c∼Ls7/3d−4/9uH−2∼T7/3μ0Ls7/3,kz,c∼Ls10/7d−1/3uH−10/7∼T5/3μ03/4Ls17/12,ωx,c∼Ls23/21d−2/9uH−10/7∼T5/3μ01/2Ls7/6. We have numerically verified all these behaviors.Additionally, we wish to consider directions other than purely x and z. We can scan the low energy modes at various mixed directions and different dˆ. It turns out that low-temperature and high-density regimes exhibit non-trivial transitioning between hydrodynamic and collisionless regimes. For some small kx, increasing |kz| might give us a non-zero real part, followed by a vanishing real part, see Fig. 3. The effect is less prominent for higher temperatures and lower particle densities. The behavior can be better understood by looking at the mixed angle spectra in Fig. 1. The transition to zero sound happens at lower momentum. The initial slope is consistent with cxcosφ from (4.14). Later on, the zero sound mode transition is complete and the slope is consistent with cq. It is interesting to note that the anisotropy really enter non-trivially in the momentum plane. It would be instructive to perform the same computation in the MT background and check how the non-trivial boundary shape of the hydrodynamic to collisionless regime scales with the anisotropy parameter a.5.2Quantitative comparisonsWe scale uH and Ls out of the conductivity quantities. We can naturally extract the xx and zz components of the (longitudinal) conductivity tensor.(5.2)σxx≡T7Ω3Ls60/7e−ϕ0σˆxxuH4/7=ω=0T7Ω3Ls60/7e−ϕ01+dˆ2uH4/7,σzz≡T7Ω3Ls64/7e−ϕ0σˆzzuH8/7=ω=0T7Ω3Ls64/7e−ϕ01+dˆ2uH8/7. The last equalities are for DC conductivity for which we see that, σˆxx=ωˆ=0σˆzz.The numerical analysis confirms the DC conductivity to great accuracy, see Fig. 4. The low frequency AC conductivity is heavily affected by the thermal effects and our analytical results turn out to be unreliable at high frequencies. The real part of the conductivities start declining at low momenta and then start growing when ω→∞ while the imaginary part becomes increasingly more negative. The behavior of σˆzz and σˆxx differ significantly which was to be expected from our analytical computations. The imaginary part of σˆzz becomes negative at very low momenta at low dˆ while the imaginary part of σˆxx becomes positive at low momenta but is negative at higher momenta. Fig. 5 contains plots of AC conductivity computations.When we scaled out uH, the diffusion coefficient for low momenta became equal for all directions. We find excellent agreement between the numerical and analytical value, as depicted in Fig. 6.6Conclusions and outlookIn this paper we performed an in-depth study of flavor physics in an anisotropic background. We investigated both the thermodynamics of massless fundamental degrees of freedom as well as their dynamical properties in a cold and dense environment. One of our intermediate goals was to fill in an important gap in the literature as anisotropic backgrounds have become more and more under the scope of recent discussions. The results that we delivered can be directly used in many contexts since we kept the metric components general.A long-term goal and definitely a much more ambitious one is to identify the physical system that could be genuinely modeled by the anisotropic setups we studied here. While the original Mateos–Trancanelli background was fairly well suited in studying heavy ion physics, here the emphasis has been on dense and cold regimes, naturally reached inside neutron stars. For example, perturbations of the self-gravitating fluids will induce anisotropic stresses. The local anisotropy of energy density and pressure may then even lead to cracking of the star [66–68]. If anisotropy is present in the ultradense regime in neutron stars, it will have significant effects on the stellar properties and structures.Recently, the first steps have been taken in holography to model dense and cold quark matter phase that could be realized inside neutron stars [69,70]. A generalization of such a study by incorporating the anisotropy in the strongly coupled quark matter phase would be interesting. This might potentially pave the way forward in understanding if anisotropies are relevant in compact objects. We hope to return to this issue in the future.AcknowledgementsWe would like to thank Matteo Baggioli for an interesting comment. The work of G. I. is supported by FAPESP grant 2016/08972-0 and 2014/18634-9. N. J. is supported in part by the Academy of Finland grant no. 1303622. J. J. is in part supported by the Academy of Finland grant no. 1297472 and the U. Helsinki Graduate School PAPU. A. V. R. is funded by the Spanish grants FPA2014-52218-P and FPA2017-84436-P by Xunta de Galicia (GRC2013-024), by FEDER and by the Maria de Maeztu Unit of Excellence MDM-2016-0692.Appendix ASome useful integralsLet us collect in this appendix some integrals which are useful in the analysis of the collective excitations of the matter in case of the fixed point Lifshitz-like metric. First of all, we define the integral Iλ1,λ2(u) as:(A.1)Iλ1,λ2(u)≡∫0uu˜λ1du˜(u˜λ2+d2)12. This integral can be explicitly performed in terms of the hypergeometric function:(A.2)Iλ1,λ2(r)=22+2λ1−λ2u1+λ1−λ22F(12,12−λ1+1λ2;32−λ1+1λ2;−d2uλ2). For large u, assuming that λ2 and λ1+1 are negative, we have the expansion:(A.3)Iλ1,λ2(u)=−1λ2B(λ1+1λ2,12−λ1+1λ2)d2λ1+1λ2−1+uλ1+1d(λ1+1)+…. Let us next define Jλ1,λ2(u) in the form:(A.4)Jλ1,λ2(u)≡∫0uu˜λ1du˜(u˜λ2+d2)32, which can also be computed explicitly:(A.5)Jλ1,λ2(u)=22+2λ1−3λ2u1+λ1−3λ22F(32,32−λ1+1λ2;52−λ1+1λ2;−d2uλ2). For large u, when λ2 and λ1+1 are both negative, we can expand Jλ1,λ2(u) as:(A.6)Jλ1,λ2(u)=−1λ2B(λ1+1λ2,32−λ1+1λ2)d2λ1+1λ2−3+uλ1+1(λ1+1)d3+….References[1]J.Casalderrey-SolanaH.LiuD.MateosK.RajagopalU.A.WiedemannGauge/String Duality, Hot QCD and Heavy Ion Collisions2014Cambridge University PressCambridge, UKarXiv:1101.0618 [hep-th][2]N.BrambillaQCD and strongly coupled gauge theories: challenges and perspectivesEur. Phys. J. C741020142981arXiv:1404.3723 [hep-ph][3]J. Zaanen, Y.W. Sun, Y. Liu, K. Schalm, Holographic Duality in Condensed Matter Physics.[4]S.A.HartnollA.LucasS.SachdevHolographic quantum matterarXiv:1612.07324 [hep-th][5]H.OoguriC.S.ParkHolographic end-point of spatially modulated phase transitionPhys. Rev. D822010126001arXiv:1007.3737 [hep-th][6]H.OoguriC.S.ParkSpatially modulated phase in holographic quark-gluon plasmaPhys. Rev. Lett.1062011061601arXiv:1011.4144 [hep-th][7]C.A.B.BayonaK.PeetersM.ZamaklarA non-homogeneous ground state of the low-temperature Sakai–Sugimoto modelJ. High Energy Phys.11062011092arXiv:1104.2291 [hep-th][8]O.BergmanN.JokelaG.LifschytzM.LippertStriped instability of a holographic Fermi-like liquidJ. High Energy Phys.11102011034arXiv:1106.3883 [hep-th][9]A.DonosJ.P.GauntlettHelical superconducting black holesPhys. Rev. Lett.1082012211601arXiv:1203.0533 [hep-th][10]N.JokelaG.LifschytzM.LippertMagnetic effects in a holographic Fermi-like liquidJ. High Energy Phys.12052012105arXiv:1204.3914 [hep-th][11]A.Ballon-BayonaK.PeetersM.ZamaklarA chiral magnetic spiral in the holographic Sakai–Sugimoto modelJ. High Energy Phys.12112012164arXiv:1209.1953 [hep-th][12]Y.-Y.BuJ.ErdmengerJ.P.ShockM.StrydomMagnetic field induced lattice ground states from holographyJ. High Energy Phys.13032013165arXiv:1210.6669 [hep-th][13]N.JokelaM.JärvinenM.LippertFluctuations and instabilities of a holographic metalJ. High Energy Phys.13022013007arXiv:1211.1381 [hep-th][14]M.RozaliD.SmythE.SorkinJ.B.StangHolographic stripesPhys. Rev. Lett.110202013201603arXiv:1211.5600 [hep-th][15]A.DonosJ.P.GauntlettJ.SonnerB.WithersCompeting orders in M-theory: superfluids, stripes and metamagnetismJ. High Energy Phys.13032013108arXiv:1212.0871 [hep-th][16]A.DonosStriped phases from holographyJ. High Energy Phys.13052013059arXiv:1303.7211 [hep-th][17]B.WithersBlack branes dual to striped phasesClass. Quantum Gravity302013155025arXiv:1304.0129 [hep-th][18]B.WithersThe moduli space of striped black branesarXiv:1304.2011 [hep-th][19]M.RozaliD.SmythE.SorkinJ.B.StangStriped order in AdS/CFT correspondencePhys. Rev. D87122013126007arXiv:1304.3130 [hep-th][20]Y.LingC.NiuJ.P.WuZ.Y.XianH.ZhangMetal-insulator transition by holographic charge density wavesPhys. Rev. Lett.1132014091602arXiv:1404.0777 [hep-th][21]N.JokelaM.JärvinenM.LippertGravity dual of spin and charge density wavesJ. High Energy Phys.14122014083arXiv:1408.1397 [hep-th][22]A.DonosC.PantelidouHolographic magnetisation density wavesJ. High Energy Phys.16102016038arXiv:1607.01807 [hep-th][23]A.AmorettiD.AreanR.ArgurioD.MussoL.A.Pando ZayasA holographic perspective on phonons and pseudo-phononsarXiv:1611.09344 [hep-th][24]S.CremoniniL.LiJ.RenIntertwined orders in holography: pair and charge density wavesarXiv:1705.05390 [hep-th][25]N.JokelaG.LifschytzM.LippertStriped anyonic fluidsPhys. Rev. D9642017046016arXiv:1706.05006 [hep-th][26]N.JokelaM.JärvinenM.LippertHolographic sliding stripesPhys. Rev. D9582017086006arXiv:1612.07323 [hep-th][27]S.CremoniniA.HooverL.LiBackreacted DBI magnetotransport with momentum dissipationJ. High Energy Phys.17102017133arXiv:1707.01505 [hep-th][28]N.JokelaM.JärvinenM.LippertPinning of holographic sliding stripesPhys. Rev. D96102017106017arXiv:1708.07837 [hep-th][29]A.AmorettiD.AreánB.GoutérauxD.MussoEffective holographic theory of charge density wavesPhys. Rev. D9782018086017arXiv:1711.06610 [hep-th][30]A.AmorettiD.AreánB.GoutérauxD.MussoDC resistivity of quantum critical, charge density wave states from gauge-gravity dualityPhys. Rev. Lett.120172018171603arXiv:1712.07994 [hep-th][31]A.DonosJ.P.GauntlettT.GriffinV.ZiogasIncoherent transport for phases that spontaneously break translationsJ. High Energy Phys.18042018053arXiv:1801.09084 [hep-th][32]B.GoutérauxN.JokelaA.PönniIncoherent conductivity of holographic charge density wavesJ. High Energy Phys.18072018004arXiv:1803.03089 [hep-th][33]W.J.LiJ.P.WuA simple holographic model of the Goldstone modes for translational symmetry breakingarXiv:1808.03142 [hep-th][34]D.GiataganasProbing strongly coupled anisotropic plasmaJ. High Energy Phys.12072012031arXiv:1202.4436 [hep-th][35]R.RougemontR.CritelliJ.NoronhaAnisotropic heavy quark potential in strongly-coupled N=4 SYM in a magnetic fieldPhys. Rev. D9162015066001arXiv:1409.0556 [hep-th][36]J.F.FuiniL.G.YaffeFar-from-equilibrium dynamics of a strongly coupled non-Abelian plasma with non-zero charge density or external magnetic fieldJ. High Energy Phys.15072015116arXiv:1503.07148 [hep-th][37]E.CondeH.LinJ.M.PeninA.V.RamalloD.ZoakosD3–D5 theories with unquenched flavorsNucl. Phys. B9142017599arXiv:1607.04998 [hep-th][38]U.GürsoyI.IatrakisM.JärvinenG.NijsInverse magnetic catalysis from improved holographic QCD in the veneziano limitJ. High Energy Phys.17032017053arXiv:1611.06339 [hep-th][39]J.M.PeninA.V.RamalloD.ZoakosAnisotropic D3–D5 black holes with unquenched flavorsJ. High Energy Phys.18022018139arXiv:1710.00548 [hep-th][40]Y.BeaN.JokelaA.PönniA.V.RamalloNoncommutative massive unquenched ABJMInt. J. Mod. Phys. A3314–152018185007810.1142/S0217751X18500781arXiv:1712.03285 [hep-th][41]T.AzeyanagiW.LiT.TakayanagiOn string theory duals of Lifshitz-like fixed pointsJ. High Energy Phys.09062009084arXiv:0905.0688 [hep-th][42]D.MateosD.TrancanelliThe anisotropic N=4 super Yang–Mills plasma and its instabilitiesPhys. Rev. Lett.1072011101601arXiv:1105.3472 [hep-th][43]D.MateosD.TrancanelliThermodynamics and instabilities of a strongly coupled anisotropic plasmaJ. High Energy Phys.11072011054arXiv:1106.1637 [hep-th][44]D.GiataganasU.GürsoyJ.F.PedrazaStrongly-coupled anisotropic gauge theories and holographyarXiv:1708.05691 [hep-th][45]N.JokelaA.V.RamalloUniversal properties of cold holographic matterPhys. Rev. D9222015026004arXiv:1503.04327 [hep-th][46]G.ItsiosN.JokelaA.V.RamalloCold holographic matter in the Higgs branchPhys. Lett. B7472015229arXiv:1505.02629 [hep-th][47]G.ItsiosN.JokelaA.V.RamalloCollective excitations of massive flavor branesNucl. Phys. B9092016677arXiv:1602.06106 [hep-th][48]L.PatinoD.TrancanelliThermal photon production in a strongly coupled anisotropic plasmaJ. High Energy Phys.13022013154arXiv:1211.2199 [hep-th][49]V.JahnkeA.LunaL.PatiñoD.TrancanelliMore on thermal probes of a strongly coupled anisotropic plasmaJ. High Energy Phys.14012014149arXiv:1311.5513 [hep-th][50]S.Y.WuD.L.YangHolographic photon production with magnetic field in anisotropic plasmasJ. High Energy Phys.13082013032arXiv:1305.5509 [hep-th][51]K.A.MamoHolographic RG flow of the shear viscosity to entropy density ratio in strongly coupled anisotropic plasmaJ. High Energy Phys.12102012070arXiv:1205.1797 [hep-th][52]P.KovtunD.T.SonA.O.StarinetsViscosity in strongly interacting quantum field theories from black hole physicsPhys. Rev. Lett.942005111601arXiv:hep-th/0405231[53]Y.BuChern–Simons diffusion rate in anisotropic plasma at strong couplingPhys. Rev. D8982014086003[54]L.ChengX.H.GeS.J.SinAnisotropic plasma at finite U(1) chemical potentialJ. High Energy Phys.14072014083arXiv:1404.5027 [hep-th][55]L.ChengX.H.GeS.J.SinAnisotropic plasma with a chemical potential and scheme-independent instabilitiesPhys. Lett. B7342014116arXiv:1404.1994 [hep-th][56]X.H.GeY.LingC.NiuS.J.SinThermoelectric conductivities, shear viscosity, and stability in an anisotropic linear axion modelPhys. Rev. D92102015106005arXiv:1412.8346 [hep-th][57]R.A.JanikP.WitaszczykTowards the description of anisotropic plasma at strong couplingJ. High Energy Phys.08092008026arXiv:0806.2141 [hep-th][58]E.BanksJ.P.GauntlettA new phase for the anisotropic N=4 super Yang–Mills plasmaJ. High Energy Phys.15092015126arXiv:1506.07176 [hep-th][59]S.JainN.KunduK.SenA.SinhaS.P.TrivediA strongly coupled anisotropic fluid from dilaton driven holographyJ. High Energy Phys.15012015005arXiv:1406.4874 [hep-th][60]D.RoychowdhuryHolographic charge transport in non commutative gauge theoriesJ. High Energy Phys.1507201512110.1007/JHEP07(2015)121arXiv:1506.00209 [hep-th][61]D.ÁvilaD.FernándezL.PatiñoD.TrancanelliThermodynamics of anisotropic branesJ. High Energy Phys.16112016132arXiv:1609.02167 [hep-th][62]D.K.BrattanR.A.DavisonS.A.GentleA.O'BannonCollective excitations of holographic quantum liquids in a magnetic fieldJ. High Energy Phys.12112012084arXiv:1209.0009 [hep-th][63]J.MasJ.P.ShockJ.TarrioA note on conductivity and charge diffusion in holographic flavour systemsJ. High Energy Phys.09012009025arXiv:0811.1750 [hep-th][64]R.A.DavisonA.O.StarinetsHolographic zero sound at finite temperaturePhys. Rev. D852012026004arXiv:1109.6343 [hep-th][65]N.I.GushterovR.RodgersR.RodgersHolographic zero sound from spacetime-filling branesarXiv:1807.11327 [hep-th][66]L.HerreraCracking of self-gravitating compact objectsPhys. Lett. A1651992206[67]L.HerreraN.O.SantosLocal anisotropy in self-gravitating systemsPhys. Rep.286199753[68]L.HerreraE.FuenmayorP.LeónCracking of general relativistic anisotropic polytropesPhys. Rev. D9322016024047arXiv:1509.07143 [gr-qc][69]C.HoyosD.Rodríguez FernándezN.JokelaA.VuorinenHolographic quark matter and neutron starsPhys. Rev. Lett.11732016032501arXiv:1603.02943 [hep-ph][70]E.AnnalaC.EckerC.HoyosN.JokelaD.Rodríguez FernándezA.VuorinenHolographic compact stars meet gravitational wave constraintsJ. High Energy Phys.18122018078arXiv:1711.06244 [astro-ph.HE]