Low-energy modes in anisotropic holographic fluids

In 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.


Introduction
Holography has shown to be a useful tool to study various gauge field theories particularly at finite density [1][2][3][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  and in particular the conductivities of charged excitations [26][27][28][29][30][31][32][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][35][36][37][38][39][40][41][42][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][46][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 in 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. The authors of [41] found a scaling solution with the following string frame metric: ds 2 =R 2 s r 7/3 (−dt 2 + dx 2 + dy 2 ) + r 5/3 dw 2 + dr 2 r 5/3 + R 2 s r 1/3 dΩ 2 where dΩ 2 S 5 is the line element of a five-sphere and R s andR s are constant radius related as R 2 s = 12R 2 s /11. The dilaton and the axion are: where β is an arbitrary input parameter indicating the amount of anisotropy. This solution also has the standard RR five-form F 5 . In section 3 of [41] the authors found a running solution which interpolates between the scaling solution written above in the deep IR and the AdS 5 × S 5 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 w 1 and w 2 and the anisotropy is induced by a RR two-form F 2 ∝ dw 1 ∧ dw 2 . 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]. The string frame metric for these backgrounds takes the form: where L s is a constant length (which in [42,43] is put to one) and B, F, and H are functions depending on the radial coordinate u. The RR five-form F 5 has the standard form and the dilaton and axion are: where the anisotropy parameter a is arbitrary as in (1.2). The functions F and B can be written in terms of the dilaton φ. 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 u = 0 (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 k 0 = | 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 S 5 is deformed, which corresponds to new internal components of the RR five-form F 5 . 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 AdS 4 × 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.
The rest of the paper is structured as follows. We will start discussing the background geometry which has a general metric possessing anisotropy, of the form (1.3). 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 (1.2) geometry 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.

Gravity background and flavor thermodynamics
We 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, where M = 0, . . . , 9 and F 1 = dχ is the axion field strength. The metric ansatz in the string frame is 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 where Ω S 5 is the volume element of a five-sphere and α is a constant determined by flux quantization.
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, u H . The equations of motion for these ansätze require α = 4e −φ(0) L 2 s = 4L 2 s and F(0) = 1/Z(0). Finally, the ansatz was further simplified by setting 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 AdS 5 metric at the UV boundary and it becomes more and more anisotropic closer to the horizon. The anisotropicity is controlled by the axion strength, a 2 . 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 The probe D7-brane will span the directions (t, x, y, z, u, Ω 3 ). We furthermore turn on a gauge field on the brane, F = A t (u)du ∧ dt. We have chosen the gauge A u = 0. The brane will obey the dynamics following from the DBI action: where the metric g is the induced metric on the brane, i.e., Most of the time, we will set L s = 1 to avoid cluttering the equations. We only focus on massless fundamentals, so it is consistent to integrate over the internal directions as the embedding does not vary inside Ω 3 . In other words, ψ = const. and θ = 0 are consistent solutions to the equations of motion.
The action evaluates to The Ω 3 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 A t is a cyclic variable so it can be easily solved from the Euler-Lagrange equations to give where the last equality is for the solutions with the restrictions (2.3). Here, d is a constant and we will show that it is proportional to the particle density.
The chemical potential is while the particle density is where Ω grand = −S on−shell is the unregularized grand potential, For further thermodynamic computations, the on-shell action needs to be regularized. The simplest way for this is to subtract the 0-density action, 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 The energy density can be found with a Legendre transformation = Ω grand V + µρ. In order to determine the pressures along the different directions, let us put the system in a box of sides L x , L y and L z . Then, the pressure along direction q can be found with 14) 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 nontrivially on L z as seen in the explicit Lifshitz scaling solution of section 4. With these, we can also compute the speed of sound with Later in Sec. 4 we will explicitly evaluate all the formulas in this section in a particular geometry.

Low-energy modes
We 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 To expand the action, we need to expand the determinant We can use the following expansion The first order terms vanish as we are fluctuating around a saddle point. The inverse of (g + F ) is 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 We get the equations of motion from the Euler-Lagrange equations. We make the assumption that a µ do not depend on the spherical coordinates. The equations are 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 a u = 0. We get an important constraint equation from this by For spatial or temporal coordinates, we have We make a further assumption. Due to rotational invariance in the xy-plane, we assume that a µ 's are independent of y. Then we Fourier transform along directions t, x, and z with The equations of motion as functions of ω and k = (k x , k z ) are where we have already used gauge-invariant quantities (3.10) Using the gauge constraint and definitions of E x and E z , we can solve the derivatives of a µ in terms of E x and E z 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 E x it is and for E z it is 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.

T = 0: zero sound mode
First, we take the limit u → ∞. We need the asymptotic behavior of the metric components, (3.14) To preserve Lorentz invariance for the (t, x, y) components, we set α b = α f . We relax the conditions on (2.3) by setting H = H 0 e −φ , Z = Z 0 e φ/2 . We further assume that α φ < 4, The equations of motion for E decouple in the asymptotic limit We require ingoing boundary conditions so the solution to these equations is where F q is an integration constant. The ω → 0 limit of these are Second, we take the low-frequency limit of equations (3.9). The equations decouple and a x and a z are easily solved. From these, we obtain E x and E z and then integrate them These are easily integrated and the gauge-invariant fields are where, in the second step, we have defined the integrals J x (u), J z (u), and I(u). In the above expressions, C x , C z , E x,0 , and E z,0 are integration constants. The next step is to approximate these expressions at the near-horizon limit We now match these expansions with the ones in (3.17). We first solve the F x and F z coefficients and then solve for E x,0 and E z,0 , which will give us two linear equations in terms of C x and C z . Imposing the Dirichlet boundary conditions (E q,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 where the coefficients are (3.23) The condition for the determinant to be zero, i.e., the dispersion relation, is given by the equation The 1st order solution to this equation gives us the zero sound mode The next order terms gives us the damping of the mode. By defining ω = c q k + δω, we can extract

T = 0: diffusion mode
This time we set T = 0 and take the near-horizon limit u → u H and the low-frequency limit ω ∼ k 2 → 0. Note that we are implicitly expecting to find a diffusive solution, i.e., ω ∼ −iDk 2 .
We first take the near horizon limit of the equations of motion where the coefficients are (3.28) We solve these equations using the Frobenius series, i.e., E q = F q (u − u H ) αq (1 + β q (u − u H ) + . . .), where α q , F q and β q are all coefficients that might depend on ω and k. With the expansion, we can solve for α q 's and β q 's where we have chosen an explicit sign for α q in order to have an infalling solution. In the expression for β q , one either chooses the first indices or the second indices for all the terms.
Taking the low-frequency limit with ω ∼ k 2 ∼ 2 , β's take the value For the other order, we once again first solve for a x and a z and then write down the E q . The solution reads which we then plug into the expression for E 's To respect our low-frequency expansion, we can neglect the ω 2 terms as a x and a z should be of the same order.

The integrated expressions for E's then read
where we have also done a near-horizon expansion.
We must now match our two solutions. First, we set (u − u H ) α = E nh and C q ∝ F q , then match the two terms in the expansions. We get the relation A non-trivial solution to Dirichlet boundary conditions is provided only when the matrix is singular, i.e., when ω = −i(k 2 which corresponds to a diffusion mode.

Two-point functions
We 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 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 where we already performed a Fourier transform. All the multiples of fields are to be interpreted as The term with derivatives with respect to u can then be written as We now do a partial integration with respect to u for the terms which include E q 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.
As a next step, we consider a low-energy limit, yielding The final steps involve expressing C q 's in terms of the boundary values and taking functional derivatives with respect to the boundary values.

T = 0
We invert equation (3.22) and get The on-shell action, in terms of the boundary values, now takes the form Now we will simply take functional derivatives of the on-shell action to obtain the currentcurrent correlators. Bear in mind that ∂ ∂aµ(k) = ∂E i (k) Second, we consider direction q = cos χx+sin χẑ, i.e. the current J q = J x cos χ+J z sin χ. In addition, for a more condensed notation, we use the notation k x = k cos ϕ and k z = k sin ϕ. The current two-point function becomes On the other hand, two perpendicular directions with q ⊥ = sin χx − cos χẑ have the twopoint function The conductivity tensor can be computed with the relation In the low-frequency limit, the longitudinal and Hall conductivities are and The i ω singularity resembles Drude conductivity and implies a delta peak for the real part of the conductivity at ω = 0.

T = 0
Inverting the matrix in (3.34) gives (3.51) which we use to express the on-shell action in terms of field boundary values . (3.52) The two-point functions can then be easily computed. We list the same components as above The corresponding DC conductivities are We see that these only depend on the near-horizon physics.

Einstein relation
Einstein relation relates conductivity to susceptibility and diffusion in a non-trivial manner, The validity of this relation has previously been checked and verified in many other holographic settings, starting with [63]. Susceptibility is computed with A simple algebraic exercise shows that the Einstein relation is indeed satisfied.

Fixed-point Lifshitz metric
No 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 the constant L s drops from the expressions and we obtain the string frame metric In the Einstein frame ds 2 E = e −φ/2 ds 2 and with r H = 0, we see that the metric exhibits a Lifshitz-like scaling with {t,x,ỹ} → k{t,x,ỹ},w → k 2/3w , and r → k −1 r. In the metric, 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) 4 N V ol(S 5 ) , β = k Lw , where L w is the period of thew coordinate. From low-energy flat space perspective, the branes are extended along the following directions (4.5) We will specialize our above solutions and computations to this special background metric. We will begin with the thermodynamic expressions and then move on to the low-energy excitations.

Thermodynamics
We start by direct evaluation of the thermodynamic quantities as calculated in the general framework in Sec. 2. First, the temperature is given by The regularized on-shell action is where the last line is the zero-temperature result. The chemical potential is where µ 0 = √ 11d 4/9 e 4φ 0 /9 Γ 2 9 Γ 5 18 12 √ 3π (4.9) is the zero-temperature chemical potential. The pressure and energy density expressions at low temperatures are In the next section, we compare our analytical expressions to numerical ones.

Numerical verification
The numerical methods we use are standard, see, e.g., [62]. All of our solutions are obtained at finite temperature. Fortunately, we can scale out some of the quantities of the background. We scale u H and φ 0 out with the substitutions With these transformations, we can effectively set u H = 1 and φ 0 = 0 in the equations of motion, leaving us with onlyd as a tunable parameter. However, these quantities will still appear in the two-point functions, e.g., conductivities. With these scalings, highd corresponds to high particle density or equivalently to low temperature and vice versa for lowd. Similar interpretations apply tok andω.

Transition from hydrodynamic to collisionless regime
In 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 ofω receives little thermal corrections when it comes to ∇kω. The case is very different for the imaginary part. We can study this more quantitatively by fixing k and d and scalingk andd appropriately to probe the effect of the temperature. For k z = 0, we fixk x = 0.02d 1/3 and, for k x = 0, we pickk z = 0.02d 2/9 . We compare this to the imaginary part of the zero sound at variousd 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 k x = 0, the thermal correction is Im ω(T ) = Im ω(T = 0)(1 + γ x T 7/3 ) and for k z = 0, it is Im ω(T ) = Im ω(T = 0)(1 + γ z T 5/3 ), 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 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 differentd. It turns out that lowtemperature and high-density regimes exhibit non-trivial transitioning between hydrodynamic and collisionless regimes. For some small k x , increasing |k z | 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 c x cos ϕ from (4.12). Later on, the zero sound mode transition is complete and the slope is consistent with c q . 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.

Quantitative comparisons
We also wish to scale out u H out of the conductivity quantities. We can naturally extract the xx and zz components of the (longitudinal) conductivity tensor.
The last equalities are for DC conductivity for which we see that,σ xxω =0 =σ zz .
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   Log Im Ω Im Ω T 0 Im Ω Im Ω T 0  becomes increasingly more negative. The behavior ofσ zz andσ xx differ significantly which was to be expected from our analytical computations. The imaginary part ofσ zz becomes negative at very low momenta at lowd while the imaginary part ofσ xx becomes positive at low momenta but is negative at higher momenta. Fig. 5 contains plots of AC conductivity computations.
When we scaled out u H , 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.

Conclusions and outlook
In 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][67][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.